Random Dynamical Systems - Arnold L.

I Random Dynamical Systems and Their Generators
1.4 Invariant Measures for l/Ieasurable RDS
1.5 Invariant Measures for Continuous RDS
2.1.3 RDS with Independent Increments . .
2.2 Continuous Time 1: Random Differential Eqs.
2.2.3 Random Differential Equations from RDS
2.3 Continuous Time 2: Stochastic Differentia1 Eqs.
2.3.2 Semirmartingales and Dynamical Systems: Stochastic Calculus for Two—Sided Time
2.3.3 Semimartingale Helices with Spatial Parameter
2.3.4 RDS from Stochastic Differential Equations
2.3.5 Stochastic Differential Equations from RDS
3.3.2 The Furstenberg—I{esten Theorem for One—Sided Time
3.3.3 The Furstenberg—I{esten Theorem for Two—Sided Time
4.3.3 Random Riomannian Motrics on Manifolds
5.6.2 Invariant Measure in the Hyperbolic Case
5.6.3 Time Reversibility and Iterated Function Systems
7.3.6 Final Global Invariant Manifold Theorem
7.5.2 Dynamical Characterization and Globalization
7.5.3 Local Manifolds for Continuous Time
8.2 Normal Forms for Random Diffeomorphisms
9.3.4 A General Criterion for Pitchfork Bifurcation
9.3.6 Discrete Time
9.4 Noisy Duffing-van der Pol Oscillator
A.2 Stochastic Processes and Dynamical Systems
B.5 Vector Fields and Flows on Manifolds
Author: Arnold L.
Tags: mathematics probability theory dynamic systems exact sciences
Year: 1997
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Text
                    Random Dynamical Systems
Ludwig Arnold
Institut fur Dynamische Systeme
Universitat Postfach
Bremen Germany
email arnold mathematik uni bremen de
Final Version August
1

Preface
Background and Scope of the Book
This book continues extends and unites various developments in the inter
section of probability theory and dynamical systems I will brie y outline
the background of the book thus placing it in a systematic and historical
context and tradition
Roughly speaking a random dynamical system is a combination of
a measure preserving dynamical system in the sense of ergodic theory
FP ttTT R RZ Zwithasmoothortopologi
cal dynamical system typically generated by a di erential or di erence
equationx fx orxn
xn to a random di erential equation
x f t x or random di erence equation xn
nxn
Both components have been very well investigated separately However
a symbiosis of them leads to a new research program which has only partly
been carried out As we will see it also leads to new problems which do
not emerge if one only looks at ergodic theory and smooth or topological
dynamics separately
From a dynamical systems point of view this book just deals with those
dynamical systems that have a measure preserving dynamical system as a
factor or the other way around are extensions of such a factor As there
is an invariant measure on the factor ergodic theory is always involved
Our book is a continuation of that by Guckenheimer and Holmes
on Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector
Fields In their own words Preface page xi their book should be seen
as an attempt to extend the work of Andronov et al i e the analysis of a
single degree of freedom nonlinear oscillator L A by one dimension i e
by adding a small periodic forcing term L A Speci cally they look at
certain equations of the form x f t x in R where t is periodic
We will go further and beyond the periodic noise to a general measure
preserving dynamical system to which the ordinary di erential equation is
iii

iv
coupled In yet other words we take the step from autonomous systems x
f x to nonautonomous systems but of the special kind x f t x
i e to those which are coupled to a dynamical bath
If the ow
t intheequationxft xisaowof
homeomorphisms of a compact space we are in the realm of skew product
ows in the sense of Sacker Sell and Johnson see e g
We go beyond this again by stripping o all the topology from
t and instead adding an invariant measure shortly by going
from almost periodic to random
We also extend and generalize Mane s book
on Ergodic Theory
and Di erentiable Dynamics He has a measure invariant with respect
to the ow t of a deterministic vector eld x f x on a manifold M
Here we have a measure on M with marginal P on invariant with
respect to the skew product ow
x
ttxwheret
is the solution ow generated by the random vector eld x f t x
From a probabilistic point of view this book o ers another look at the
quite classical subject of random di erence equations and of random and
stochastic di erential equations i e ordinary di erential equations driven
by real or white noise
During the last to years an impressive structure called stochas
tic analysis has been erected part of which is a theory of di erential
equations with semimartingale rather than only Gaussian white noise or
Wiener driving processes providing us with a uni ed theory of random
and stochastic di erential equations
Around it was discovered by Elworthy Baxendale Bismut Ikeda
Watanabe Kunita and others see e g
that a
stochastic di erential equation generates for free a much richer structure
than just a family of stochastic processes each solving the stochastic di er
ential equation for a given initial value It gives us in fact a ow of random
di eomorphisms We can now bridge the gap between stochastic analysis
and dynamical systems by proving that a random or stochastic di erential
equation generates a random dynamical system
This makes it possible to re evaluate and improve all the classical results
which are based on one point motions and Markov transition probabilities
on stochastic stability existence of invariant measures etc by Kushner
Khasminskii
Bunke
and many others In I have de
scribed the extension of the horizon when going from Markov processes to
stochastic ows and cocycles
The present book also adds a new chapter to the volume by Horsthemke
and Lefever
entitled Noise Induced Transitions and re interprets their
ndings Their noise induced transitions are nothing but bifurcations on the

v
static level of the Fokker Planck equation We will also study bifurcation
scenarios on the dynamic level
The book closest to ours in spirit and content is the one by Kifer
on
Ergodic Theory of Random Transformations He however deals exclusively
with the i i d case i e with the case of iterations of random mappings
chosen independently with identical distribution In this case the orbits
in state space form a Markov chain We go beyond that by allowing a
stationary stochastic sequence of mappings to be iterated keeping the i i d
case as an important particular case
It is a characteristic feature of the theory of random dynamical systems
that every problem involves some ergodic theory and ergodic theorems
The most crucial and most important ergodic theorem applies to the lin
earization of smooth random dynamical systems It is traditionally called
the Multiplicative Ergodic Theorem and was proved by Oseledets
in
This theorem provides a random substitute of linear algebra and
hence makes a local theory of smooth random dynamical systems possible
Without it the whole eld in particular this book would not exist
Structure of the Book
As this is the rst monograph on random dynamical systems my main
intention is foundational This forces me to adopt a systematic maybe
sometimes even pedantic style and put my emphasis on theory rather
than applications I hope nevertheless to present a useful reliable and
rather complete source of reference which lays the foundations for future
work and applications
Part I Random Dynamical Systems and Their Generators introduces
the subject matter settles the subtle perfection question develops the the
ory of invariant measures Chap and gives a hopefully ultimate treat
ment of the problem of which random dynamical systems have in nitesimal
generators Chap
Part II Multiplicative Ergodic Theory is the heart of the book I rst
present and prove the classical Multiplicative Ergodic Theorem for products
of random matrices in Rd Chap then present its various modi cations
and the concept of random norms which turns out to be basic Chap In
Chap the multiplicative ergodic theory of related linear random dynam
ical systems obtained by taking the inverse the adjoint and exterior and
tensor products is studied The same is done with the systems induced by
a linear random dynamical system on the unit sphere the projective space
and Grassmannian manifolds culminating in the Furstenberg Khasminskii
formulas for Lyapunov exponents Finally a multiplicative ergodic theorem
for rotation numbers is proved Chap
1

vi
Part III Smooth Random Dynamical Systems addresses the three most
fundamental problems regarding nonlinear systems The rst one is the
construction of invariant manifolds Chap We adopt the new method
of Wanner
which provides a uni ed approach towards invariant mani
folds and the Hartman Grobman theorem The second basic problem is the
simpli cation of a random dynamical system by means of a smooth coordi
nate transformation normal form problem Chap I nally present the
state of the art of random bifurcation theory Chap which is still in its
infancy and is not much more than a collection of numerical examples
Part IV Appendices collects some facts from measurable dynamics
Appendix A and smooth dynamics Appendix B
This book is a research monograph which belongs to the richly struc
tured interface of probability theory and dynamical systems Therefore
substantial mathematical knowledge is required from the reader
The book as a whole is probably not suitable for a course There are
however various possibilities to use subsets of the text as the basis of a
graduate course or seminar or for private study For example
i The multiplicative ergodic theorem Extract the basic de nitions
from Chap then read Chaps and
ii Smooth random dynamical systems Read Chaps
and then
choose one or several of the Chaps
or
iii Stochastic bifurcation theory Read Chaps
and then go to
Chap
Omissions
The exclusion of the following topics from this book is partly compensated
by some recent publications which augment this book and complete the
overall picture of the subject
I completely omitted topological dynamics of random dynamical sys
tems and refer instead to the book by Nguyen Dinh Cong
Fortunately I do not need to include Pesin s theory probably the deep
est of recent developments in random dynamical systems as it is beautifully
and completely presented in the book by Liu and Qian
With many scruples I decided to omit the beautiful geometry of
stochastic ows see the work of Baxendale
Baxendale
and Stroock Carverhill Carverhill and Elworthy Elworthy
Elworthy and Rosenberg
Elworthy and Yor
El
worthy Le Jan and Li
Elworthy and Li
Kunita
section
Li
Liao
and the references therein The
subject is worth a book of its own
1

vii
I also did not include the theory proper of products of random matrices
On the one hand this area with its elaborate methods and numerous ap
plications is so vast that it would easily ll a volume in itself On the other
hand the subject is already quite well documented Besides the books by
Bougerol and Lacroix
and Hognas and Mukherjea
there are nu
merous contributions survey articles and further references in the three
proceedings volumes
and
Finally I omitted in nite dimensional random systems and instead refer
the reader to the work of Crauel and Flandoli
Flandoli and
Schmalfu
Mohammed
Mohammed and Scheutzow
Schaumlo el
in addition to many others
Acknowledgements
Beginning with my collaboration with Peter Sagirow in the late s in
Stuttgart I have always maintained strong and extremely fruitful contacts
with engineers to whom I am indebted for numerous suggestions and prob
lems In addition to my nestor Peter Sagirow I would like to explicitly men
tion Walter Wedig S T Ariaratnam Mike Lin and N Sri Namachchivaya
My contacts with them gave me the badly needed reassurence that I was
doing something that would be useful
Some of my former students and collaborators also deserve an extra
special acknowledgement as they took very active part in the unfolding of
the subject and taught me much of what I know today Wolfgang Kliemann
Volker Wihstutz and Hans Crauel
I would also like to thank all those who proof read parts of preliminary
versions in particular Peter Baxendale Thomas Bogenschutz Hans Crauel
Matthias Gundlach Stefan Hilger Peter Imkeller Peter Kloeden who
very e ciently served as my default native English speaker Liu Pei
Dong Hans Friedrich Munzner Nguyen Dinh Cong Gunter Ochs Klaus
Reiner Schenk Hoppe Michael Scheutzow N Sri Namachchivaya Bjorn
Schmalfu Stefan Siegmund and Thomas Wanner
I am grateful to Gabriele Bleckert and Hannes Keller who produced the
gures Gabriele Bleckert also did the extensive numerical studies of the
noisy Du ng van der Pol oscillator in Chap
Last but not least I would like to thank my wife Birgit for her under
standing and support during the six very hard years of writing
Bremen August
Ludwig Arnold

Contents
Preface
iii
I Random Dynamical Systems and Their Genera
tors
Basic De nitions Invariant Measures
De nition of a Random Dynamical System
Local RDS
Perfection of a Crude Cocycle
Invariant Measures for Measurable RDS
Invariant Measures for Continuous RDS
Polish State Space
Compact Metric State Space
Invariant Measures on Random Sets
Markov Measures
Invariant Measures for Local RDS
RDS on Bundles Isomorphisms
Bundle RDS
Isomorphisms of RDS
Generation
Discrete Time Products of Random Mappings
One Sided Discrete Time
Two Sided Discrete Time
RDS with Independent Increments
Continuous Time Random Di erential Eqs
RDS from Random Di erential Equations
The Memoryless Case
Random Di erential Equations from RDS
ix

x
CONTENTS
The Manifold Case
Continuous Time Stochastic Di erential Eqs
Introduction Two Cultures
Semimartingales and Dynamical Systems Stochastic
Calculus for Two Sided Time
Semimartingale Helices with Spatial Parameter
RDS from Stochastic Di erential Equations
Stochastic Di erential Equations from RDS
White Noise
An Example
The Manifold Case
RDS with Independent Increments
II Multiplicative Ergodic Theory
MET in Euclidean Space
Introduction
Lyapunov Exponents
Deterministic Theory of Lyapunov Exponents
Singular Values
Exterior Powers
Furstenberg Kesten Theorem
The Subadditive Ergodic Theorem
The Furstenberg Kesten Theorem for One Sided Time
The Furstenberg Kesten Theorem for Two Sided Time
Multiplicative Ergodic Theorem
The MET for One Sided Time
The MET for Two Sided Time
Examples
MET on Bundles
Temperedness Lyapunov Cohomology
Tempered Random Variables
Lyapunov Cohomology
MET on Manifolds
Linearization of a C RDS
The MET for RDS on Manifolds
Random Di erential Equations
Stochastic Di erential Equations
Random Lyapunov Metrics and Norms
The Control of Non Uniformity in the MET
1

CONTENTS
xi
Random Scalar Products
Random Riemannian Metrics on Manifolds
MET for Related Linear RDS
Inverse and Adjoint
MET on Linear Subbundles
Exterior Powers Volume Angle
Exterior Powers
Volume and Determinant
Angles
Tensor Product
Manifold Versions
A neRDS
Representation
Invariant Measure in the Hyperbolic Case
Time Reversibility and Iterated Function Systems
RDS on Homogeneous Spaces
Cocycles on Lie Groups
Group Valued Cocycles and Their Generators
Cocycles Induced by Actions
RDS Induced on Sd and P d
Invariant Measures
Furstenberg Khasminskii Formulas
Spectrum and Splitting
RDS on Grassmannians
Invariant Measures
Furstenberg Khasminskii Formulas
Manifold Versions
Sphere Bundle and Projective Bundle
Grassmannian Bundles
Rotation Numbers
The Concept of Rotation Number of a Plane
Rotation Numbers for RDE
Rotation Numbers for SDE
III Smooth Random Dynamical Systems
Invariant Manifolds
The Problem of Invariant Manifolds
Reductions and Preparations
1

xii
CONTENTS
Reductions
Preparations
Global Invariant Manifolds
Construction of Unstable Manifolds
Construction of Stable Manifolds
Construction of Center Manifolds
The Continuous Time Case
Higher Regularity
Final Global Invariant Manifold Theorem
Hartman Grobman Theorem
Invariant Foliations
Topological Decoupling
Hartman Grobman Theorem
Local Invariant Manifolds
Local Manifolds for Discrete Time
Dynamical Characterization and Globalization
Local Manifolds for Continuous Time
Examples
Normal Forms
Deterministic Prerequisites
Normal Forms for Random Di eomorphisms
The Random Cohomological Equation
Nonresonant Case
Resonant Case
Normal Forms for RDE
The Random Cohomological Equation
Nonresonant Case
Resonant Case
Examples
Normal Form and Center Manifold
The Reduction Procedure
Parametrized RDE
Small Noise A Case Study
Normal Forms for SDE
The Random Cohomological Equation
Nonresonant Case
Small Noise Case
1

CONTENTS
xiii
Bifurcation Theory
Introduction
What is Stochastic Bifurcation
De nition of a Stochastic Bifurcation Point
The Phenomenological Approach
Dimension One
Transcritical Bifurcation
Pitchfork Bifurcation
Saddle Node Case
A General Criterion for Pitchfork Bifurcation
Real Noise Case
Discrete Time
Noisy Du ng van der Pol Oscillator
Introduction Completeness Linearization
Hopf Bifurcation
Pitchfork Bifurcation
General Dimension Further Studies
Baxendale s Su cient Conditions for D Bifurcation
and Associated P Bifurcation
Further Studies
IV Appendices
A Measurable Dynamical Systems
A Ergodic Theory
A Stochastic Processes and Dynamical Systems
A Stationary Processes
A Markov Processes
B Smooth Dynamical Systems
B Two Parameter Flows on a Manifold
B Spaces of Functions in Rd
B Di erential Equations in Rd
B Autonomous Case Dynamical Systems
B Vector Fields and Flows on Manifolds
Bibliography
1

xiv
CONTENTS

Part I
Random Dynamical
Systems and Their
Generators
1

Chapter
Basic De nitions
Invariant Measures
Summary
In this rst chapter we introduce the concept of a random dynamical system
and study its invariant measures objects which characterize the possible
long term behavior of the system
The de nition of a random dynamical system or cocycle is given in Sect
and basic properties are derived Theorem
The local concept
allowing for explosion is introduced in Sect
Sect deals with the key but tricky technical problem of perfecting
a crude cocycle as generated by a stochastic di erential equation We
give a satisfactory answer in Theorem
and its two corollaries This
section can be omitted at rst reading
The basic Sects to
are devoted to the study of invariant mea
sures of random dynamical systems We describe invariance in terms of the
factorization of the measure Theorem
For a Polish state space we
introduce a topology of weak convergence of measures which permits us to
carry over the Krylov Bogolyubov procedure Theorem
and to prove
that each continuous random dynamical system on a compact state space
has at least one invariant measure Theorem
for a useful generaliza
tion to a random compact set see Theorem
In Sect we relate our general de nition of an invariant measure to
the classical one for Markov processes Corollary
Theorem
in Sect stating that all invariant measures for con
tinuous random dynamical systems with state space R are random Dirac
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
measures will be frequently quoted later as we will explicitly work out sev
eral one dimensional examples
Finally in Sect we introduce the bundle version of a random dy
namical system and provide several notions of isomorphism which are used
later to identify di erent systems of similar structure
De nition of a Random Dynamical Sys
tem
Imagine a mechanism which at each discrete time n tosses a possibly com
plicated many sided coin to randomly select a mapping n by which a
given point xn is moved to xn
n xn The selection mechanism is
permitted at time n to remember the choices made prior to n and even to
foresee the future The only assumption made is that the same mechanism
is used at each step This scenario called a product of random mappings
is one of the prototypes of a random dynamical system see Fig
Figure Product of random mappings
In continuous time the random selection at time t would be from a
set of di erential equations or vector elds x f t x again with the
stipulation that the statistics of f t be independent of t
We will now give a formal de nition of a random dynamical system
which is tailor made to cover the most important families of dynamical
systems with randomness which are currently of interest in particular ran
dom and stochastic ordinary and partial di erential equations
We will often describe the above situation by saying that a certain type
of noise in uences or perturbs a dynamical system
A random dynamical system thus consists of two basic ingredients
a model of the noise
a model of the system which is perturbed by the noise
Throughout the book noise will always be modeled by a metric i e
measure preserving dynamical system in the sense of ergodic theory see
Appendix A for all basic notions and some examples and the system will
in most cases be modeled by a di erence or a di erential equation or its
solution ow respectively
Dynamics studies those properties of a collection of self mappings of
some space which become apparent asymptotically through iteration This
collection of maps is algebraically always a semigroup often a group T

DEFINITION OF A RANDOM DYNAMICAL SYSTEM
which we call time Throughout the book time T always stands for the
following additive semigroups or groups
T R Two sided continuous time
TRftRtgsometimesTR
R One sided
continuous time
TZf
g Two sided discrete time
T Z f gsometimesT Z
ZorTN
f g One sided discrete time
We will now give a hierarchy of de nitions
De nition Random Dynamical System A measurable ran
dom dynamical system on the measurable space X B over or covering
or extending a metric dynamical system F P t t T with time T is
a mapping
T
XXtx
tx
with the following properties
i Measurability is B T F B B measurable
ii Cocycle property The mappings t
t
X Xform
a cocycle over i e they satisfy
idX for all
ifT
ts
ts
s forall st T
Here
means composition which canonically de nes an action on the left
ofthesemigroupofselfmappingsofXonthespaceX ie f g x
f g x This fact creates as we will see a certain break of symmetry in
the theory
If we want to emphasize that equation
holds identically we call
a perfect cocycle We call a crude cocycle if
holds for xed s
and all t T P a s where the exceptional set Ns can depend on s We
call a very crude cocycle if
holdsfor xedst TPas where
the exceptional set Ns t can depend on both s and t
Note that axiom
of De nition
is not redundant However if
the mappings t X X are known to be invertible
implies
Dynamical system s and Random dynamical system s are henceforth often
abbreviated as DS and RDS respectively
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
It is very useful to imagine an RDS move on the trivial bundle X
as Fig depicts While is shifted by the dynamical system in time s
to the point s on the base space the cocycle s moves the point
xinthe berfg Xover tothepoint s xinthe berfs g X
over s The cocycle property is also clearly visible on this bundle
Figure A random dynamical system as an action on a bundle
De nition Continuous RDS A continuous or topological RDS
onthetopologicalspaceXoverthemetricDS FP t t T isamea
surable RDS which satis es in addition the following property For each
the mapping
TXXtx
tx
is continuous
De nition Smooth RDS A smooth RDS of class Ck or a Ck
RDS where k
on a d dimensional C manifold X is a topo
logical RDS which in addition satis es the following property For each
t T the mapping
t
t
XXx
tx
is Ck i e k times di erentiable with respect to x and the derivatives are
continuous with respect to t x
De nition Linear RDS A continuous RDS on a for simplic
ity nite dimensional vector space is called a linear RDS if t L X
foreacht T
where L X is the space of linear operators of X
If we endow the vector space X with its natural manifold structure then
L X C X X Hence a linear RDS is automatically C
Notations i We often omit speci cally mentioning the underlying metric
DS FP t t T orabbreviateitas andspeakofan RDS over
thus identifying an RDS with its cocycle part Whenever we speak of a
Ck RDS we assume k
ii We denote by C X X or Homeo X the semigroup or group of con
tinuous mappings or homeomorphisms of a topological space X endowed
with its compact open topology If X is a locally compact Hausdor space
this is a Hausdor topological semigroup or group and the evaluation map
ping f x f x is continuous
1

DEFINITION OF A RANDOM DYNAMICAL SYSTEM
iii Finally we denote by Ck X X or Di k X the semigroup or group
of Ck mappings or Ck di eomorphisms of a manifold X respectively en
dowed with its compact open topology for the de nition see Appendix B
for X Rd and B for the manifold case This is a Polish topological
semigroup or group and the evaluation mapping f x f x is Ck with
respect to x In the manifold case also Homeo X is a Polish group
Remark i If T is discrete measurability of t x
tx
is equivalent to measurability of x
t xforeach xedt T
continuity of t x
t xforeach
is equivalent to continuity
ofx
t xforeach xedt
T and the Ck smoothness of
x
t xisjustwithrespecttoxforeach xed t T
ii A measurable continuous Ck RDS with continuous time T is also
a measurable continuous Ck RDS if restricted to discrete time T Z
iii A measurable continuous Ck RDS with two sided time T is also
a measurable continuous Ck RDS if restricted to one sided time T
T Riv We stress that we never allow our exceptional sets in the de nition
of a cocycle to depend on x X In fact it is one of the basic problems of
a theory of RDS in an in nite dimensional space X that
tsx
ts
sx
often holds only outside a set of measure zero which depends on s t T
andonx X SeeegSkorokhod
Chap I and Chap IV
for examples of linear stochastic di erential equations in Hilbert space
v Example Deterministic DS and DS in the sense of ergodic theory
are particular cases of RDS Indeed if is independent of then the
RDS decouples into a metric DS F P t t T and a deterministic
measurable continuous Ck DS on X
In case time T is a group the underlying metric DS is invertible with
t
t Equations
and
then force the cocycle to be
invertible too More precisely we have the following far reaching conse
quences of the cocycle property
Theorem Basic Properties of RDS with Two Sided
Time SupposeTisagroup ie T RorZ
i Let be a measurable RDS on a measurable space X B over
Then for all t
T
t is a bimeasurable bijection of X B
and
t
ttforalltT
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
or equivalently
t
tt
forall t T
Moreover the mapping
tx
t
x
is measurable
ii Let be a continuous RDS on a topological space X Then for all
t T wehave t HomeoX If
T Zor
T R and X is a topological manifold or
T R and X is a compact Hausdor space
thentx
t
x is continuous for all
iii Let beaCkRDSonamanifoldX Thenforall t T
t
DikX Moreover tx
t
x is Ck with respect to x
for all
Proof iWeusethefactthatg f ifandonlyiff g idand
gfidPutt sin
and use
to obtain
idX
ss
s forall s
Now use
fors tand
t and relation
to obtain
idX t
ttforallt
yielding for s t equation
The mapping t x
t
x
t t x is measurable since it is the composition of the measurable
mapst x
ttxandtx
t x The measurability
of the inverse mapping with respect to x follows from that of t x
t
x by freezing t
ii Equation
says that the left hand side is continuous with
respect to x since the right hand side is by De nition
Thus
t
Homeo X In case the continuity of t x
t
xis
trivially satis ed In case observe that t x t t x is a continu
ous by De nition
and bijective by part i of this proof mapping
of R X onto itself This mapping is thus a homeomorphism by Brouwer s
theorem see Dieudonne
p
sotheinverse tx t t
x
is continuous in particular t x
t
x is continuous Case
1

DEFINITION OF A RANDOM DYNAMICAL SYSTEM
A continuous bijection of a compact space into a Hausdor space is a
homeomorphism see Dunford and Schwartz
p
We apply this
to
K X K XwhereK Risacompactintervaland
txttx
iii The Ck di eomorphism property of t
follows again from
equation
and De nition
For the last statement we use the
facts that the derivative of a di eomorphism is nonsingular and that the
derivative of the inverse is given by the formula
Dt
xDtyjyt
x
Hencetx Dt
x is continuous since
t x D t x is continuous by assumption
tx
t
x is continuous by part ii of this proof we are in
the case where X is a C manifold and
g g iscontinuousinGldR
Similarly for higher order derivatives
Remark i It is somewhat surprising that under the assumptions
of part ii of the above theorem the function
t
t
x
ttx
is continuous in t although t x was assumed to be only measurable in
iiLet beacontinuousRDSwithtimeT R IfXisnotlo
cally Euclidean or compact Hausdor we can in general not conclude that
tx
t
x
t t x is continuous This is due to the ap
pearance of the shift operator t in formula
for the inverse This
suggests replacing the continuity requirement in De nition by the fol
lowing weaker for discrete time equivalent one For each t
T
the mapping
tXXx
tx
is continuous We still could conclude that t is a homeomorphism In
fact this weaker assumption su ces for most things we do with continuous
RDS The reason we stay with the stronger version of a topological RDS
as given in De nition is that we automatically obtain such continuous
RDS when solving random or stochastic di erential equations see Sects
and

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Remark RDS as a Skew Product Given an RDS Then
the mapping
x
ttx
txtT
is a measurable DS on
X F B exercise which is called the skew
productofthemetricDS FP t t T andthecocycle t onX
Conversely every such measurable skew product DS de nes a cocycle
on its x component thus a measurable RDS We can consequently use
RDS cocycle and skew product synonymously
Remark RDS have Stationary Increments Let be a
measurable RDS For n N and ti T for i
n where
tt
tn the X X valued random variables
ttt
tntntn
are called the forward increments If time is two sided the cocycle prop
erty yields
tss
t
s
fort s
so that the quantities in
are indeed multiplicative increments
The invariance of P implies that the joint distribution of the incre
ments does not change if the ti are replaced by ti h T i e each RDS
has stationary increments In particular
LtssLtsforallts
where L denotes the probability distribution of the random variable
Similarly for backward increments if time is two sided
Remark Backward Cocycles A family t of mappings
of a space X which satis es all of the requirements of an RDS except that
instead of the forward cocycle property
it satis es the backward
cocycle property
ts
t
st
is called a backward cocycle Since composition is canonically a left action
on X backward cocycles are somewhat unnatural or acausal since in
the rst step one applies a mapping which depends on the second step
Backward cocycles typically occur as companions of cocycles For ex
ample if T is two sided then t is a cocycle over if and only if
t
t
1

LOCAL RDS
is a backward cocycle over and since is a cocycle backward cocycle
over if and only if
is a cocycle backward cocycle over
t
t
t
tt
is a backward cocycle over Remark can also be de ned for one
sided time and non invertible but invertible as t
tt
which is important for the study of random attractors see for example
Arnold Demetrius and Gundlach Crauel and Flandoli Schmalfu
The di erences between the cocycle and the two backward cocycles
and in asymptotic behavior for t
are quite dramatic The reason
isinthebundlepicturethat t mapsx fg Xto t x
ftgXattimetwhile t mapsxftgXwhichis
movingwitht to t x fg X the xed berattime Similarly
t mapsxftgXattimetto t xfgXattime
see Fig
In general in contrast to autonomous systems it makes a big di erence
in the non autonomous case with which we are concerned here between
moving points from to t and moving them from t to Only in the
second case will the result be in the same ber for all t hence can be
studied for t
This is the reason why the backward cocycles and
will prove to be of fundamental importance for the construction of the
invariant objects measures attractors of the cocycle
In Sect forward and backward cocycles are considered from the
perspective of actions of groups on spaces
In the deterministic autonomous case forward cocycles and backward
cocycles coincide namely with a ow
Figure A cocycle and corresponding backward cocycles and
Local Random Dynamical Systems
It is well known that a deterministic vector eld in general only de nes a
local ow due to the possibility of explosion exit from the state space in
nite time see Appendix B and B Explosion is a continuous time
phenomenon which originates from the fact that we can de ne a DS via an
in nitesimal generator with the potential for explosion built in In the
random case we will nd a similar situation It will turn out that the solu
tion of a random or stochastic di erential equation will generally explode
in nite time so we need the concept of a local RDS which allows for this
possibility We restrict our attention to time T R and the continuous Ck
case
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
DenitionLocalRDS SupposeT Rand FP t tR
is a two sided metric DS A local continuous Ck RDS over on a topological
space manifold X is a measurable mapping
DXtx
tx
where D R
X is a measurable set with the following properties
For all
i The random domain
DftxRXtxDgRX
is non void and open and
D
Xtx
tx
is continuous Ck meaning that it is k times continuously di erentiable
with respect to x
ii Foreachx X
DxftRtxDgftRtxDgR
is an open interval containing hence can be written as
Dx
x
xR
iii
satis es the local cocycle property
idX
and for all x X and all s D x we have the following property
tDs
s x ifandonlyift s D x equivalently
DxsDs
sx
In this case we have
tsxts
sx
The local continuous Ck RDS is said to be global if D R
X
Remark i x and
x are respectively the forward
and backward explosion times of the orbit
x starting at time t
in the position x The set D x R is automatically open by part i of
De nition
as a section of the open set D
ii A local RDS which is global in the sense of De nition
is an
RDS in the sense of De nitions
and
respectively The following
conditions are obviously equivalent for a local RDS to be global
1

LOCAL RDS
D x Rforall x
X
x
and
x
for all x
X
Dt
Xforallt R where
DtfxXtxDgX
is the in general possibly empty set of initial values x X for which
the trajectories still exist at time t As a section of D D t
isopen If t sthenbydenitionDt
Ds
and
Dt
Ds
iii We have
tDx
xDt
sinceDt fxt D xgandD x ftx Dt gSeeFig
Figure The domain of a local random dynamical system
Theorem Basic Properties of Local RDS Let be a local
continuous Ck RDS over the metric DS F P t t R with time T
R on a topological space manifold X Then
i and are measurable
Furthermore for all
the following assertions hold
iix
x
is lower semicontinuous and x
x
is upper semicontinuous
iiiForalltx D
DxtDttx
iv Forallt R
tDt
Rt
Dtt
is a homeomorphism Ck di eomorphism and
t
ttDtt
Dt
v If X is a manifold then the mapping
tx
t
x
ttx
is continuous Ck
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Proof i For t xed
fx
xtgfxtxDg
is measurable as the section of a measurable set Similarly for t
fx
xtgfxtxDg
hence the
are measurable
ii For xed andt
fx
xtgfxtxDgDt
is open hence
is lower semicontinuous Analogously for
iii This is
since tx D ifandonlyift D x
iv It su ces to prove the inversion formula
By part iii of
De nition
wehavefort D x
tDttx
Dx
which is always the case Thus the local cocycle property holds for t and
t yielding
xx
tt
tx
Similarly with t and t interchanged
xxt
ttx
We conclude from either of these two equations for t D t
Rt thatRt
DttandDt
Rtt Taken
together
Rt
D t t forallt
from which the inversion formula follows
v Fix and consider the mapping t x
ttxfromD
into R X This mapping is continuous with respect to t x by de nition
and bijective by part iv of the theorem The set D
R Xisopenby
de nition and connected since D x is an interval containing As we
have restricted ourselves to the case where X is a manifold we can as in
the global case of Theorem
appeal to the principle of the invariance
of domain cf Dieudonne
p
which says in particular that the
mapping t x t t x is a homeomorphism so
tx
t
x
ttx
is continuous That it is Ck follows from the formulae for the derivatives
of the inverse of a Ck di eomorphism as in the proof of Theorem

PERFECTION OF A CRUDE COCYCLE
Remark Globality Test for Positive Time Whether a local
system is global or not can be checked using positive time only as follows
Assume we have already veri ed that
x
for all x
X
iethatDt
Xforallt and
Then
x
for all x
X
t X X is surjective for all t
R
ieRt
Xforall t
R By Theorem iv this is
equivalent to D t
X for times t
We stress however that
and
are two independent
criteria Many nonlinear RDS relevant in applications are only forward
global but explode backwards in time e g the noisy Du ng van der Pol
oscillator see Sect and Schenk Hoppe
Perfection of a Crude Cocycle
In Sect we will see that it follows from the uniqueness of the solution of
a stochastic di erential equation that the solution is a crude cocycle The
question is how close is a very crude cocycle to a perfect one More
speci cally can a very crude cocycle be perfected i e is there a
which is indistinguishable from see Appendix A for which the cocycle
property holds identically This is a technical point of great importance
For example only for perfect cocycles does the formula
t
tt
hold and the skew product
is a ow if and only if is perfect
The strongest argument however in favor of perfection is the fact that
the multiplicative ergodic theorem for a perfect linearized cocycle holds on
an invariant set of full measure see Chap This makes constructions
based on it like invariant manifolds normal forms stochastic bifurcations
etc much easier to handle
Perfection techniques were rst developed for crude multiplicative func
tionals by Walsh
and then applied to the perfection of crude helices
by de Sam Lazaro and Meyer
However these methods are strictly
limited to cocycles where t
R i e where composition is addi
tion and cannot be applied to cocycles consisting of mappings which are
elements of more general groups This was clearly seen by Bismut p
who doubted the possibility of perfection in general
Perfecting a crude cocycle needs quite sophisticated changes on sets
of probability zero in order not to arrive at contradictions since the cocycle

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
property couples those at which we have to change with those where
it is already correctly de ned For example assume that is an almost
perfect cocycle i e satis es the cocycle property identically outside a null
set N F The rst idea for constructing a perfect would be to put
t
t forallt Tif Nand t
idX say for
allt Tif
N Suppose for some
N there exists an s T
for which
s N If were a perfect cocycle we would have
idX ts idXthust
id X for all t T which is typically
a contradiction
Fortunately the perfection problem has a satisfactory solution in all
relevant situations For discrete time a simple universal answer is possible
Theorem Perfection for Discrete Time Let be a very
crude measurable continuous Ck cocycle over with discrete time T Then
there exists a measurable continuous Ck cocycle over which is perfect
and indistinguishable from i e there exists an N F with P N
and
ft
t forsomet TgN
Proof Let Ns t be the set for which
does not hold for some
xedst T LetN stTNstand
N c Consider the set of full
measure
tTt
IfT Zthen s
i e is invariant and
t
t
tT
idX
tT
is obviously a perfect cocycle which is indistinguishable from since
on Hence we can choose N c
If T Z extension from N to Z is always possible by putting
idX then s
i e is forward invariant De ne the
random variable
minft
tg
which is the rst entrance time of into Note that a t
tandbt
for all t
this statement is
vacuous for
De ne
t
t
tT
idX
t
t
t
1

PERFECTION OF A CRUDE COCYCLE
If
then the second line of
applies Since
on
and are indistinguishable It can be easily checked that is a perfect
cocycle of the same regularity category as exercise
For continuous time the perfection of a crude cocycle is a much more subtle
problem The following theorem of Scheutzow see Arnold and Scheutzow
and its proof were inspired by a theorem of Zimmer
Theorem
B on the perfection of measurable cocycles
We will discuss the perfection problem for the following more general
setting
The group R is replaced with a locally compact Hausdor topo
logical group G Our proof relies on the fact that such a group
has a Haar measure
G
t
t isanactionofGontheleft onthe
set ie itsatises
eG
for all
eG the identity of G
ts
t s forallts G
The probability measure P invariant under is replaced with a
nite measure which is quasi invariant under i e A F and
A implies t A forallt G
The cocycle as a family of self mappings t X X of some
space X is replaced with a group valued cocycle over i e a function
G
H with values in some topological group H satisfying
the perfect or crude or very crude cocycle property For all t s G
ts
ts
s
for all
or Ns or Ns t respectively Note that
and the fact that we are in a group implies that
eG
eH eH the identity of H
for all
or almost all
respectively
Theorem Perfection for Continuous Time T G As
sume that H and G are Hausdor topological groups with a count
able base of their topologies with respective Borel algebras H and G In

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
addition assume that G is locally compact and let F be a nite
measure space G
be G F F measurable such that
eG
for all
eG the identity of G
and
ts
t s foreveryst G
Further assume that is non trivial and quasi invariant under i e A
F A implies t A forallt Gandlet G
H
satisfy
i is G F H measurable
ii
G H is continuous for almost every
iii Very crude cocycle property For every s t G there exists Ns t F
such that Ns t
and
ts
ts
s for Nst
Then there exists G
H which satis es i
ii
G H is continuous for all
iii Perfect cocycle property
ts
ts
s foreveryst G
In particular eG
eH for all
iv
and are indistinguishable Moreover if is continuous for
all
then
Nft
t forsomet GgF
and N
Proof Step Preparations By ii there exists N F N
such that f
is not continuousg N Rede ning on N to be
identically equal to eH we see that we can assume that satis es i ii
and iii
Let G G be countable dense and de ne Ns
tGNstfors G
ThenNs F Ns
Since every t G is the limit of a generalized
sequence from G since
is continuous and since the limit of a gen
eralized sequence is unique in the Hausdor space H we have
1

PERFECTION OF A CRUDE COCYCLE
iii Crude cocycle property
ts
ts
s foreveryts G Ns
Hence we can and will assume that satis es i ii and iii
Let be a probability measure on G G which is equivalent to a Haar
measure and assume without loss of generality that is a probability mea
sure for the construction of such an equivalent probability measure nd a
positive function which is integrable with respect to the original measure
exercise
De ne
Mfs G
ts
ts
s forallt Gg
f s MforaasGg
f
s
for aasGg
Step We show that
F
and that is
invariant under i e
implies t
forallt G
Note that since is continuous H is Hausdor and by the de nition
of a Haar measure charges all non empty open subsets of G the set M
remains unchanged if we replace for all t G by for a a t G
The map
st
ts
s
ts
Ast
is G G F H measurable since for topological spaces T T U U with
a countable base the Borel algebra of T U coincides with T U see
Dudley
p
hence continuity entails measurability Since H is
Hausdor feHg is closed hence Borel and
AeHGGF
Using iii and Fubini s theorem we get
AeH
Again
by Fubini s theorem using
s MZGAeHstdt
weobtainM G F
M
Similarly
F and
Further
F follows from Fubini s theorem and the relation
ZGZGMs udsdu
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
and
again by Fubini s theorem and also since is equivalent to
a Haar measure
Finally is invariant under since is equivalent to a Haar measure
Step De ne
t
ts
s
s
s
s
eH
Remember that for
s
for aa s GWeshowthat
is well de ned
Assume that
s
and u
By the de nition of
and since is equivalent to a Haar measure we have for all t G and
r Nsu whereNsuisa nullset
ts
s
tr
r
rs
s
and
tu
u
tr
r
ru
u
Now for r N s u and consecutively using
for t
to trans
form s
s
for t
to transform r
r
to transform r u
u
and nally
to transform
rs
s
tr
r
we deduce that
ts
s
s
s
tu
u
u
u
Hence is well de ned
Step Obviously
G H is continuous for all
iv
follows since and agree on the set
Nc
eG of full measure
where NeG is the exceptional set constructed in step for s eG
Step We show that satis es the perfect cocycle property iii Fix
st Gand
The assertion is clear for
so we assume
hence s
Picku Gsuchthat u
andus
thisholdsevenfor aa u G Thenbythedenitionof
ts
tsu
u
u
u
s
su
u
u
u
and
ts
tsu
u
su
u
so iii follows
1

PERFECTION OF A CRUDE COCYCLE
Step It remains to show i for De ne
Bst
ts
s
s
s
s
eH
otherwise
Then B is G G F H measurable recall that
F and is jointly
measurable According to step t s
s
s
s
does
not depend on s as long as
and s
the latter happening
for a a s G by the de nition of Since the countable base of H
generates H and separates points H H is isomorphic as a measurable
space to a subset of see Zimmer
p
Consequently
tZGBstds
for all t so by Fubini s theorem is G F H measurable
If
is continuous for all
then
ft
t forallt Gg
ft
t forallt GgF
thus proving the last statement of iv
Remark i If G R and in ii and ii continuous is
replaced by right continuous or left continuous or cadlag caglad
see Appendix A then Theorem
remains true
ii IfG R andacomplete ltration Ftt RorFts s tisgiven
on F see Sect
and if in addition is adapted i e t is
Ft measurable or t
s
is Fts measurable then obviously
is also adapted Moreover if H is also metrizable then both and are
in fact progressively measurable
We now apply this general theorem to our situation and obtain the following
corollaries
Corollary Perfection of a Very Crude Continuous Cocy
cle Let T R Let be a very crude continuous cocycle over the metric
DS on the Hausdor topological space X where X is either
compact metric or
locally compact and locally connected with a countable base e g a
manifold
A topological space is called locally connected if the connected open sets form a base
of the topology
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Then can be perfected to a continuous cocycle which is indistinguishable
from
Proof By the de nition of a continuous cocycle and by Theorem
Rt
t
Homeo X is continuous for all
It is known
see Arens that under the above conditions H Homeo X
endowed with the compact open topology is a Hausdor topological group
which has a countable base in fact is even a Polish group
Corollary Perfection of a Very Crude Ck Cocycle Let
T R and let be a very crude Ck cocycle where k
Then
can be perfected to a Ck cocycle which is indistinguishable from
Proof It is known that Di k X
k
endowed with the Ck
compact open topology is a Hausdor topological group which has a count
able base in fact is even a Polish group For more details see Appendix
B or Baxendale p
Remark i If takes values in a subgroup H of Homeo X or
Di k X endowed with its respective relative topology then the perfected
also takes values in H
ii One sided time T R The fact that we need to assume that time
is a group leaves the perfection problem unsolved here for the case T
R However in case is generated by a random or stochastic di erential
equation the latter driven by a continuous semimartingale helix we can
always assume basically without loss of generality that T R see Sects
and
If a stochastic di erential equation in Rd is driven by a semimartingale
helix which is not necessarily continuous we have a genuinely one sided
situation since the solution semi ow only takes values in the semigroup
C Rd Rd or Ck Rd Rd The perfection problem in this case is considered
by Kager
and Kager and Scheutzow
Invariant Measures for Measurable Ran
dom Dynamical Systems
Let be a measurable RDS over Suppose the probability measure on
X F B is invariant for the skew product corresponding to i e
t
forallt T As
where
X
x
is the projection onto we have
t
forallt T
1

INVARIANT MEASURES FOR MEASURABLE RDS
Hence the marginal of on F is invariant As our philosophy
is that noise is something given to us and not at our disposal we require
that
P which leads to the following de nition
De nition Invariant Measure for RDS Given a measurable
RDS over a probability measure on X F B is said to be an
invariant measure for the RDS or invariant if it satis es
t
forallt T
P
De ne
PP X f probabilityon XF B withmarginalPon Fg
and
IPfPPX
invariantg
which are both convex sets if we de ne
and t
t
t
Remark i For two sided time it su ces to require condition
only for t since t
tt
ii For discrete time condition follows from
iii Invariant measures which are nite but not nite are interesting
in stochastic bifurcation theory see Sect
Note however that an RDS in general does not come equipped with an
invariant measure it comes only with a invariant P on F In fact
nding and studying the invariant measures of an RDS which are lifts of a
givenPon F to XF B willbeoneofthemajortasksofour
theory of RDS
Note also that an invariant measure is not a product measure in general
see the examples below
Suppose PP X We call a function
B
a
factorization or disintegration or sample measure of with respect to
Pif
forallB B
B is F measurable
forPaa
B
B is a probability measure on X B
forallA F B
AZZXA x dxPd
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
We write symbolically
ddx
dxPd
Condition is also equivalent to the following one For all f L
ZXfdZZXf x dx Pd
Introducing the sections A fx
x Ag can be written as
AZAPd
The following proposition gives su cient conditions for the existence and
uniqueness of a factorization
Proposition Existence and Uniqueness of Factorization
Let PP X Suppose
i B is countably generated
ii the marginal X on X B can be compactly approximated
Then a factorization of exists and is P a s unique
Conditions i and ii are satis ed for any if X B is a standard
measurable space in particular if X is a Polish space with its Borel
algebra
For a proof see Ganssler and Stute
p
Let E F be a sub algebra and PP X If the conditions
i and ii of Proposition
are satis ed then the factorization of the
restriction of to E B with respect to the restriction of P to E is said
to be the conditional expectation of with respect to E and is denoted by
E jE Note that
EjEBEBjE Pas
foranyB B
Lemma Factorization of Image Measure Let be a mea
surable RDS on a standard space and let PP X If is measurably
invertible e g if T is two sided then the factorization of t is given by
t
tt
t
t
t Pas
where the second equality holds for two sided time
See Appendix A for the de nition of a standard measurable space
1

INVARIANT MEASURES FOR MEASURABLE RDS
Proof We check equation
on product sets F B where F F
B B We obtain
tFBZtFt
BPd
ZFt
t
BPd
ZF
tt
t BPd
using the invariance of P for the last equation
The next theorem rewrites invariance of in terms of its factorization
Theorem Invariance in Terms of Factorization Let be a
measurable RDS on a standard space X B and let PP X Then
i IP ifandonlyiffor allt T
EtjtF
t Pas
ii If is measurably invertible e g if T is two sided then t F
Fforallt Tand IP ifandonlyifforallt T
t
t Pas
Proof i It su ces to check equation
for product sets F B
F FB B Wehaveforthosesets
tFB
tFBZtF
t BPd
ZtFt
BPd
and
FBRFBPdZFBtPd
ZtFtBPd
Thus if is invariant then for each xed B and t the t F
measurable function t B is a version of the conditional expectation
E t Bjt F oftheFmeasurablefunction t
Bre
member that t F F
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
The exceptional set here can depend on B and t Since B is countably
generated we can nd a universal exceptional set rst for a countable gen
erating algebra then by the extension theorem for all of B in continuous
time still depending on t outside of which
holds
Conversely if we start with
we arrive at the invariance of
ii In the case that t is measurably invertible we have t F F
and thus
t
B
tBPas
form which we continue as in i
Remark i If property
is satis ed is also called equiv
ariant with respect to over
ii Of course
is su cient for
and thus for the invariance
of also in the case where is not necessarily invertible
iii Let IP with marginal
X EonXThen
and the invariance of P imply that
tEt
forallt T
iv A invariant measure is called a random Dirac measure if there
exists a random variable x
X with
x P a s Invariance in
the case of two sided time then reads as
tx
xt PasforalltT
i e the orbit of the cocycle starting at the random initial value x is a
stationary stochastic process in X Relation
is also su cient for the
invariance of for one sided time
This situation in which is supported by the graph of a random variable
will be encountered quite frequently See Sect for a ne RDS and
Remark
for nonlinear RDS and Sect and Chap for various
examples
Example Invariant Product Measures When is a product
measure P
invariant
i In the case that is measurably invertible in particular if T is two
sided
Pas ifandonlyif t
Pasforallt Tie
almost all mappings t leave the measure invariant which will be
a rare case
ii If is not necessarily invertible P is invariant if and only if
EtjtF
Pas
1

INVARIANT MEASURES FOR MEASURABLE RDS
forallt T SupposeT Z orR and t and t Fareindepen
dent for each t T Since then t
BRXBtxdxand
t F are independent the last condition becomes in this case
EtBZXPftxBgdxB
in short
Et
meaning that is invariant on the average with respect to For RDS
which are either iterations of i i d mappings see Sect
or solutions
of classical stochastic di erential equations see Sect
the above
mentioned independence condition can be satis ed by a proper choice of
the set up Moreover the one point motions are a Markov family with
transition probability
PtxBPftxBg
Then
readsasZXPtxB dx B
Hence in these cases P is invariant if and only if is invariant on
the average if and only if is a stationary measure of the Markov transition
probability P of the one point motion of
The next lemma shows that if we are willing to accept an extension of the
underlying DS and if we only deal with one invariant probability we
can assume without loss of generality that is a random Dirac measure
Lemma Every Invariant Measure is Dirac on Extension
Let IP for a measurable continuous Ck RDS If we extend the
RDS by
FP ttT
XFB
ttT
t
t
then is a measurable continuous Ck RDS on X over and the measure
de ned through its factorization dx
x dx wherex
x
x
x is invariant IP
Proof is invariant if and only if t f f for all measurable
bounded functions f on X where t y
txty
Butputtingfx fxx
tftfff

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Invariant Measures for Continuous RDS
Polish State Space
We can sometimes construct or at least can assure existence of invariant
measures for continuous RDS
Let in this whole subsection unless stated otherwise X be a Polish
space covering in particular any locally compact Hausdor space with a
countable base see e g Bauer
such as Rd a manifold or a
compact metrizable Hausdor space
By Proposition
any probability measure PP X has thus
a P a s unique factorization
ddx
dxPd
and can be naturally identi ed with this factorization Let Cb X be the
Banach space of real valued bounded continuous functions on X with sup
normkfkb supx Xjf xj
Call a function f Cb X measurable if x f x is mea
surable and de ne
LP CbX ff CbX measurablekfk Zkf kbdP g
where as usual f and g are identi ed if kf gk
Remark i Since X is separable kf kb is measurable
ii The joint measurability of f could be rewritten using the following
well known lemma see Castaing and Valadier Lemma
Lemma Suppose f X Z where X is separable metric Z
ismetric f x ismeasurableforeachx X andx f x is
continuous for each
Then
x f x is measurable
iiiForeachf LP CbX and PP Xwehavef L
and putting as usual f R fd
jfjkfk
Further f f islinearforeach and f is a neforeachf
De nition Topology of Weak Convergence in PP X
We call the smallest topology in PP X which makes
f con
tinuous for each f LP Cb X the topology of weak convergence on
PP X Anetf gconvergesinthistopologyto if f f
foreachf LP CbX

INVARIANT MEASURES FOR CONTINUOUS RDS
Suppose PP X and P X are endowed with their respective topologies
of weak convergence Then the projection X PP X P X de ned
by assigning to each PP X its marginal
X
E onX
is continuous
For a thorough study of the topology of weak convergence we re
fer to Crauel
In particular Crauel gives criteria for compactness
Prokhorov theory and proves that PP X endowed with the topol
ogy of weak convergence is itself a Polish space if and only if the probability
space F P is countably generated Theorem p
It follows from the next lemma that the topology of weak convergence
is Hausdor
Lemma Let X be a metric space and
PP X Then
ZfdZfd forallfLP CbX
Proof We prove only the non trivial direction It su ces to show that
AF
A F forallA FandallclosedF Thisisimpliedby
Lebesgue s theorem if we can exhibit a sequence fn x
A gnx
LP CbX withgn F Butgnx
ndxF wheredisthe
metric on X is such a sequence since d x F
xF
Let be a measurable RDS with corresponding skew product x
tx
t t x t actsonfunctionson Xvia
tfxftx
It also acts on measures on X F B via
tA
tAAFB
or
tf
tffL
Proposition Let be a continuous RDS on a Polish space X
Theni The mappings f t f t T are a commuting family of continu
ous linear mappings of LP Cb X to itself isometries if T is two sided
ii The mappings
t t T area commutingfamilyofa ne
mappings of PP X to itself which are continuous with respect to the
topology of weak convergence
All statements of Sects
and remain true if the assumption that is a
continuous RDS is replaced with the more general assumption that is a measurable
RDS of continuous mappings
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Proof iForxedt tf x ft t xismeasurable
and for xed continuous with respect to x Since k t f kb
kf t kb and the last inequality is an equality for two sided time since
t is bijective
k tfk kfk
with equality for two sided time This implies that t is bounded Lin
earity is clear
ii Clearly t is a ne since by de nition t
t
t for all
We prove that t is continuous
Take a net
ie f fforeachfLP CbX Then
alsotf
t fsincebydenition t f
tf
and tf LP CbX bypartioftheproposition
Corollary Let be a continuous RDS on a Polish space X Then
IP is a possibly empty convex and closed subset of PP X
Proof Convexity and closedness follow from the fact that
tis
a ne and continuous on PP X respectively
Remark Suppose P is ergodic Then the set of extreme elements
of the convex set IP coincides with the set of ergodic elements i e
those for which there is no non trivial invariant set for
see De nition
and any two distinct extreme elements are mutually singular See
Crauel Lemma
So far we do not know whether or not there are in fact elements in IP
The following procedure sometimes produces such elements
Theorem Krylov Bogolyubov Procedure for Continuous
RDS Let be a continuous RDS on a Polish space X De ne for an
arbitrary PP X andN TN
N
NPN
n
n
T discrete
NRN t dt T continuous
similarly for N if T is two sided Then every limit point of N for
N
in the topology of weak convergence or for N
for two sided
time is in IP
and any IP arises in this way
Proof For the nal assertion put
which produces the constant
sequence N
1

INVARIANT MEASURES FOR CONTINUOUS RDS
We treat only the continuous time case Assume Nk
This im
pliesthatforeach xedt T t Nk
t since
tis
continuous
Letnowf LP CbX t
xedandNk t Sincet
t f is measurable and bounded by kfk hence locally integrable
the integral in
is meaningful We have
jtNkf Nkfj NkZNkt
Nk Zt
tfdt
t
Nkkfk k
hence t f f ByLemma
t
Since t
is
arbitrary is invariant this is true also for two sided time by Remark
i
Remark i If in Theorem
is measurably invertible in par
ticular if time is two sided then the factorized form of N in
is
N
NPN
n
nn
n
NPN
n
n
n
NRNtt
tdtNRNt
tdt
where the second expression is valid for two sided time This follows from
Lemma
We can see that for two sided time the random Krylov Bogolyubov
procedurebuilds upa measure N inthe berf g X ie attimet
by rst transporting for each xed t the measure
tfromftg
X by means of the mapping t
to a measure t
tin
f g X and then by averaging all those measures in f g X with respect
tot N seeFig
ii Particular case If
X a random variable
then N B is the proportion of time from N which the orbit t
t
t fg XspendsinthesubsetB B
iii The Krylov Bogolyubov procedure remains valid if the initial mea
sure is replaced with a sequence n and if in
N is formed with
n Figure The mechanism of the Krylov Bogolyubov procedure
Compact Metric State Space
If the Polish space X is a compact metric space we can use classical duality
theory
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
For a compact metric space X the Riesz representation theorem asserts
that Cb X M X where M X is the Banach space of signed measures
on X B of nite total variation this holds for any compact space As a
consequence
LPCbXLPMX
with the duality pairing given by
hfiZZXf x dxPd
where LP M X is the space of P essentially bounded M X valued
measurable functions Here measurability is meant with respect to the
Borel algebra of the weak topology on M X see Ledrappier
p
or Bourbaki chapitre x N
We identify PP X
with LP P X by identifying PP X with its factorization
LP PX
PPX
LP PX
By the duality
and the identi cation
the topology of weak
convergence of PP X as de ned in De nition
coincides with the
weak topology in LP P X
Theorem Existence of Invariant Measures Let X be a
compact metric space Then
iPP
X
LP P X is a convex compact subset of
LP MX
ii If is a continuous RDS on X then the convex compact set of its
invariant measures IP is non void
Proof i Since by Alaoglu s theorem for each Banach space B the
closed unit ball of B is compact in the B topology of B the closed
setsLP PX andIP areweak compactinB LP MX
ii By Proposition ii the mappings
t t Tarea
commuting family of a ne weak continuous mappings of LP P X
PP X to itself Hence by the Markov Kakutani xed point theorem
see Dunford and Schwartz
p
there is an element for which
t
forallt TieIP
Remark i The Markov Kakutani theorem can be applied to a
not necessarily measurable cocycle of continuous maps
ii By the Krein Milman theorem see Dunford and Schwartz
p
IP is equal to the closed convex hull of its extremal points See
Remark
1

INVARIANT MEASURES ON RANDOM SETS
iii For a compact metric space X Cb X is separable Hence
LP Cb X is separable if and only if the probability space F P is
countably generated see Appendix A Assuming this the topology of
weak convergence of the compact Hausdor space PP X will be even
metrizable Hence Choquet s theorem see Phelps
applies to IP
and gives the following ergodic decomposition theorem Suppose P is er
godic then for each
IP there exists a unique Borel probability
measure Q such that Q E E the extremal points of IP
such that
f REefdQeforallf LP CbX
Invariant Measures on Random Sets
Again in this section let X be a Polish space
It often happens that we want to construct a invariant measure for
which A for some A F B equivalently
A
Pas
where torecallA fx X x Ag Bisthe sectionofA
LetA PX
A be a function whose values are subsets of
X Such a function is uniquely determined by its graph
graphA f x
XxAgX
Conversely every subset A
X de nes such a function via A
Letforametricspace Xd thedistanceofxandBbedenedby
dxB inffdxy y Bg
The following notion of a set valued random variable will turn out to be
crucial
De nition Random Set The set valued map A P X
A where A is closed compact for all
is called a random
closed compact set if for each x X the map d x A is measurable
A random open set is a set valued map U such that U c is a random closed
set
IfAisarandomclosedsetthensoisAc theclosureofAc IfUisa
random open set then U is a random closed set If A is a random closed
set thenintA theinteriorofAisarandomopenset IfC andC are
random compact sets then so is C C see Crauel Chap for all
these statements
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
The following important facts are quoted from Castaing and Valadier
Theorem III p and Theorem III p
Recall that the algebra Fu of universally measurable sets associated
with the measurable space F is de ned as
Fu QFQ
where the intersection is taken over all probability measures Q on F
and FQ denotes the completion of F with respect to Q
Proposition Let the set valued map A P X take values
in the subspace of closed subsets of a Polish space X Then
iAisarandomclosedsetifandonlyifforallopensetsU Xthe
set f A U g is measurable
ii If A is a random closed set then
graphA F B
iii Conversely if F contains the algebra Fu of universally measur
able sets in particular if F is P complete then graph A F B implies
that A is a random closed set
The property of A being a random closed set is thus slightly stronger than
graph A being measurable and A being closed
Proposition Measurable Selection Theorem Let the set
valued map A P X take values in the subspace of closed non void
subsets of a Polish space X Then A is a random closed set if and only if
there exists a sequence an n N of measurable maps an
X such that
A closure fan
n Ngforall
In particular if A is a random closed set then there exists a measurable
selection i e a measurable map a
Xsuchthata Aforall
De nition Support of a Measure Let PP X
i issaidtobesupportedbythesetC F BandCsupports if
C equivalently if
C
Pas
De ne
PPCfPPXCg
1

INVARIANT MEASURES ON RANDOM SETS
ii The support of is the set valued map
C
supp
Nc
X
N
which by de nition takes its values in the subspace of closed non void
subsets of X Here N F is a P null set outside of which is a probability
measure
Corollary The support C of PP X is a random
closed set In particular graph C F B
Proof Since for any non void open set U X and Nc U
if and only if U supp
fCUgNfNcsuppUg
NfNcUg
which is in F by the de nition of Hence C is a random closed set with
C
by Proposition i
Lemma Let be a net in PP X converging to in the
topology of weak convergence Then for each random closed set C
limsup C C
and for each random open set U
liminf U U
Proof Let C be a random closed set Then d x C is measurable
for each x and clearly x d x C is continuous for each Hence
fnx
ndxC
LP CbX
fn C andfn C Foreach xedn
limsup C lim fn fn
consequently
lim sup C inf
nfn C
The last statement of the lemma follows by taking complements
The support supp of a measure on X B is the intersection of all closed sets
which support We have supp
which in fact holds for any topological space
with a countable base
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
The nal justi cation of the notion of a random set is given by the following
consequence of the previous lemma
Corollary i The set PP C of measures supported by a random
closed set is convex and closed
ii Let be a continuous RDS and C be a random closed set Then
the set
IPjC IP PPC
of invariant measures supported by C is convex and closed
Proof i If
and C then by Lemma
also C
ii IP jC is the intersection of two convex and closed sets
The key question is of course whether IP jC is non void To apply the
random Krylov Bogolyubov procedure or the Markov Kakutani xed point
theorem we have to assure that
t maps PP C into itself This is
guaranteed by an invariance property of C which we are going to introduce
now
De nition Invariant Sets of RDS Let be a measurable
RDS and C
Xaset
a C is called forward invariant if for t
C
tCtPas
equivalently
tCCtPas
b C is called invariant if for all t T
C
tCtPas
for two sided time equivalent to
tCCtPas
Remark i If there is no exceptional set in the de nition of a
forward invariant set we call it strictly forward invariant A set C is
strictly forward invariant if and only if C t C for all t
and
strictly invariant if and only if C t C for all t T
ii For two sided time it su ces to assume strict invariance for t
onlyiii The de nition of strict backward invariance is obtained from the
forward de nitions by replacing t
with t
iv The term positively invariant is also used in place of forward
invariant

INVARIANT MEASURES ON RANDOM SETS
Exercise Algebra of Invariant Sets Prove the following
random analogues of some properties of deterministic invariant sets see
Bhatia and Szego Sect II
A Let be a measurable RDS Then
arbitrary unions and intersections of strictly forward invariant sets
are strictly forward invariant
countable unions and intersections of forward invariant sets are for
ward invariant
C is strictly forward invariant
Cc is strictly backward in
variant
C is strictly invariant
Cc is strictly invariant
The strictly invariant measurable sets hence form a sub algebra of F B
B Let be a continuous RDS Then
If C is strictly forward invariant then so is C
If time is two sided and if C is strictly forward invariant then so
isCintC and C
The proof consists in applying the operations int and on the
relation de ning the invariance of C
Proposition Let be a continuous RDS and IP Then
the random closed set C supp is
i forward invariant if is invertible
ii invariant if time is two sided
Proof i For any continuous f X X and any Borel measure
f supp supp f
For invertible
is invariant if and only if t
t
PasHence
tCCtPas
ii For a homeomorphism f f supp supp f
Proposition Let be a continuous RDS on a Polish space X
i Let C F B be a forward invariant set Then for each t
tPPC PPC
ii If in addition C is a forward invariant random closed set then any
limit point of the random Krylov Bogolyubov procedure with an arbitrary
initial measure PP C is in IP jC
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Proof i Take PP C Since C is forward invariant
Cx
CttxPas
thus
tCZZXCt txdxPd
ZZXCt txdxPd
ZCPd
C
ii By Corollary i PP C is closed
Theorem Existence of Invariant Measures on Random
Sets Let be a continuous RDS on a Polish space X Let K
PX
K
be a forward invariant random compact set Then
PP K is compact and the compact subset IP jK of invariant measures
supported by K is non void
Proof If X is compact metric then PP K is convex and compact in the
topology of weak convergence and
tt
is a commuting
family of a ne continuous mappings of PP K into itself by the previous
theorem Hence the Markov Kakutani theorem applies
For the much more di cult case of a non compact X we refer to Crauel
Corollary
Markov Measures
This section is based on the work of Crauel
De nition Past Future of an RDS Let be a measurable
RDS on a standard space X B with two sided time We call a sub
algebraF Fpastof ifitsatisesforallt
t is F measurable
tFF
The past F is called exhaustive if F
tFt
F
Analogously F F is called future of if it satis es for all t
t is F measurable
tFF
The future F is called exhaustive if F
tFt
F

MARKOV MEASURES
The smallest possible but in general not exhaustive choice for F is of
coursethepastFftxt xXg
and the future
Fftxt xXg
generated by
The concept of past and future of an RDS with two sided time allows
one to restrict the RDS to RDS
and
with one sided
timeT T R overtheDS F PjF
t t T where replacing
F by F amounts to throwing away information in a controlled way
Theorem One to One Correspondence of Invariant Mea
sures for Two Sided and One Sided Time i Let be a continuous
RDS on a Polish space X B with two sided time T with exhaustive future
F F Let be the corresponding RDS with one sided time T intro
duced above Then there is a one to one correspondence between the set of
invariant measures IP and the set of invariant measures IP
given by
IP
EjF
IP
and
IP
lim
t
tt
tIP
The limit in
existsPas andE jF
In particular IP is ergodic if and only if E jF IP
is ergodic
ii A completely analogous statement holds for the restriction to an
exhaustive past F
Proof We rst prove
and assume that satis es
t
ttT
We now assume t and take the conditional expectation on both sides
of the last equation with respect to t F which yields
EtjtFEtjtF
t
Conditioning both sides on F leaves the right hand side unchanged since
t F F while the left hand side can be rewritten as
EEtjFjtFEtjtF
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
where we used the fact that t is F measurable Hence
IP
We now prove
Weshowthatforany xedB Bbutweomit
the argument
t
tt
t
is a martingale with respect to the ltration Gt t T where Gt
t F NoticethatGt G F F
Indeed for
st
E tjGs
Et
t jGs
sEts
t jGs
sEtsjtsF
t
s
ts
t
s
where we have used the cocycle property and the fact that s is
Gs measurable for the second equality sign Gs
ttsFfor
the third and the assumed invariance of for the fourth
Since t B
the martingale is uniformly integrable It hence
convergesPas andinL P
lim
t
t
Pas andL
andtEjGt
The limit is F measurable by construction hence F measurable since
F is exhaustive
Notice that the limit in
is the berwise expression of
lim
t
t
Since s PP X PP X iscontinuousforalls Thereweuse
the assumption that X is Polish and t is continuous the invariance
of the limit follows
It remains to show that if we start with IP go to
EjF
IP
and then construct IP by the above procedure then
By the de nition of t the Gt measurability of t
and the
invariance of
tEjGtt
While the left hand side converges to the right hand side converges to
E jG E jF by the exhaustiveness of F

MARKOV MEASURES
Remark If in the last theorem F is not exhaustive the proof
shows that the one to one correspondence is between those IP which
are F measurable and IP For a general IP only E jF
can be recovered from
E jF by the above procedure
Similarly if F is not exhaustive
De nition Markov Measure Let be a measurable RDS
with two sided time with past F and future F
A probability measure PP X for which the factorization
is F measurable or F measurable i e E jF
Pas iscalled
a Markov measure More speci cally an F F measurable is called a
forward backward Markov measure
There is the following useful improvement of Theorem
Theorem Existence of Invariant Markov Measures Let
be a continuous RDS on a Polish space X with two sided time past F and
future F Let
K
be a forward invariant F measurable ran
dom compact set Then the set IP F jK of invariant forward Markov
measures supported by K is compact and non void
In particular each limit point of the forward Krylov Bogolyubov pro
cedure with a forward Markov initial measure supported by K yields a
invariant forward Markov measure supported by K
Similarly if K is a backward invariant F measurable random com
pact set then the set IP F jK of invariant backward Markov measures
supported by K is compact and non void
Proof Let be the set of forward Markov measure supported by K
Then is non void convex and compact in the topology of weak conver
gence Crauel Theorem
Further by our assumptions
t
which follows from the formula
t
tt
t
Lemma
and the de nition of F
Similarly for the backward case
The reason for naming those measures Markov measures and for the at
tributes forward and backward is foreshadowed by the following corol
lary to Theorem
and will be discussed in Subsects for T Z
and forT R
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
Corollary Past and Future Independent Let be a con
tinuous RDS on a Polish space with past F and future F Suppose that
F and F are independent Then there is a one to one correspondence be
tween the set IP F
of invariant forward Markov measures of and the
subset of those elements of IP
which are product measures
P
Analogously for the invariant backward Markov measures
Invariant Measures for Local RDS
Let be a local continuous RDS with two sided time T R on a Polish
space X
An invariant measure for a local RDS is formally de ned as for a
global one e g by the equivariance property
t
t PasforalltR
Can we have invariant measures in the presence of an explosion
De nition Set of Never Exploding Initial Values De ne
E
tRDt
to be the set of never exploding initial values or initial values for which
the trajectories live from eternity to eternity i e those x for which
DxR
Note that E
X is not necessarily connected and could be empty We
can write
E
E
E
E
tRDt
where E are limits of a decreasing family of open sets
Theorem Let be a local continuous RDS on a Polish space
ThenigraphE F BandE isaG setforall
ii E is strictly invariant
tEEforalltR
and any other strictly invariant set is contained in E
iii E supports all invariant measures i e
IP IPjE
This is only needed here to assure existence and uniqueness of the factorization of
PPX
1

INVARIANT MEASURES FOR LOCAL RDS
Proof i We have in fact E
tZDt whenceE isaGset
Now
graph E
fxxEg
tZfxxDtg
tZDtFB
since D t the t section of the measurable set D R X is measurable
ii Lett Rbe xed Wehavex E ifandonlyifx Ds t
foralls Rifandonlyifs t D xforalls R Bythelocal
cocycle property this is equivalent to s D t t x for all s R
hence equivalent to t x D s t for all s R meaning that
t x E t Thisproves
tE
E t foreach t
hence E is strictly invariant
iii Since t
Dt
D t t we necessarily have
Dt
foreacht RPastorender t
meaningful
P a s Since E is the intersection of countably many sets of full
measure This implies E
Pas
Remark i restricted to E is a nontrivial global bundle RDS
in the sense of Sect
with bers E
XBy
E and
E t are topologically equivalent via the homeomorphism t
ii int E and E E are also strictly invariant
iii If there is a sequence
tt
t
with tjnj
as jnj such that D tn
Dtn thenE
E and
hence E
E
E are closed
iv See the appearance of E in the theory of random attractors Sub
sect
Local RDS with State Space X R
It is an elementary fact that every ergodic invariant measure of a local
dynamical system generated by a C vector eld x f x in R is a Dirac
measure at an equilibrium point This follows for example from Hale and
Kocak Lemma It basically carries over to local continuous RDS
with state space R
Theorem Invariant Measures in X R Let be a local
continuous RDS with time T R and state space X R Then

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
i is monotone with respect to initial conditions i e for each t
andx x Dt suchthatx x
tx
tx
ii For each t and D t is an open interval
iii For each E is a possibly empty interval
iv Each ergodic invariant measure is a random Dirac measure
x
Proof i Since t
t x is continuous x
xbuttx
t x implies the existence of a t for which t x
tx
As a consequence t
Dt
R t is not injective which is a
contradiction
ii If the open set D t is not an interval there exist x
xx
with x x D t but x D t This contradicts the injectivity of
some t
asintheproofofi
iii The set E
n ZD n is the intersection of open intervals
hence is an interval
iv Communicated by Hans Crauel Suppose is an ergodic
invariant measure Let x be the smallest median of i e the in mum
of all points x for which
x
x
the set of medians is a compact interval and x is measurable
Consider
C
x
for which C
by de nition We claim that C is an invariant
setIndeed iimpliesthatxisamedianof ifandonlyif t xisa
median of t
Since is invariant
xt
tx
implying t C
C t Since is assumed to be ergodic
C
Pas
Using the same argument for C
x
we obtain for
fxgC
C
fxg
thus
x Pas
We will encounter many applications of this useful theorem see Exercises
Subsect
and Sect
1

RDS ON BUNDLES ISOMORPHISMS
Random Dynamical Systems on Subsets
and Bundles Isomorphisms
Bundle RDS
We will now introduce two useful extensions of our notion of an RDS which
both have to do with replacing the trivial bundle X in the rst case
with a subset of X which is not a product set in the second case
with a nontrivial bundle We will for convenience call both generalizations
bundle RDS
RDS on Subsets
The restriction of an RDS onto a strictly forward invariant or strictly in
variant subset of X see Sect
creates an RDS on this subset Here
is a formal de nition see Bogenschutz
De nition RDS on a Subset Let C
XCFBbe
a subset with C fx
x Cg forall
An RDS on the
subset C or a bundle RDS with bers C is a measurable mapping
TCCtx
tx
ttx
suchthatthemappings t C C t x
t x form a cocyle
i e satisfy identically
idC
ts
ts
s
If is an RDS on the subset C then C is obviously strictly forward in
variant for one sided time and strictly invariant for two sided time with
t
tt
An invariant measure for an RDS on C has to satisfy C see
Sect
In particular if X is a Polish space and C is a random closed
set then PP C is closed
We also saw in Sect that to each local RDS there corresponds a
natural global bundle RDS on the set E of the never exploding initial
values
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
RDS on Measurable Bundles
Nontrivial bundle DS arise naturally already in the deterministic case A
smooth DS on a manifold X generates through its linearization T a DS
on the tangent bundle T M typically a non trivial bundle The chain rule
is equivalent to T being a skew product x v
txT txv which
is linear on bers i e the mappings
TtxTxXTtxX
form a cocycle of linear mappings over This motivates our de nition of
an RDS on a bundle for which we need the following measurable version
of the notion of a bundle
De nition Measurable Bundle i A measurable bundle
Y
with typical ber X consists of a measurable space Y Y a mea
surable space F whose one point sets are measurable a measurable
space X B a measurable onto map Y
and a global measurable
trivialization i e a bimeasurable bijection Y
X such that
id
equivalently preserves bers see Fig
In
particular for any
j
fgX
is a bimeasurable bijection in the respective trace algebras
Figure Measurable bundle
ii If the typical ber X has the additional structure of a d dimensional
vector space topological space smooth manifold we assume the same
structure for all bers
and
is assumed to preserve this
structure i e is a vector space isomorphism homeomorphism di eomor
phism We then speak of a linear continuous smooth measurable bundle
Notice that there is no probability measure involved in this de nition
DenitionBundleRDSiLet FP ttTbeamet
ricDSandlet Y
be a measurable bundle with typical ber X A
measurable bundle RDS over is a measurable DS on Y which covers
equivalently preserves bers i e it satis es
t
t
forall t T
is equivalent exercise to the statement that the ber mappings
t
tj
t
1

RDS ON BUNDLES ISOMORPHISMS
constitute a cocycle i e
id
ts
ts
siiIfY
is a linear continuous smooth measurable bundle and
if the ber mappings
t
t
respect this structure i e are linear continuous Ck in the two sided
time case they are even linear isomorphisms homeomorphisms Ck di eo
morphisms we speak of a linear continuous Ck bundle RDS
The theory of invariant measures developed in Sects to for RDS
on the trivial bundle X carries over in an obvious manner to RDS on
measurable bundles
Linear bundle RDS are extensively treated in Sect
Remark RDS as DS with a Metric Factor An even more
general notion of an RDS would be a DS with a metric factor i e
there is a measurable mapping which satis es
For Lebesgue spaces can be represented as a skew product over see
Sinai
p for details
Isomorphisms of RDS
One of the basic tasks of the theory of RDS is classi cation For this task
we need concepts enabling us to decide when two RDS can be considered
basically identical or isomorphic Our concepts for RDS are made to be
consistent with and to extend those concepts already available in ergodic
theory metric isomorphism and in the theory of topological or smooth
dynamical systems topological or smooth equivalence or conjugacy
We will certainly not be willing to identify two RDS
if the metric
DS and over which they are de ned are not isomorphic Hence the
rst requirement is that both systems have the same time T and the
metricfactors iFiPi ittT i
are metrically isomorphic
see Appendix A We thus can and will assume that for the purpose of
classicationallRDSaredenedoverthesameDS FP t t T
The second requirement depends on the category of RDS under consid
eration For any category we assume that the two RDS are measurably
isomorphic or cohomologous in the sense of the following de nition
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES
De nition Measurable Isomorphism of Bundle RDS
Given two measurable bundle RDS on Y and on Y over They
are called measurably isomorphic if there exists a measurable isomorphism
of the corresponding bundles i e a bimeasurable bijection Y Y
which covers the identity id i e
id
such that
t
t forallt T
For RDS i on subsets Ci
Xi replace Yi by Ci in the de nition
Measurably isomorphic is an equivalence relation
The de nitions immediately yield exercise the following proposition
Proposition Cohomology of Cocycles Two measurable bun
dle RDS and are measurably isomorphic via if and only if the
corresponding cocycles
it
itji
i
it
are cohomologous with cohomology
j
which means that
t
t
t
For RDS i on subsets Ci
Xi replace i by Ci
See Fig for the diagram of cohomology
Figure Cohomology of cocycles
Remark Origin of the Notions Cocycle etc The no
tions cocycle cohomology and also coboundary a cocycle which is cohomol
ogous to the trivial cocycle t
id X have their origin in homological
algebra and are fundamental in algebraic ergodic theory see e g Schmidt
and Zimmer
InthecaseofagroupT ZorRandacocy
cle taking values in a topological group H the cocycle property is exactly
the condition that be a Borel cocycle on T in the sense of Eilenberg
MacLane with values in the T module of measurable functions from to
H where the T action is de ned by Utf f t Similarly coho
mology in the sense introduced in Proposition
is cohomology in the
sense of Eilenberg MacLane using only Borel cochains See Zimmer
pp
for further references

RDS ON BUNDLES ISOMORPHISMS
If we have invariant measures for the RDS or more structure on the
bers we take these into account with the following more re ned notion of
isomorphism
De nition Metric Linear Isomorphism Topological Ck
Equivalence
i Two measurable bundle RDS and with invariant measures
and are called metrically isomorphic if there is a measurable isomor
phism in the sense of De nition
which satis es
Metrically isomorphic is an equivalence relation
ii Two linear continuous Ck bundle RDS and on linear con
tinuous smooth bundles Y Y are called linearly isomorphic topologi
cally Ck equivalent if there is a measurable isomorphism for which
for all
j
is a linear isomorphism homeomorphism Ck di eomorphism Linearly
isomorphic topologically Ck equivalent is an equivalence relation
Exercise Suppose two measurable bundle RDS and having
invariant measures and are metrically isomorphic with isomorphism
Suppose further that the typical bers are standard spaces so that the
two invariant measures uniquely factorize as
id dxi
idxiPd
Then
Pas
There has been great progress in the classi cation problem in recent years
For the metric isomorphism problem see Gundlach
For the topolog
ical classi cation of RDS see Subsect
and Nguyen Dinh Cong
for a complete treatment For a random version of the Hartman Grobman
theorem see Sect For smooth equivalence normal form problem see
Chap The linear classi cation problem is discussed in Sect
1

CHAPTER BASIC DEFINITIONS INVARIANT MEASURES

Chapter
Generation
Summary
In this foundational chapter we will associate in nitesimal generators with
all reasonably regular RDS
The task is easy for discrete time where all RDS have obvious generators
their time one maps Sect
Products of i i d random mappings and
their associated Markov chains are studied in more detail in Subsect
In the continuous time case we ask for the stochastic equivalent of the
deterministic one to one correspondence between dynamical systems and
vector elds symbolically written as
dynamical system exp vector eld
There are two basically di erent answers
In Sect we obtain the following one to one correspondence Ev
ery wise random di erential equation of the form xt f t xt gen
erates an RDS which is absolutely continuous with respect to time t see
Theorem
for the local case and Theorem
for the global case
Conversely every RDS for which t
t x is absolutely continuous is
a solution of such a random di erential equation Theorem
All we
need for the proof is a new look at classical results for ordinary di erential
equations
It seems surprising at rst sight that there is a large and very im
portant class of RDS for which t
t x is though continuous not of
bounded variation and yet have generators The key extra property is that
t
t x is a semimartingale The generators are stochastic di erential
equations Sect
1

CHAPTER GENERATION
As we are again aiming at a general one to one correspondence between
those RDS and their generators we have to use Kunita s general con
cepts extended to two sided time Our main results are Every stochastic
di erential equation driven by a semimartingale helix additive cocycle
generates a semimartingale RDS Theorem
for the global case The
orem
for the local case Conversely every semimartingale RDS is
generated by a stochastic di erential equation driven by a semimartingale
helix Theorem
This one to one relation can be succinctly written
as
semimartingale cocycle exp semimartingale helix
The proof relies crucially on our Perfection Theorem
Criteria for when a classical i e Brownian motion driven stochastic
di erential equation generates a continuous or Ck RDS are given in Subsect
Subsect
is devoted to the detailed treatment of a one dimensional
example
In the nal Subsect
we sketch the route historically and system
atically that lead from Markov processes via classical stochastic di erential
equations to random dynamical systems As a climax we characterize those
invariant measures which are related to the solutions of the Fokker Planck
equation Theorem
Our hope is that this chapter can serve as a reference text for the gen
eration problem
Discrete Time Products of Random
Mappings
One Sided Discrete Time
Let be a measurable continuous Ck RDS on X over with time T
Z Introduce the time one mapping
XX
By repeatedly applying the cocycle property
we obtain
n
n
n
idX
n
The RDS is measurable if and only if x
x is measurable It
is continuous Ck if and only if x
x is in addition continuous Ck
Recall that an RDS on N can be trivially extended to Z by putting
idX
1

DISCRETE TIME PRODUCTS OF RANDOM MAPPINGS
Conversely given a family of mappings
X X such that
x
x is measurable in addition x
x is continuous Ck
Then de ned by
is a mesurable continuous Ck RDS We say that
is generated by
Hence every one sided discrete time RDS has the form
ie is
a product of a stationary sequence of random mappings or an iterated
function system or a system in a random environment see Fig
Figure Product of random mappings
To emphasize the dynamical perspective we can write the discrete time
cocycle n x as the solution of an initial value problem for a random
di erence equation
xn
nxnn
xxX
The sequence of random points n x n in state space X is the orbit
of the point x under the RDS
Example Linear and A ne RDS Suppose a generator
takes values in some sub semigroup S of the semigroup with respect to
composition of all self mappings of X Then the cocycle will stay there
for all time Cases of particular importance are for X Rd
i Linear RDS products of random matrices Let S Rd d be the
semigroup of all d d matrices with matrix multiplication as composition
A linear RDS has thus the form
nAn
A
Ak
Ak
where A
Rd d is measurable The theory of products of random
matrices with the multiplicative ergodic theorem see Chap is the core of
the theory of RDS with many fundamental papers such as Furstenberg and
Kesten
Furstenberg Oseledets Ruelle Guivar ch and
Raugi
Goldsheid and Margulis
See also the books by Bougerol
and Lacroix Part A and by Carmona and Lacroix Chap IV
ii Products of positive random matrices Let S be the semigroup of
matrices A aij d d with nonnegative or positive entries For a ran
dom Perron Frobenius theory and a survey of earlier results see Arnold
Demetrius and Gundlach
iii A ne RDS Let S be the semigroup of all a ne mappings of Rd
hence
xAxbwithA
Rdd andb
Rd
measurable A ne RDS are iterated function systems in the classical sense
They are important for encoding and visualizing fractals see Hutchinson
Barnsley and Sect

CHAPTER GENERATION
Two Sided Discrete Time
For T Z the time one mapping
and the time minus one
mapping
are by the cocycle property related by
so the generator
X X is invertible for all The repeated
application of the cocycle property forwards and backwards in time gives
n
n
n
idX
n
n
n
This de nes a measurable RDS if and only if
x
xandx
x
are measurable Moreover the RDS is continuous or Ck if and only if
Homeo X or Di k X respectively
Conversely given for each an invertible mapping
XX
such that the two mappings in
are measurable in addition
HomeoX orDikX Then denesvia
a measurable in
addition continuous or Ck RDS
Consequently all two sided discrete time RDS have the form
The corresponding random di erence equation
xn
nxnnZxX
can now be solved forwards and backwards in time
Suppose that a generator takes values in some subgroup of the group
of invertible self mappings of X Then the cocycle will stay there for all
time The cases X Rd
GldR or
an invertible a ne
mapping are of particular importance see Example
Exercise A ne Equation For T Z and X Rd let x
A x b be the time one mapping of the a ne cocycle We have
xAxb
xA
xb
and by induction
nx
n
xPn
jj
bj
n
x
n
n
xPjnj
bj
n
1

DISCRETE TIME PRODUCTS OF RANDOM MAPPINGS
where is the linear cocycle generated by A Let now A be deterministic
and have the form
AAA
with A having its spectrum inside and A outside the unit disc in C and
with log jbj L P Put b bb Then the unique invariant measure
is the random Dirac measure x
with base point
x
x
x
PnAnbn
PnAnbn
A ne cocycles will be treated systematically in Sect
Exercise In the proofs of Lemma
and Proposition
we
will make use of the following observation Let
xAxFx
x
Rd
be measurable and A
GldR Then isthetimeone map
ping of a two sided measurable RDS if and only if x h x
I A F x is jointly measurably invertible It is moreover the
time one mapping of a two sided continuous RDS if and only if x F x
is continuous and h
is continuous
RDS with Independent Increments
We now have a closer look at those RDS on X which are products of i i d
random mappings equivalently have independent increments see Remark
It turns out that they have the extra property that their k point
motions n x
n xk n TareaMarkovfamilyinXkforeachk N
forwards in time for T Z respectively forwards and backwards in time
for T Z We will in particular study the relation between invariant
measures for the RDS and stationary measures for the one point Markov
transition operators P
One Sided Time
This case has been systematically dealt with by Kifer
Theorem Markov Chain Corresponding to RDS i Let
be a measurable cocycle with time T Z which is a product of i i d random
mappings n
n Then for any xed x X or a random variable

CHAPTER GENERATION
x independent of n n Z the orbit xx
n ofxn
nxn x
x the
one point motion is a homogeneous Markov chain with state space X and
transition probability
PxBPf xBg
More generally the k point motions are homogeneous Markov chains for
anyk N
ii If is continuous and X is a metric space then the Markov chain
is Feller i e the transition operator f P f de ned by
PfxZXfyPxdy Ef x
maps Cb X to itself
Proof i We only treat the case k and show rst that P de ned by
is a stochastic kernel
aForeach xedxPx isaprobabilityonXBsinceitisthe
distribution of the random variable
x
b P B is measurable for each B B Indeed denoting the mapping
x
xby wehave
PxB PAx A
BFB
LetA fA F B PAx ismeasurableinxgItcanbeeasilyseenthat
A contains product sets is an algebra and a monotone system Hence it is
a algebra andA F B
For the proof of the Markov property
Pxx
n BjFk Pxx
n Bjxx
kFk
xx
xx
kxXkn
we need the following well known lemma
Lemma Let F P be a probability space X B a measurable
space h X
R bounded measurable C F a sub algebra
X C B measurable and h x and C independent for each x X Then
EhjCEhjH
Hx Ehx
We apply the lemma with h x
Bn
kxCFk
xx
k which yields
Ehxx
k jFk
Pxx
n BjFk
Pxx
n Bjxx
k
Pn
kz Bjz xx
k
Pkn xx
kB
1

DISCRETE TIME PRODUCTS OF RANDOM MAPPINGS
This proves the Markov property as well as a representation for the transi
tion probability
In particular for k n we obtain
Pxx
n Bjxx
nzPnzBPfzBgPzB
i e the Markov chain is homogeneous due to the fact that the n are
identically distributed
In fact xx
n x X is a Markov family with common transition probability
P ii See Kifer
pp
While the passing from a product of i i d random mappings to a Markov
chain is unique the solution of the inverse problem of constructing a
measurable continuous Ck RDS of i i d random mappings with a pre
scribed transition probability is typically not unique and so far largely
unsolved The general obvious reason for non uniqueness is that a tran
sition probability only determines the statistics of the one point motion of
a possible RDS while the mapping
also describes the simultaneous
motion of k points for arbitrary k N
However there is always the following a rmative abstract answer to
the inverse problem
Theorem RDS Corresponding to Markov Chain Let
X B be a standard measurable space and P x B be a stochastic ker
nel Then there exists a algebra H on the semigroup H of measurable
selfmappings ofXsuchthat fx fx isH B Bmeasurable and
a probability m on H with P x B mff f x Bg Consequently the
coordinate mappings
n
non FPnnZ where HZ
F HZ P mZ theshifton generateanRDSonXofiid
random mappings for which the transition probability is the prescribed one
For a proof and for more literature see e g Kifer
pp
Results on
the representation of a Markov chain on a manifold by smooth maps were
obtained by Quas
An up to date survey on the representation
question is given by Dubischar
If the state space X is compact metric then every continuous RDS has
at least one invariant measure see Sect
and every Feller transition
probability P has at least one stationary measure see Appendix A For
an RDS which is a product of i i d random mappings with one sided time
Z we have the following simple one to one correspondence discovered by
Ohno
between invariant product measures P on X and
stationary measures of the corresponding Markov chain on X
1

CHAPTER GENERATION
Theorem One to One Correspondence Between Invariant
Product Measures and Stationary Measures Let be a measurable
RDS of i i d random mappings
n with discrete time T Z on a
standard measurable space X B in its canonical representation i e over
the canonical metric DS F P n n Z introduced in Theorem
Theni n and nF are independent for all n Z
ii A measure P
PP X is invariant if and only if
PX isstationaryforPxB Pf x Bg
iii A invariant measure
P is ergodic if and only if is
ergodic
Proof i is obvious The proof of ii uses the characterization of in
variant measures given in Theorem
and relies on the independence of
n
and nF Details were already given in Example
A proof of iii is given by Kifer Theorem
We stress that even for a product of i i d random mappings in canonical
respresentation there exist in general invariant measures which are not
product measures see Exercise
and Subsect
The colored noise version where
n gnandnisasta
tionary Markov chain was treated by Arnold and Kliemann
and Crauel
among many others
Two Sided Time
Let be a product of i i d random mappings n n Z with two sided time
T Z on a standard space X B Assume that has a canonical rep
resentation i e
G is measurable where G G is a measurable
groupofselfmappingsofXie fg f gandf f aremeasur
able acting measurably on X i e f x f x is measurable
GZ
F GZP mZwherem L
is the two sided shift and
We will now utilize the theory of past and future of an RDS developed
in Sect
For in a canonical representation a natural choice of past and future
clearly is
F
nn
n
nZ
and
F
nn
n
nZ
BothF andF areexhaustiveF F f gandF andF are
independent Further
n andnF aswellas n and nF are
independent for all n
1

DISCRETE TIME PRODUCTS OF RANDOM MAPPINGS
We recall from Sect
that IP F is the set of invariant for
ward backward Markov measures of and that is the restrictions of
to one sided time Z where F is replaced by F
The following theorem is an immediate consequence of Corollary
Theorem One to One Correspondence Between Markov
Measures and Stationary Measures Let be a continuous RDS with
two sided time T Z on a Polish space which is a product of i i d random
mappings in canonical representation Then
i There is a one to one correspondence between the set IP F
of
invariant forward Markov measures of and the set
fIP
PgfP
P
g
i e the set of stationary measures of the forward one point Markov transi
tion probability P x B Pf x Bg The correspondence is given
byE
and
limn
nn
In particular a invariant forward Markov measure is ergodic if and
only if E
is ergodic
ii Similarly there is a one to one correspondence between the set
IPF
of invariant backward Markov measures of and the set of sta
tionary measures of the backward one point Markov transition probability
PxBPf
xBgPf
xBg
The two sided RDS typically has invariant measures which are not
Markov measures see Exercise
Since IP and IP
areinaone
to one correspondence by Theorem
the one sided time restrictions
of also typically have more invariant measures than those corresponding
to stationary measures of P
The two transition probabilities P and P are obviously not inde
pendent of each other but their precise relation is still unclear We have
x yifandonlyif
y x which for a countable state space
yields necessarily
P yfxg P xfyg forallxy X
i e the transition matrices P are doubly stochastic with P
P
The inverse problem Is there an RDS of i i d mappings on X which
reproduces prescribed transition probabilities P is largely unsolved
For a countable state space condition
is also su cient for the repre
sentation by bijections see Dubischar
1

CHAPTER GENERATION
Continuous Time Random Di eren
tial Equations
Generators of deterministic continuous time dynamical systems are au
tonomous ordinary di erential equations or vector elds see Appendix
B In case of an RDS we should expect generators to be certain non
autonomous di erential equations whose right hand side depends on
In this section we treat the easy real noise case in which the generator
is indeed a certain family of ordinary di erential equations with parameter
i e it can be solved pathwise for each xed as a deterministic
nonautonomous ordinary di erential equation All we need is a new way
of looking at well known classical results which we have for convenience
collected in Appendix B
RDS from Random Di erential Equations
LetT RX Rd and beametricDSWewillnowestablisha
basically one to one correspondence between local continuous Ck RDS
over which are absolutely continuous with respect to t and random
di erential equations xt f t xt driven by The correspondence is
given by
t x xZtfs
s xds
which is valid in the local case for all t D x an open interval of R
containing and in the global case for all t R If
holds we say
that t
t x solves or is a solution of the random di erential equation
xt f t xt sometimes called solution in the sense of Caratheodory
or that the random di erential equation generates Note that
implies
x xforall andx
If the solution is di erentiable with respect to t and satis es for t
Dx
d
dttxfttx
xx
it is called a classical solution of xt f t xt Non classical solutions
are important since they allow us to consider discontinuous noise
Theorem Local RDS from RDE Let f Rd Rd be
measurable consider the pathwise random di erential equation
xtftxt
Random di erential equation s is henceforth abbreviated as RDE
1

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
andforxedletftx ft x
iIff LlocRC for all then
uniquely generates
through its maximal solution a local continuous RDS over
ii Iff LlocRCk forall forsomek then
uniquely
generates a local Ck RDS over
iii If f C for all then the continuous RDS from i is a
classical solution i e it satis es
It is locally Lipschitz with respect
to tx andC withrespecttot
iv Iff Ck forall forsomek thentheCkRDS from
ii is a classical solution It is C with respect to t x
Proof The existence uniqueness and regularity proofs are wise adap
tations of the deterministic proofs see e g Amann pp
The only non standard assertion is the local cocycle property which we
verifyhere egforthecasesi ii Forthispurpose x x letD x
be the interval of maximal existence of t x and take s D x We
will show that
stDx
tDs
sx
and in that case we have
tsxts
sx
We distinguish between the four cases
st
s t analogous to
st
s t analogous to
Case step Lett D s
s x Hence
ts
sx
s xZtfus
us
s xdu
xZsfu
u xdu
Zst
sfu
uss
s xdu
where we have used the fact that s D x and made the substitution
u s u Hence both integrals in the last lines are nite We thus have
For the de nition of the space Lloc R C etc see Appendix B
1

CHAPTER GENERATION
found that the function
ux
ux
us
uss
sxsust
satis es
tsxxZtsfu
u xdu
This proves that t s D x so by uniqueness of the solution
tsxtsxts
sx
Case step Lett s D x Then
tsxxZtsfu
u xdu
xZsfu
u xduZt s
sfu
u xdu
s xZtfus
usxdu
where all integrals are nite Putting
sx
s xand
ux
u s x wehavefoundthat
t x xZtfu
u xdu
ietDxDs
s x so by uniqueness of solution
txts
sxtxtsx
Case step Lets t andt D s
sx
Subcase
s t Since
stsstDxautomatically
The local cocycle property then follows from the uniqueness of the solution
Subcase
stSincetst
stDs
sx
automatically and the local cocycle property follows from the uniqueness
of the solution
Similarly for the remaining cases
As a particular case of the last theorem we obtain su cient conditions for
the generation of a global RDS by an RDE
1

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
Theorem Global RDS from RDE Assume the situation of
Theorem
i If f Lloc R Cb for all then equation
uniquely gener
ates a continuous RDS over
iiIff LlocRCk
b for all and some k then
uniquely
generates a Ck RDS over
iii If f Cb for all then the continuous RDS from i is a
classical solution of
ivIff Ck
b for all andsomek thentheCkRDSfrom ii
is a classical solution of
v In cases ii and iv consider the Jacobian of t at x
Dtx
txi
xj
GlRd
Then D is a Ck RDS on Rd Rd over uniquely generated by
xt vt f t xt Df t xt vt Hence D uniquely solves the varia
tional equation
DtxIZtDfs
s xDs xds
thus is a matrix cocycle over the skew product t x
ttx
Finally the determinant det D t x satis es Liouville s equation
detD t x expZttraceDf s
s xds
and so is a scalar cocycle over
Proof The proofs of these facts are wise versions of their determinis
tic counterparts The cocycle property of D over and of D and
det D over follow again from the uniqueness of the wise solution of
the respective equation
Remark i In both theorems we could have required that f has
the corresponding property for all in a invariant set of full measure
Outside this set put t
id Rd for all t R to obtain a perfect cocycle
ii Note that it does not in general su ce to have conditions on f
for each xed
iii The su cient condition for a global continuous RDS in Theo
rem i can be replaced by the following more general one f
LlocRC and kf xk kxk
where t
t andt
t are locally integrable

CHAPTER GENERATION
Remark Variational Equation for Local RDS The varia
tional equation is valid also for the local Ck RDS of Theorem ii and
iv aslongasx Dt thedomainof t Itisgloballyvalidforall
t Rprovidedx E
t RDt seeSects
and for details
because E is strictly invariant and since D t is open t can be
di erentiated at x E for all t
Su cient Existence and Uniqueness Conditions
We will now present static su cient conditions for f Lloc R C
etc the validity of which can often be read o from the right hand side of
xtftxt
Lemma Let FP ttRbeametricDSandg L FP
Then the measurable stationary stochastic process t g t is locally
integrable on an invariant set of full measure and for all a b R
Zb
agtdtL FP
Proof The set of s for which t g t is locally integrable is clearly
measurable exhaust R by countably many nite intervals and invariant
due to the ow property of and the shift invariance of Lebesgue measure
We show that this set has full measure By Fubini s theorem utilizing
Ejgtjm
forallab Rwitha b
Zb
aEjgtjdt mb a EZb
ajg tjdt
implying that Z b
ajg tjdt
Pas
With this lemma we can deduce immediately the following theorem from
Theorems
and
Theorem Su cient Conditions for Local RDS Assume
the situation of Theorem
and
with a measurable f
iIff LP C oriff LP Ck thentheset ofthose
sfor whichf f
LlocRC orf LlocRCk is
invariant and has full measure After having rede ned f outside by
f
the random di erential equation
uniquely generates a
local continuous or Ck RDS
1

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
iiIffLPCboriffLPCk
b thenf LlocRCb
orf LlocRCk
b
on an invariant set of full measure and after pos
sibly rede ning f outside this set as in i the random di erential equation
uniquely generates a continuous or Ck RDS More generally this
istrueiff LlocRC oriff LlocRCk and
kf xk kxk
LFP
Proof We use the de nition of the spaces Lloc R C etc and Lemma
forg kf kKetc
Remark i The following conditions are equivalent to the condi
tionf LP C
Forsomex RdEkfxk
for all R
kfxfykLRkxykkxkkykR
andELR
ii The following conditions are equivalent to the condition f
LP Cb
Forsomex RdEkfxk
forallxy Rd
kfxfykLkxyk
andEL
Example Linear and A ne RDE i Linear RDE Let the
measurable function A
RddsatisfyA L FP Then
ftx At xsatisesf LlocRCb Pas hencext At xt
generates a unique up to indistinguishability C even linear RDS
satisfying
t
IZtA s
sds
and
det t
expZttraceA s ds
1

CHAPTER GENERATION
Also di erentiating t
t
I yields see Sect
t
IZt
sAsds
ii A ne RDE The equation
xtAtxtbtAbLFP
generates a unique up to indistinguishability C RDS The variation of
constants formula yields
tx
txZtt
u budu
txZttuubudu
where is the matrix cocycle generated by xt A t xt Consequently
the RDS consists of a ne mappings see Sect
Exercise RDE with Polynomial Right Hand Side Let
fxPN
j aj xj be a random polynomial with N
If
aj L F P for all j then the random di erential equation
xt
NX
jajtxjt
uniquely generates a local C RDS in R
The case
xtatxtbtxN
t
can be explicitly solved by transforming the equation to an a ne equation
viayxN
N Write down explicitly all ingredients of the local
RDS in particular the domains D t and ranges R t of the set of
never exploding initial values E and the invariant measures The cases
N and N are treated by Xu
and in Subsect
For the
white noise case see Subsects
and
The Memoryless Case
In most applications equation
has the particular form
xtftxtgt
xt
1

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
where t is a nice stationary stochastic process on some state space
E E In other words we assume that the right hand side of
is
memoryless in the sense that only the value of the perturbation at time t
enters into the generator at t
We assume for simplicity that E Rm everything remains valid if E is
a manifold or a Polish space We are interested in allowing discontinuous
processes such as jump Markov processes A good class of processes to
work with is the one whose paths are cadlag i e are right continuous
with left hand limits see Appendix A for details This class is well
suited to deal with semimartingales and Markov processes since under
mild conditions those processes have cadlag versions
We will now describe the canonical setting in which we study
We choose the measurable DS constructed in A with sample space equal
to the Polish space
D R Rm with its Borel algebra F and the
classical shift t
t P can be any invariant measure
De ne
Rm
P
Then the coordinate process
t
t
t
t
is the canonical measurable stationary stochastic process with cadlag paths
and prescribed distribution P Now let
gRmRdRd
be continuous this could be easily generalized and put f x
g
xSincenowftx gtx gt
x wehavean
RDE of the form
Thefunction t x g t
x is measurable t g t
xis
cadlagforall x andx g t
x is continuous for all t
We next give su cient conditions on the function g assuring that
generates an RDS
Theorem Su cient Conditions for Local RDS i Sup
pose is cadlag and g C Then g
LlocRC forall
and the random di erential equation xt g t
xt has a unique
maximal solution which is a local continuous RDS The function
is di erentiable with respect to t at those times at which is continuous
Ifg Ck theng
Lloc R Ck and the local RDS is Ck
ii Suppose is continuous and g C Then g
C
for all
andxt g t
xt has a unique maximal classical solution
which is a local continuous RDS
Ifg Ck theng
Ck andthelocalRDSisCk
1

CHAPTER GENERATION
Proof We just deal with the case i and show that g C implies
g
Lloc R C Indeed
sup
txabKkgt
xkC
since g is continuous and is cadlag thus bounded on compact sets Fur
ther by de nition g C means that g is locally Lipschitz with respect
to x which is equivalent to being uniformly Lipschitz with respect to x on
compact sets K K Rm Rd But since is cadlag a b
K
K a compact set in Rm Consequently the local Lipschitz constant of
g
can be estimated as follows
sup
tabxyKxykgt
xgtyk
kx yk
sup
KxyKxykgxgyk
kxyk L
As a result g
LlocRC
Theorem Su cient Conditions for Global RDS Suppose
iscadlaggC orgCk and
kg xk kxk
where t
tt
t
are locally integrable this will hold on an
invariant set of full measure if Rm R Rm R are continuous or
if
LRmBm
P L then the if is continuous
classical solution of the random di erential equation xt g t
xtisa
global continuous or Ck RDS
Example Noisy Parameters in ODE Consider an ordinary
di erential equation xt g xt whose right hand side depends on a pa
rameter Rm In numerous applications it has to be assumed that such
a parameter is noisy This amounts to replacing the xed with a sta
tionary stochastic process t We therefore arrive at xt g t
xt
Assume in particular g x A x b with continuous functions
ARm RddbRm RdWehaveg Cb Ift iscadlagthen
there is a global C RDS of a ne mappings generated by
xtAt
xtbt
and given by the formula of Example ii

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
Random Di erential Equations from RDS
We now deal with the inverse problem of when for a given RDS on
Rd over with time T R there exists a random di erential equation
xt F t xt which generates We will also determine the only possible
forminwhichFiscoupledto namelyFt x f t x
Theorem RDE from RDS i Let be a local continuous
RDS for which t
t xisdierentiableatt forall x Put
fxd
dttxjt
Then f is measurable t
t xisdierentiableforallt D x and
d
dttxfttx
xx
i e is a classical solution of xt f t xt and f is uniquely determined
by ii Let be a local continuous RDS for which t
t x is absolutely
continuous with respect to t D x for all x Then there exists a
measurable function f Rd Rd for which for all x
t x xZtfs
sxdstDx
ie is a solution ofxt f t xt Thefunctionfis uniqueinthe
sense that if f is another generator then for all x f t t x
f t t xforLebesguealmostallt D x
Proof i We x x and choose an arbitrary t D x Since
D xisopen wehavet h D xforallhwithjhj ht x By
thedenitionofalocalRDSt h D x
hDttx
and
thxhttx
Subtracting t x from both sides dividing by h taking the limit for
h and taking into account that is di erentiable at t yields
d
dttxd
dhhttxjhfttx
ii By assumption there is a g such that
txxZtgsxdsGtx
1

CHAPTER GENERATION
g can be chosen to be measurable in fact put for s D x
gs x limsup
n
GsnxGsx
n
This function g is measurable it is a version of the Radon Nikodym density
of G for all x and the limit exists for Lebesgue almost all s D x
The local cocycle property of over translates into the local helix
property of G over
ForsDxwehavetsDx
tDs
x and
GtsxGts
xGsx
Inserting this into
yields
gs x limsup
n
Gns
x
n
gs
x
Now de ne f x g x which is certainly measurable By
gsxfs
sx
ie wehaveconstructedanffor whichforall x andt D x
t x xZtfs
s xds
meaning that has generator f and f is unique in the sense claimed
The nal result is that the following classes of objects are basically in one
to one correspondence
solutionsofRDEoftheformxt ft xt ie driven bythe
metric DS note that t f t
can also be viewed as a stationary
stochastic process taking values in the space C etc of vector elds
continuous Ck RDS which are di erentiable or absolutely continu
ous with respect to t
The Manifold Case
If X M is a d dimensional C manifold all statements made so far
in this section except the statements about the generation of global RDS
remain valid if properly interpreted see Appendix B for more details
For ease of reference we collect some basic facts in the following theorem
1

CONTINUOUS TIME RANDOM DIFFERENTIAL EQS
Theorem Local RDS from RDE on Manifold Let f
XkM
k
be measurable and consider the RDE
xtftxt
onMLetforxed tft ft
Xk M satisfy
f LlocRCk forall
a su cient condition for
isf LP Ck
Theni Equation
uniquely generates a local Ck RDS over If
f C k for all then is a classical solution of
and is C with
respect to t x
ii The derivative T T M T M is a local Ck RDS which uniquely
solves
vtTft vt
where Tf is the natural lift off to TM
iii If M is a Riemannian manifold and if we use the Riemannian
connection on M to decompose TvT M into horizontal and vertical compo
nents then T f v has horizontal component f x and vertical com
ponent rf x v where r denotes the covariant derivative and equation
is equivalent to the coupled system of the original RDE
and
the variational equation
vtrfttxvt
Here the absolute derivative vt is taken along the integral curve t
tx
in the direction of the vector eld f t see Appendix B
Furthermore we have Liouville s equation for det T t x For t
Dx
detTtx
expZtdivf s
s xds
expZ t trace rf s
s xds
iv The RDS is global in case M is compact but f still satis es
or by embedding M into RN f is the restriction of a random
vector eldf onRNforwhichf LlocRCk
b
1

CHAPTER GENERATION
Continuous Time Stochastic Di er
ential Equations
Introduction Two Cultures
We can go a big step beyond Sect and can assign generators to an
important and large class of RDS which are just continuous but not abso
lutely continuous in fact nowhere di erentiable and locally not of bounded
variation with respect to t Such generators cannot be pathwise di erential
equations but will turn out to be so called stochastic di erential equations
An SDE contains stochastic integrals one of the central objects of stochas
tic analysis which are de ned only as limits in probability and not as wise
limits In particular wise deterministic results cannot be utilized as in
Sect
to establish the existence and uniqueness of an RDS generated
by an SDE
More speci cally the class of RDS which have an SDE as generator
consists of those which have the additional statistical property of t
t x being a semimartingale for each xed x The generating SDE
will turn out to be driven by a semimartingale with stationary increments
called a helix This establishes a basically one to one correspondence
which can be succinctly expressed as
semimartingale cocycle exp semimartingale helix
where exp symbolizes an engine the implementation of which is the
subject of this section
For example consider the simplest possible SDE
dxt dWt W Brownian motion see Appendix A
ItgeneratestheC RDS t x x Wt
of random translations
over the canonical DS describing Brownian motion But t
t xis
known to be nowhere di erentiable and of unbounded variation
The general case is conceptually and technically very complicated and
can only be fully appreciated with a good knowledge of stochastic analysis
which we assume for this section The theory of SDE and stochastic analysis
have been laid down in numerous excellent textbooks of which we just men
tion a few more recent one s in alphabetic order Belopolskaya and Dalecky
Elworthy
Gard
Gihman and Skorohod
Ikeda and
Watanabe
Karatzas and Shreve
Malliavin
Protter
Revuz and Yor
Rogers and Williams
Stroock and Varadhan
Stochastic di erential equation s is henceforth abbreviated as SDE
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Two Cultures
In the theory of RDS two well established mathematical cultures meet
overlap and sometimes collide
Dynamical systems the ow point of view Typically T R or Z
unless mappings are non invertible which typically happens only for dis
crete time or in nite dimensional state space On the measurable level
ergodic theory assumes an invariant measure where invariance is de ned
with respect to the mappings of the ow Time is purely algebraic and
non physical There is no such thing as past present future or
evolution
Markov processes stochastic analysis Here time is almost exclu
sively one sided i e T R or Z or part of it Markov processes are
de ned and studied via their transition semigroup forward in time The
necessity to really construct stochastic processes with prescribed transition
semigroup their existence follows from Kolmogorov s fundamental theo
rem motivated the creation of the theory of di erential equations with
white noise input in other words the theory of SDE and nally stochastic
analysis Continuous time is R and a ltration Ft t i e an increasing
family of sub algebras of the completion of F collects the information
available at time t Everything at least until recently has to be adapted
i e Ft measurable Invariant measure now means that the measure is
stationary with respect to the Markov transition semigroup The ltration
gives a direction to time and allows one to speak of the past etc of t thus
getting closer to a physical concept of time
In short the key object of continuous time T R dynamical systems
is a ow t t R of mappings and in ergodic theory a measure P invariant
with respect to it while the chief ingredient of stochastic analysis is a
ltration Ft t on T R which allows to study evolution in time
The rst people who clearly spelled out this gap between ergodic the
ory and stochastic analysis and made a rst attempt to bridge it were
de Sam Lazaro and Meyer
with their theory of ltered ows
FP t t Ft t forevenearlierworkseeKrengel p
J de Sam Lazaro and Meyer wrote on p in
Mais la theorie
ergodique est plus proche dans bien des cas de la theorie des groupes que
de celle des processus stochastiques car il y manque l idee probabiliste es
sentielle celle d une evolution dans le temps Such a concept of evolution
in time is the ltration Ft
The work of de Sam Lazaro and Meyer was continued by Protter
who was able to characterize the class of semimartingales with stationary
but not necessarily independent increments These semimartingales qual
ify as integrators driving noise in SDE which generate RDS The work

CHAPTER GENERATION
of de Sam Lazaro Meyer and Protter can be considered as a preparation
for our systematic study of the relations between RDS and SDE presented
below We closely follow Arnold and Scheutzow in our presentation
which is based on the fundamental work of Baxendale Le Jan and
Watanabe
and Kunita
In order not to overburden matters we rst deal with the global case in
X Rd and discuss the local and manifold case later
The gate from stochastic analysis to dynamical systems was really
opened around
when several people Elworthy
Baxen
dale Bismut
Ikeda and Watanabe
Kunita
Meyer
and others realized and proved that writing down a classi
cal SDE means much more than originally intended by the fathers of
stochastic analysis An SDE does not only generate a Markov family of
solution processes indexed by the initial time and initial position but it
generates a two parameter ow s t of random homeomorphisms or dif
feomorphisms However this is still a static object and not yet an RDS
To prove that t is in fact a cocycle over the DS by which the driving
noise is modelled more work is necessary
There are certain obstacles in our way While we can and will assume
that our metric DS has two sided time T R classical stochastic calculus
as already mentioned above has been almost exclusively developed for one
sided time T R and a one parameter ltration Ft t R Hence the
rst task we will have to accomplish in Subsects
and
is the
extension of stochastic calculus to two sided time T R Quite naturally
we will work with a two parameter ltration Fts s t which we will have to
make consistent with
Besides its aesthetic appeal two sided time is fundamental for the the
ory of smooth RDS as it allows a better multiplicative ergodic theorem
providing an invariant splitting of subspaces rather than only an invariant
ag hence making general invariant manifolds normal forms bifurcation
theory etc possible
Semimartingales and Dynamical Systems
Stochastic Calculus for Two Sided Time
Let F P denote from now on a complete probability space
De nition Two Parameter Filtration Assume Fts s t R
s t is a two parameter family of sub algebras of F with the following
properties
Fts Fv
uforustv
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Fts
u tFu
s Fts Fts
usFtu Fts fors t
Fts contains all P null sets of F for every s t
Then Fts s t is called a two parameter ltration on F P We
de ne
Ft
s tFts Fs
t sFts
As we want to solve an SDE from an initial time forwards as well as back
wards in time we follow Kunita
and introduce the following notions
De nition Forward Backward Semimartingale Let Fts
s tbealtrationon FPandletF RR
Rd
st
Fs t be measurable and jointly continuous with respect to
stforall
a F is called an Fts forward semimartingale if for each s R
Fsst tisanFst
s t semimartingale
b F is called an Fts backward semimartingale if for each s R
Fsst t isanFs
s t t semimartingale
c F is called an Fts semimartingale or just semimartingale if F is both
a forward and a backward semimartingale
Remark i As in the classical case the class of Fts forward re
spectively backward semimartingales remains invariant under an equiva
lent change of the underlying probability P
ii Fs
s is non trivial in general
De nition FilteredDS Let F P t t R beametricDS
letFbethePcompletion ofF andletFts s t be a ltration on
FP Wecall
FP ttRFtsstalteredDSifforallstuRs twehave
uFts Ftu
su
Remark One can construct a ltered DS from a metric DS in the
following way in complete analogy with the construction of de Sam Lazaro
and Meyer
p in the one parameter case
1

CHAPTER GENERATION
Let F P t t R beametricDS FthePcompletionofF andlet
F and F be two sub algebras of F representing past and future
respectively see Sect both containing all P null sets of F such that
t F F forallt
and
t F F forallt
Then de ne
Ft
tF Fs
sF FtsFsFtforst
Clearly Ft is increasing in t and Fs is decreasing in s Further Ft and Fs
and hence Fts contain all P null sets of F By de nition
u Fts
uFs
uFt
usFutF
Ftu
suforalluRst
It is proved in
p thatFtt
is right continuous It follows
immediately that F P t t R Fts s t is a ltered DS which moreover
satis es
Fts
Fts Fts fors t
It will turn out that generators of cocycles are again cocycles with compo
sition of mappings replaced by addition in a vector space Additive cocycles
were called helices by de Sam Lazaro and Meyer the expression goes
back to Kolmogorov
Here is an abstract de nition already used in
the Perfection Theorem
DenitionHelix Let FP ttRbeametricDSand
HagroupFR
H is called a perfect helix or perfect
cocycle if
Fts
Fts Fs
forallst Randall
F is called a very crude helix or a very crude cocycle if for every
stR
only holds up to a P null set which may depend on s and
t

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Wewillonlyusetheterm helix ifHisAbelian egH Rd
whereas we will continue to use cocycle if H is a non Abelian group of
self mappings of a space with respect to composition Note that in contrast
to De nition
an abstract cocycle helix in the sense of De nition
automatically satis es F
eH for all
eH being the identity of
the group H since we have assumed that F takes values in a group
Proposition Let H be a group
iAssume FP ttRisametricDS
and F
H satis es
Fth FhtFt
foralltth
Then F can be uniquely extended to a
helix F If holds only up to a P null set depending on h and t then
F can be extended to a very crude helix F If H is a topological group
and F is continuous cad cadlag cag caglad for all
then the same is
true for F
iiAssume F P ttRFts s tisalteredDSFisanHvalued
very crude helix and H is some algebra on H such that the composition
map is measurable If for some
F h is Fh H measurable
forall h thenFisFtsadaptedieFt Fs isFtsH
measurable for all
st
Proof iDeneFs
Fs for
s
and by induction over
k
Fks
Fsk
Fk
Fk
ifk N
Fks
Fsk
Fk
Fk
ifk N
for
s
Note that F is well de ned It is straightforward to check
that F is a helix resp a very crude helix If F satis es equation
without an exceptional null set then it is clear that F is the only function
which extends F to a helix on R The last assertion is also clear
ii For
h
using the assumption that F h is Fh H
measurable and the de nition of a ltered DS we see that F h t is
Fth
t H measurable for any t R By the very crude helix property and
the completeness of Ft h
t thesameistrueforFt h Ft
For
h we have
FthFt
FthFt
Ft
Ft
which is Ft h
t H measurable The assertion now follows by induction

CHAPTER GENERATION
Next we study processes which are both semimartingales and helices at the
same time
De nition Semimartingale Helix Suppose we are given a
lteredDS F P tt R Fts s t An Rd valuedhelixFiscalled
forward resp backward resp semimartingale helix if
Fst
Ft Fs
is an Fts forward resp backward resp semimartingale
The next proposition shows that there is a canonical way in which a semi
martingale with stationary increments can be ameliorated to be a semi
martingale helix over a ltered DS
Proposition Let E be a locally compact Hausdor space with a
countable base let C E Rd be the space of continuous functions from E
to Rd endowed with the metrizable topology of uniform convergence on
compact sets and let C R C E Rd be the space of continuous C E Rd
valued functions on R which are zero at zero also equipped with the topology
of uniform convergence on compact sets
Let Fts s t be a ltration on the complete probability space F P
AssumeF R E
Rd tx
F t x is jointly continuous
intxforall
Fx
forallx E
and that
Fstx Ftx Fsx
is an Fts forward resp backward resp semimartingale for every x E
Further assume that F has stationary increments in the sense that the law
offFt hx Ftx
x Eh RgonCRCERd doesnot
depend on t R
Thenthereexistsa lteredDS F P tt R Fts s t withforward
resp backward resp semimartingale helices F x
x E which are
also jointly continuous in t x for every
such that the laws of F and
FonC RCERd coincide
Proof De ne
CRCERd
and let F be the Borel algebra of which coincides with the algebra
generated by the evaluations Note that the continuity and measurability
assumptions imply that F is an F valued random variable Let P be
itslaw Deneashift by
ts
ts
t
tsR
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Clearly s t
t s forall
s t R P is invariant under
since F has stationary increments Further R
isBFF
measurable since it is continuous and and R are both topological spaces
with a countable base see Dudley
p Dene
Ft
ttR
ThenFisan H
C E Rd valued helix since
Fth
Ft
thtthFht
F and F have the same law by construction Let F resp F be the
algebra generated by the P null sets of F the P completion of F and
ss
resp
ss
De ne Fts as in Remark
ieFts
is the completion of v
u s u v tbyallFnullsets In
particular Fs
sistrivialforalls R
It remains to show that F x is an Fts forward resp backward resp
semimartingale for every x E We only prove the forward statement
the backward one is analogous The mapping
F FFtsstP
FFtsstP
isFFmeasurableFP PandGts F Fts Fts t s Fix
s R Stricker s theorem see Protter
p
implies that
Fs ux Fsx
u
isalsoaGsu
s
u semimartingale for
every x E Finally Theorem
in Jacod p
shows that
Fs ux Fsx
u
isanFsu
s
u semimartingale This
proves the proposition
The next proposition connects for the helix case our de nition of a semi
martingal with the usual one parameter de nition
Proposition Givena lteredDS F P tt R Fts s t and
FR
Rd
F is a forward semimartingale helix if and only if F is a helix and
F j is an Ft t semimartingale in the usual sense
Similarly F is a backward semimartingale helix if and only if F is a
helix and F j is an F t t semimartingale in the usual sense
Finally F is a semimartingale helix if and only if F is a helix F j is
an Ft t semimartingale and F j is an F t t semimartingale
both in the usual sense
Proof We only prove the forward statement By Proposition ii
and the fact that s
FFttP
FFst
s tPisan

CHAPTER GENERATION
isomorphism by De nition
Jacod Theorem
p
and
the helix property of F imply that if F j is an Ft t semimartingale
thenFs u FsuisanFsu
s
u semimartingale
Example A forward semimartingale helix need not be a back
ward semimartingale helix As an example consider Brownian motion
on R de ned on the canonical space and choose for s t the ltration
Fts Wu Wv
v u t completed by null sets Clearly
W is a forward martingale helix but not a backward semimartingale helix
Next we introduce the forward and backward local characteristics of a
forward backward semimartingale helix
Proposition Givena lteredDS F P tt R Fts s t
i Let F be an Rd valued forward semimartingale helix Let
FtMtBtt
be the canonical decomposition of the Ft t semimartingale F j cf
Theorem
Then there exist
a strictly increasing continuous real valued Ft adapted process A t
t withA
an Ft predictable Rd valued process b t t
an Ft predictable process a t t
taking values in the set of
non negative de nite d d matrices
such that for all t
we have
BtZtbsdAs
Qij t hMi
Mj itZt
aijs dAs
ii Analogously let F be an Rd valued backward semimartingale helix
consider the canonical decomposition
FtMtBtt
of the Ft t semimartingale F j backward direction Then there
exist

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
a strictly increasing continuous real valued Ft adapted process A t
t withA
an Ft backward predictable Rd valued process b t t
an Ft backward predictable process a t t taking values in the
set of non negative de nite d d matrices
such that for all t
we have
BtZtbsdAs
Qij t hMi
Mj itZt
aijs dAs
Proof i Take
At
dX
iZtjdBi s jhMiit t t
bit
lim sup
st
Bit Bis
At As
and
aij t
lim sup
stQijt Qijs
At As
Then bi resp aij is a version of the Radon Nikodym derivative of Bi
resp Qij with respect to A according to Lebesgue s di erentiation the
orem see e g Carmona and Nualart pp f
All claims follow immediately
ii Take the same de nitions as in i with quantities replacing all
corresponding quantities and replacing
De nition Local Characteristics of Semimartingale He
lix Let FP ttRFtsstbealteredDSandletFbeanRd
valued forward resp backward resp semimartingale helix Then the
quantities a b A resp a b A are called the forward resp
backward local characteristics of F
In case of a semimartingale helix we can and will piece the quantities
and the quantities together to yield the canonical decomposition
FtMtBttR
1

CHAPTER GENERATION
and a triple a b A now de ned on all of R and called the local character
istics of F They satisfy for all t R and
Bt Ztbs dAs
Qijt hMi Mj itZt
aijs dAs
Remark i We de ned the local characteristics of the for
ward semimartingale helix F as the local characteristics of the Ft
semimartingale F j Using Jacod s result we can argue as in the
proof of Theorem
thatforeachs RtheFs t
s t semimartin
gale F s t F s t has the canonical decomposition
Fst Fs
Mst Bst
where Mst
Mts Bst
Bts
are both Fs t
s measurable and the local characteristics a s b s A s are
relatedto a b A via
ast
ats bst bts
and
Ast Ats
Similarly for a backward semimartingale helix
ii Given a semimartingale helix F it is clear that the quadratic vari
ation processes
Qt
Qt
t
Qt
t
is a very crude helix and hence by Theorem
has a version which is
a helix Further Q t Q s is Fts measurable A and B however generally
do not enjoy the very crude helix property nor are A t A s and
B t B s Fts measurable in general Hence the canonical decomposition
of a semimartingale helix does not in general consist of helices One may
ask whether we have chosen the wrong de nition As an alternative
one could try to decompose the forward semimartingale helix F into F
M B where M and B are a forward local martingale helix and an Fts
adapted helix of locally bounded variation resp Then hMi should also be
an Fts adapted helix Unfortunately such a decomposition does not exist
in general

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Semimartingale Helices with Spatial Parameter
In this subsection we develop stochastic calculus for semimartingale helices
depending on a parameter x Rd Our exposition is largely analogous to
that of Kunita and similar to that of Carmona and Nualart but
di ers conceptually in two respects
We consistently consider two sided time
We assume the helix property
To save space we only formulate our statements for the forward as well as
backward semimartingale case
De nition Semimartingale Helix with Spatial Parame
tersLet FPttRFtsstbealteredDSdNkZ
Assumethatforeachx Rd Fxt isanRdvaluedsemi
martingale helix and let
Fxt
Mxt Bxt
tR
be its canonical decomposition F is called Ck
b semimartingale helix if for
P almost all
M t andB t areinCk
b forallt R
for j j k the spatial derivative Dx M x t is continuous with
respectto xt andforeachxalocalmartingaleie Mxt t
isanFtt localmartingaleandMx t t isanFtt
local martingale and Dx B x t is continuous with respect to x t
and for each x of locally bounded variation in t
We will also use the corresponding notion with Ck
b replaced with Ck
Remark If the semimartingale F x t with spatial parameter
x is only assumed to be a very crude helix but if F is continuous with
respect to x t for P almost all
then F is a very crude helix with
values in the topological group C which has a countable base The
reasonisthatfor xedst R
Fxt s
Fxts Fxs
holdsforallx Rd onthecomplementof y QdNsty whereNstyisthe
set of measure zero where the helix property fails
By Theorem
F can be perfected i e there is a semimartingale
helix F indistinguishable from F

CHAPTER GENERATION
For a semimartingale helix with parameter x Rd the local characteristics
abA seeDenition
will of course also depend on x However
in order to develop a reasonable theory of SDE driven by semimartingale
helices we need to have more uniformity of the local characteristics with
respect to x We follow Kunita
pp and and make the following
assumption
Assumption Uniformly Controlled Local Characteris
tics We assume that for the semimartingale helix F x t there exists a
continuous adapted without loss of generality strictly increasing process
At withA
such that
there exists a measurable function b x t which for every x is a
predictable process such that for all x Rd t R and
BxtZtbxs dAs
there exists a measurable function a x y t which for every x y
Rd is a predictable process such that for all x y Rd t R
Qxyt hMxt Myt iZt
axys dAs
The triple a b A is called the local characteristics of F We will say that
F is uniformly controlled by A
Remark i If the local characteristics of F are a b A then by
Remark
the local characteristics of
Fxst Fxs
Fxts Mxts Bxts
areasbsAs
ii Carmona and Nualart p
introduce a similar assumption
and say F is uniformly controlled by A which we adopt in our case above
iii Le Jan and Watanabe
p
restrict themselves just to the
caseAt
t
iv Let F be a C semimartingale helix The fact that Q x y t is
uniformly controlled holds without loss of generality and follows from Ku
nita Exercise ii p
Kunita s argument needs a countably
generated F which is the case for the canonical set up of Theorem
By exploiting the joint continuity of M x t it is not hard to see that the
result remains true without the hypothesis that F be countably generated
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
v A su cient condition for B to be uniformly controlled is that t
B t is locally of bounded variation as a function with values in the
Frechet space C a similar condition is given by Carmona and Nualart
pp and
To describe further regularity conditions for the local characteristics of a
semimartingale helix we introduce the following spaces
De nition Let k Z and
De ne
Ck
bfgCRdRdRdkgkk g
where
kgkk
sup
xy kgxyk
kxk kyk X
jjk
sup
xykDxDygxyk
and in case
kgkk kgkk
sup
xxyyX
jjkkgxy gxy gxy gxyk
kx xkky yk
De ne Ck analogously
De nition Let k Z
andletA R Rbea
continuous increasing function with A
DeneLlocRdACk
bto
be the set of measurable functions f Rd R Rd for which
ftCk
b foreveryt R forAalmostallt wouldsu ce
for every t R
jZtkf skk dAsj
With the system of seminorms
LlocRdACk
b is a Frechet space
In addition we have the continuous inclusions
LlocRdACk
b
LlocRdACk
b
LlocRdACk
b
The spaces Lloc for Ck Ck
b and Ck are analogously de ned

CHAPTER GENERATION
Stratonovich Stochastic Integrals with Respect to a Semimartin
gale Helix
Let now F x t be a C semimartingale helix with local characteris
tics abA satisfyinga LlocRdAC forsome
and b
Lloc R dA C meaning that it holds for all
Letfstbea
semimartingale in the sense of De nition c
Forward Stratonovich Stochastic Integral Let s t Then
Ist Zt
sFfsu du
lim in pr
nX
k fFfstk tk Ffstk tk
Ffstk tk Ffstk tkg
where the limit in probability runs through a sequence of partitions of s t
for which the mesh
maxkntktk
This limit exists
and is a forward semimartingale
Remark For s r t the integral
Zt
rFfsu du
alsomakessensesinceFxs r u Fxsfsr u u
are
Fru
s
u semimartingales
Backward Stratonovich Stochastic Integral Let t s Then
Ist Zs
tFfsudu
lim in pr
nX
k fFfstk tk Ffstk tk
Ffstk tk Ffstk tkg
where the limit in probability runs through a sequence of partitions of t s
forwhichmax k n tk tk
This limit exists and is a backward
semimartingale
The stochastic integrals can of course be de ned with respect to semimartingales
which are not necessarily helices see Kunita
pp
but this is not needed
here
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
We stress that there is no relation whatsoever between the forward and
backward integrals
Collecting both cases into one we conclude that Is t s t R is a semi
martingale
The helix property of F allows us to reduce one endpoint of the integral
to zero
Proposition Let s R be xed
iLett Then
Zst
s FfsuduZt
sFfss udu Pas
where sFxt
Fxts Incasefss t ftsforall
tandNs
Zst
s Ffsudu
sZtFfu du Pas
ii Let t Then
Zs
stFfsuduZt sFfss udu Pas
Incasefss t ftsforallt and Ns
Zs
stFfsudu
sZtFfudu Pas
This follows immediately from the de nition of the corresponding integrals
RDS from Stochastic Di erential Equations
We will now establish the essentially one to one correspondence between
semimartingale helices and semimartingale cocycles
De nition Semimartingale Cocycle Let be a cocycle
over a ltered DS for which
Gsxt
ts
xx
is a Ck semimartingale Then is called a Ck semimartingale cocycle
Put
Gxt
Gxt
txx

CHAPTER GENERATION
Lemma Given a metric DS The following statements are
equivalent
i isacocycleover ie t s
ts
s
iiGsxt Gxtss
iii G is a helix over the skew product ow
ie
GxtsGsxtsGxs
The proof follows from the de nitions
Corollary Given a cocycle over a ltered DS is a Ck
semimartingale cocycle if and only if G t t
isanFttCk
semimartingale and G t t is an F t t Ck semimartingale
The canonical decomposition
Gsxt
Msxt Bsxt
satisesfor allst R
Ms xt
MxtssBsxt
Bxtss
whereGxt
M x t B x t is the canonical decomposition of
G
The proof is the same as for the helix case see Remark
In a rst step we move from a semimartingale helix to a semimartingale
cocycle via a Stratonovich SDE driven by the helix
Theorem Global RDS from Stratonovich SDE Given a
lteredDS FP ttRFtsstLetFbeaCk
b semimartingale helix
over where k and
Assume that F has local characteristics
aF bF AF satisfying aF Lloc R dAF Ck
b
bF LlocRdAFCk
b
andcF LlocRdAFCk
b where cF is the Stratonovich Ito correction
term given in
below Then there exists a unique up to indis
tinguishability global Ck RDS over which for any
isaCk
semimartingale cocycle which solves the SDE
dxt Fxt dt
i e more speci cally for each xed x Rd
tx x RtFsxds
t
RtFsxds t
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
this holds for xed t and x outside a P null set Nt x The semimartingale
cocycle has the local characteristics A AF
a xyt
aFtxtyt
and
bxtbFtxt
cFtxt
where
cFxt
dX
j
aj
F
xjxytjyx
Proof
Existence and uniqueness We can use Kunita s results in
we need the global versions of Theorems
and
for the
Stratonovich case which Kunita did not state explicitly but could be de
rived from Theorem
and Corollary for the Ito case taking into
account the correction term cF The key point is that our driving process
Fsxt
Fxt Fx s isaforwardaswellasbackwardsemi
martingale whose local characteristics as bs As
s a sb sA satisfy
the corresponding regularity conditions for each s R provided they do for
s
which we have assumed
Consequently there exists a Ck semimartingale of Ck di eomorphisms
st
st Rwhichforeachx Rd satises
stxxRt
sFsuxdu st
Rs
tFsuxduts
Two parameter ow property There is a version of which satis es
ss
id
and
st
ut
su
identically Indeed since our mappings are invertible with st
ts it
su ces to prove the ow property for the case s u t which was done
by Kunita
ppf
Crude cocycle property We claim that
t
t
satis es the crude cocycle property in the following form
ts
ts
s forallts R
Ns
1

CHAPTER GENERATION
where Ns is a P null set which possibly depends on s the dependence on t
can be removed by utilizing the continuity with respect to t
In view of
is equivalent to
sst
t s forallst R
Ns
Consider rst the case t We have for xed s R
sstxxZst
sFsuxdu
Zt
sFssuxdu
by Proposition i where sF x t
Fxts
Now obviously tx uniquely solves dx F x d t if and only if
t s x uniquely solves dx
sF x dt Henceforeach xeds R
and for all t
sst
tsforNs
The case t is analogous using the backward integral and Proposition
ii
Perfection The topological group Di k Rd of Ck di eomorphisms
of Rd endowed with the usual complete metric is a Polish group see Kunita
p
and hence has a countable base Theorem
applies
providing us with an indistinguishable perfect cocycle
Local characteristics By de nition the local characteristics of are
those of the semimartingale
Gxt
tx x RtFsxds
t
RtFsxdst
Decompose G as
Gxt
Mxt Bxt
First consider the case t
By Theorems
and Lemma
of Kunita
MxtZtMF sxds
with
QxytZt
aFsxsysdAFs
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
and
BxtZtbF s xs dAFsZt
cFsxsdAFs
with cF given by Kunita
p
For the case t we use formula
onp
of
and obtain
rst
MxtZtMF sxds
and then with our convention having a t
and hMit increasing on all
ofR
QxytZt
aFsxsysdAFs
also for t
Further with our convention Rt f dg R t f dg for t
BxtZtbF s xs dAFsZt
cFsxsdAFs
also for t
Remark Uni ed SDE for Two Sided Time So far all of
our stochastic integrals have lower limits less than or equal to upper limits
For a more symmetric formulation we introduce as in the case of ordinary
Lebesgue integral the following convention For each xed s R and any
semimartingale fs t put
Zt
sFfsudu Rt
sFfsudust
Rs
tFfsuduts
Then
can be simply written as
txxZtFsxdstR
in complete analogy to the deterministic case
Remark SDE Based on Ito Integral There is a similar but
less symmetric theory based on the Ito instead of Stratonovich forward
and backward stochastic integrals see Kunita
We will consider it
only for the classical SDE see Subsect

CHAPTER GENERATION
Local RDS from Stochastic Di erential Equations
Theorem Local RDS from Stratonovich SDE Let
FPttRFtsstbealteredDSandletFbeaCk
semimartingale helix over where k
and
Assume that F
has local characteristics aF bF AF satisfying aF Lloc R dAF Ck
and bF Lloc R dAF Ck Then there exists a unique up to indistin
guishability local Ck RDS over which for any
is a local Ck
semimartingale that is the unique maximal solution of the SDE
dxt Fxt dt
More precisely
i a The forward explosion time
x is an accessible stopping
time in the following sense
For each xed x
x is an Ft t stopping time
there exists a sequence n x
x of Ft t stopping times
which are lower semicontinuous with respect to x and for which
nx
xforallxPas
b For each n N and x Rd the stopped process t
t
nx
x is the unique up to indistinguishability solution of the for
ward SDE
tnxxxZtFs nxxds
t
ii a The backward explosion time
x is a backward accessible
stopping time in the following sense
For each xed x
x is an Ft t backward stopping time
there exists a sequence n x
x of Ft t backward stop
ping times which are upper semicontinuous with respect to x and for
which n x
xforallxPas
b For each n N and x Rd the stopped process t
t
nx
x is the unique up to indistinguishability solution of the back
ward SDE
tnxxxZtFs nxxds
t
We refrain from giving details of the proof of this theorem as it can be
derived from the previous proof combined with the usual truncation tech
niques see Kunita
pp
For the de nition of a local Ck semimartingale see Kunita
p
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Stochastic Di erential Equations from RDS
We nally deal with the inverse problem Given a cocycle when is there
ahelixFwhichgenerates ie suchthatd t F t dt
Theorem Stratonovich SDE from RDS Given a ltered
DS Let be a Ck RDS over such that
Gsxt
ts
xx
stxx
Msxt Bsxt
is a Ck semimartingale with local characteristics a b A satisfying
a LlocRdA Ck b LlocRdA Ck forsomek and
Then there exists a unique up to indistinguishability stochastic process
Fxt whichforsome
isaCk
semimartingale helix such
that is generated by F i e and F satisfy
txxZtFsxdstR
The local characteristics of F are obtained from those of as follows
AFA
aFxyt
at
xtyt
bFxtbt
xt dxt
where
dxt
dX
jxjajt
xt ytjyx
Proof Consider rst the forward case and put for s t
FsxtZt
sGs suxdu
Gs x t is a Ck forward semimartingale by assumption and we have a
LlocRdA Ck b LlocRdA Ck Furtherhsxt
stxisa
Ck forward semimartingale for some
Kunita
Theorem
and its proof with local characteristics of corresponding regularity
Hence Theorem i of applies and says that Fs x t is a Ck
forward semimartingale for some
with local characteristics of
corresponding regularity
1

CHAPTER GENERATION
We can thus use Fs x t as a forward integrator Then Theorem ii
of Kunita
yields
Gsxt stx xZt
sFs suxdu
st
in particular
Gxt
tx xZtF sxds
t
soFs xt isaforwardgeneratorof st s t
We now consider the backward case t s and put
FsxtZs
tGs suxdu
This is now a Ck backward semimartingale with local characteristics of
corresponding regularity which satis es
Gsxt stx xZs
tFs suxduts
in particular
Gxt
txxZtFsxdst
soFs xt isabackwardgeneratorof st t s
We next show that in fact up to indistinguishability
Fsxt
Fxt Fxs
st
where F is a forward semimartingale helix
As a consequence of the ow property we have see Kunita
p
Fsxu
Fsxt Ftxu
stu
Putting
Fxt
Fxt
we obtain
Fsxt
Fxt Fxs
st
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
SincebyLemma iiGsxt s Gxt s wehave
FsxtsZst
sGs suxdu
Zst
s
sGsuxdus
Zt
sGssuxdu
Again the cocycle property s s u
u s and Proposition i
lead to
Fsxst
sZtGu xdu
sFxt
stNs
As a result
Fxst Fxs
Fxts
stR
Ns
By Proposition i F can be extended to a continuous crude helix
on R also called F
Now Theorem
applies with H
Ck
andFcanbe
perfected the perfect version is again denoted by F
By Proposition
F is a forward semimartingale It satis es for
anys t
Fsxt
Fxt Fxs
Fxtss
We have thus found a forward semimartingale helix F which generates
the forward semimartingale cocycle st s t In particular
tx xZtF sxds
t
In step we have constructed the backward generator Fs x t t s
By the same arguments as in step we prove that there is in fact a backward
semimartingale helix F on R for which for each t s
Fsxt
Fxt Fxs
Fxtss
In particular
txxZtFsxdst
1

CHAPTER GENERATION
It remains to show that F and F are indistinguishable i e
Fxt
Fxt forallxt Pas
But this is exactly what Kunita
proves on pp
for the Ito
case and hence involving a correction term which disappears here We have
thus found an F which is a Ck
semimartingale helix for some
and is the generator of our Ck semimartingale cocycle
Finally we calculate the local characteristics of F from those of
For t to which we restrict ourselves
FxtZtGs xds
ZtM s xdsZtb s xsdAs
dX
jhZtM
xjsxdstxji
Hence the canonical decomposition of F x t MF x t BF x t is given
by
MFxtZtM s xds
BFxtZtb s xsdAs
dX
j
hZt M
xj s xdsZt
ssxGsxdsji
where we have used formula
onp
of Kunita
Consequently
QFxytZt
a
s
xsysdAs
and
BFxtZtfb s
xs
dX
jxjajs
x s ysjyxgdAs

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Summarizing our results we have proved that the following classes of ob
jects are basically the same
solutions of SDE dxt F xt dt driven by a semimartingale helix
over a ltered DS
semimartingale cocycles or RDS over
The engine semimartingale cocycle exp semimartingale helix that
we have implemented in this section converts vector eld valued additive
helices into multiplicative helices with values in a group of di eomorphisms
and vice versa This mechanism is the most general one for the generation
of RDS as semimartingales are known to be the most general class of
reasonable integrators
White Noise
Let F P t t R be the canonical metric DS describing Rm valued
Brownian motion Wt
t see Appendix A
LetFbetheP
completion of the Borel algebra F De ne
FtsWuWvsuvtNst
where N are the null sets of F In particular Fs
s is trivial for each s
Lemma i We have
Fts Fts Fts
sutFu
s
Fts Fts Fts
sutFtu
anduFts Ftu
suforalluRHence F ttRFtsstisaltered
DSiiWtisanRm
valued helix
iiiaForalls RWstWst isanFst
s t Brownian
motion in particular an Fts forward martingale b For all s R
WstWstisanFs
s t t Brownian motion in particular an Fts
backward martingale
Hence Wt is a forward as well as backward Fts martingale helix with
local characteristics A t b and a thus
hWi
tWjti tI I m munitmatrix
Recall that t
t is not B F F measurable but only B F F measurable
ThisiswhywehavetokeepF fortheDS
1

CHAPTER GENERATION
Proof i Since W Ws is left continuous at t and adapted Wt Ws is
Fts measurable so Fts Fts Fts Similarly for the argument s for xed
t Since Wt W is right continuous at s and adapted Fts Fts
As for the right continuity of Fts with respect to t we can appeal to
a well known theorem stating that for a Feller Markov process with right
continuous paths on a Polish state space the completed natural ltration is
right continuous see e g Doob p
ii Clear by de nition of
iiiaFixs RBydenitionWst Wst isanFst
st
Brownian motion if for each t Ws t Ws is Fs t
s measurable and for
eachu tFsu
s and Ws t Ws u are independent which is the case
Inparticular Wst Wst isanFst
s t martingale since for each
tandut
EWst WsjFs u
s
EWstWsuWsuWsWsuWsPas
Case b is analogous
Note that the canonical ltered DS describing Brownian motion could
also have been constructed via Theorem
RDS from Classical Stratonovich Stochastic Di erential Equa
tions
Letf Ck
bffmCk
bk
and let
Fxt
tfx
mX
j Wjt fjx Bxt Mxt
ThisisaCk
b semimartingale helix with non random local characteristics
At
t
axyt mX
jfjxfjy bxtfx
and Stratonovich Ito correction term
cxt
mX
j
dX
ifijx xifjx
If only f Ck f fm Ck then F is only a Ck semimartingale
helix with the same local characteristics Clearly a Lloc R dt Ckb if

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
and only if a Ckb
which is implied by our assumption f fm
Ckb Furtherb LlocRdtCkb ifandonlyifb f Ckb and
similarly for c Hence we can apply Theorem
to obtain the following
basic theorem for classical Stratonovich SDE
Theorem RDS from Classical Stratonovich SDE Let
fCk
bffmCk
b
and Pm
jPd
i fij xifj Ck
b for some
k and
Then
i The classical Stratonovich SDE
dxt fxtdt mX
jfjxt dWjt
mX
jfjxtdWjttR
with the convention dWt dt generates a unique up to indistinguisha
bility Ck RDS over the ltered DS describing Brownian motion For any
is a Ck semimartingale cocycle and t x
t x belongs
toC k forall
and
ii The RDS has stationary independent multiplicative increments
ieforallt t
tn the random variables
t
t
t
t
tn tn
are independent and the law of t h t is independent of t ho
mogeneous Brownian motion in the group Di k Rd
iii IfD t x denotestheJacobianof t atxthen D is
a Ck RDS uniquely generated by
together with
dvt
mX
jDfjxtvtdWjt tR
Hence D uniquely solves the variational Stratonovich SDE on R
DtxImX
jZtDfj sxDsxdWjs tR
and is thus a matrix cocycle over
Finally the determinant det D t x satis es Liouville s equation on
R
detD tx exp mX
jZttraceDfj s x dWjsAt R
and hence is a scalar cocycle over
1

CHAPTER GENERATION
Proof Assertions i and iii are particular cases of Theorem
Assertion ii follows since the random variables in question are measurable
with respect to Ftt Ftt
Ftn
tn hence independent
Remark By applying the Stratonovich rule for calculating
stochastic di erentials to D t x D t x I we obtain a forward
Stratonovich SDE for the inverse of the Jacobian
Dtx
ImX
jZtDsx Dfj sxdWjs
Example A ne SDE Consider on T R the a ne SDE
dxt
mX
jAjxtbjdWjtAjRddbjRd
Sincefj x Ajx bj Cb andPm
j Aj x Cb it uniquely generates
a global C RDS which consists of a ne mappings given by the variation
of constants formula
txtx
mX
jZt
s bj dWjsA
where is the fundamental matrix of the corresponding linear SDE
dxt
mX
j Ajxt dWjt
which is a linear RDS over
Remark Cocycles with Independent Increments We can
also recover the general results of Baxendale
and Kunita Sect
Under assumptions on the local characteristics similar to ours above a
Ck
b valued helix F with independent increments i e homogeneous Brow
nian motion in the linear space Ck
b generates a Ck valued cocycle with
independent increments i e homogeneous Brownian motion in the group
Di k Rd Conversely every Ck valued cocycle with independent incre
ments is generated by a Ck valued helix with independent increments It
is exactly this converse that forces one to go beyond the classical model

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
of nitely many vector elds and consider more general vector eld
valued driving processes
In short the result can be symbolically written as
cocycle with indep increments exp helix with indep increments
For completeness we also state the local result
Theorem Local RDS from Classical Stratonovich SDE
Letf Ck andf fm Ck for somek and
Then the
classical Stratonovich SDE
uniquely generates a local Ck RDS
over the ltered DS in the sense of Theorem
For any
is a local Ck semimartingale cocycle and t x
t xisinC k
for all
and
Criteria for Global RDS from SDE
De nition Strict Completeness of RDS The local RDS
of Theorem
is called strictly forward complete if
Pf
x
for all x Rdg
and strictly backward complete if
Pf
x
for all x Rdg
It is called strictly complete if it is strictly forward complete and strictly
backward complete For these notions see Kunita
p
The local RDS of Theorem
is clearly indistinguishable from a global
RDS if and only if it is strictly complete
The only general conditions known for globality in the case of X Rd
are the global Lipschitz conditions for the local characteristics built into
the assumptions of Theorem
There is a weaker version of completeness for solutions of SDE related to
the one point motions of the local RDS is called forward complete or
conservative regular non explosive see e g Khasminskii Chap III
ifP
x
for all x and backward complete if P
x
for all x In contrast to the deterministic case the fact that the
local solution does not explode P a s for each xed initial value in general
does not imply that the local RDS is global Examples are given by Leandre

CHAPTER GENERATION
and Carverhill and Elworthy Possibly the simplest example in
Ris
dxt
xx
xx dWt
xx
xxdWt
or dzt
zt dWtincomplexnotation z x ix W W iW
with solution t z z
zWt
Many important nonlinear physical and engineering systems such as
the noisy Du ng van der Pol oscillator are strictly forward complete but
not backward complete see Schenk Hoppe
and Sect
While forward backward completeness is determined by the transition
probabilities of the corresponding Markov process and therefore by the
forward backward generators L strict completeness is not so determined
Indeed Carverhill constructed a pair of SDE with the same forward
generator but one generating a strictly forward complete RDS and the
other not doing so this example is reproduced in Elworthy
p
By collecting the exceptional sets for a countable dense set in Rd we can
conclude from forward backward completeness that the open set D t
isdenseinRdPas fort t
For X R and two sided time T R forward backward complete
ness of a local continuous RDS imply its strict forward backward com
pleteness and thus that it can be perfected to a global continuous RDS
Indeed for each t the open and P a s dense set D t is an interval see
Theorem ii and the only open and dense interval in R is R itself
Consequently D t
RPasforallt R Nowthecontinuityof
tx
t ximpliesthatDt
Rforallt R Pas seeKunita
pp
Rede ning on the exceptional set as t
idR
for all t gives a crude global cocycle which can be perfected
The last remark and Feller s well known necessary and su cient non
explosion criteria for one dimensional di usions can be combined to give
the following conditions which often can be veri ed in speci c examples
Theorem Criterion for Global RDS from Scalar SDE
Let
dxt bxtdt xt dWt bxt
xt xtdt xtdWt
beanSDE onX RwithtimeT R Assumeb Ck
Ck
k
and x forallx R ThenthelocalCkRDS
generated by the above SDE is global if and only if are natural i e
non exit and non entrance boundary points The points
are non exit
The Ito SDE is de ned below
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
boundary points i e the RDS is forward complete and thus strictly forward
complete if and only if K
where
K xZx
yexpZy bdzZy
z expZz bdu dz dy
The points
are non entrance boundary points i e the RDS is backward
complete and thus strictly backward complete if and only if K
where
K xZx
y expZy bdz Zy
z expZz bdudz dy
For a proof see Kunita
pp
RDS from Ito Stochastic Di erential Equations
For convenience we include the corresponding basic statements about the
generation of RDS by classical Ito SDE
Recall that for a continuous process fs u s u t adapted to Fu
s
and a scalar Wt the forward Ito integral is de ned by
Zt
sfsudWu liminpr
nX
k ftk Wtk Wtk
and that the forward Stratonovich integral which needs a semimartingale
integrand see Subsect
is related to the forward Ito integral by
Zt
sfdWZt
sfdW hfWit hfWis
where hf W it denotes the quadratic covariation process of the semimartin
gales f and W The backward Ito integral of a continuous process fs u
t u s adapted to the ltration Fs
uisdenedby
Zs
tfsudWu liminpr
nX
k ftk Wtk Wtk
and the backward Stratonovich integral is related to it by
Zs
tfdWZs
tfdW hfWis hfWit
We stress that in each of the four stochastic integrals introduced above the
lower integration limit is always less than or equal to the upper limit
1

CHAPTER GENERATION
Letf Ck
bffmCk
b
and Pm
jPd
i fij xifj Ck
b for
some k and
Then the following forward backward Stratonovich
and Ito SDE are equivalent in the sense that they generate the same Ck
RDS
dxt
mX
jfjxt dWjt dxtfxtdt mX
jfjxtdWjt
where
fxfx
mX
j
dX
ifijx xifjx
Here is the Ito counterpart of Theorem
Theorem RDS from Classical Ito SDE Let f Ck
b
ffmCk
b
and Pm
jPd
i fij xifj Ck
b forsomek and
Then there is a unique up to indistinguishability Ck RDS over
the ltered DS describing Brownian motion which solves again using the
convention dWt dt
txxPm
jRtfj sxdWjs
t
Pm
jRtfj sxdWjs t
where fj fj for j
m and
fxfx
mX
j
dX
ifijx xifjx
For any
is a Ck semimartingale cocycle and t x
tx
isinC k forall
and
The RDS has stationary indepen
dent increments
Further the Jacobian of and its determinant satisfy the same proper
ties as in Theorem
where the variational Ito SDE is now obtained
from linearizing
We close with the following well known conditions under which a forward
Ito SDE generates a continuous or Ck RDS These conditions are in general
too weak to conclude the existence of a backward generator
Theorem RDS from Classical Forward Ito SDE a Let
fj Cb more generally fj Cb C for j
m Then there

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
is a unique up to indistinguishability continuous RDS over the ltered
DS describing Brownian motion which solves the forward Ito SDE
txx
mX
jZtfj sxdWjs t
For any
is a C semimartingale cocycle and t x
t xis
inC forall
and
The RDS has stationary independent
increments
bIff fm Ck
b forsomek and
then is Ck
An Example
If the parameters and in xt
xt
xN
tN
are perturbed by
white noise of intensities and respectively we obtain
dxt
xt
xN
tdt xtdWt
xN
t dWt
This equation can be explicitly solved by transforming it to an a ne equa
tionviay x N N Weobtainthea neSDE
dyt
N ytdt dt
NytdWt
dWt
with solution
ytyeNtWtyZt
eN
sWsds
dWs
Transforming back via x
N y N gives the following explicit ex
pression of the local C RDS for the original SDE
tx
xet Wt
N xNRteN
sWsds
dWs
N
We can read o from this expression the explosion time
x asthe
rst time at which the denominator which is positive for small t becomes
zero hence determining the domain D t and range R t of t
Dt
Rt
We rst observe that for
or for
the explosion time
x is nite with positive probability for any x
and
This implies that the set of never exploding initial values which also
carries all invariant measures is trivial for these cases E f g In

CHAPTER GENERATION
particular is the only invariant measure and the only solution of the
Fokker Planck equation
It thus su ces to consider the case
and
Put for de nite
ness and
Then the local C RDS for the SDE
dxt
xt xN
tdt xtdWt
is explicitly given by
x
tx
xetWt
N xNRteN
sWsdsN
We now determine the domain D t and the range R t of t
Dt
Rt
Case N Even
We have
Dt
dt
t
R
t
dt
t
where
dt
N RteN
sWsdsN
and
Rt
Dtt
rt
t
R
t
rt
t
where
rtdtt
et Wt
N RteN
sWsdsN
We can now determine
E
tRDt
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
and obtain
E
dfg
d
where
d
NReN
sWsdsN
The result for
is a consequence of the following fact
Lemma For any R
ZeWsdsZeWsds
Proof Put
andN ft Wt gThen
ZeWsdsZN sds LebN
By the arc sine law
lim sup
t
LebfstWs g
t
Pas
so that Leb N
Pas
By monotone convergence d for
and d for
The ergodic invariant measures which by Theorem
are random
Dirac measures are
i for
and
d
which are both F
measurable i e Markov measures the density of E is determined by
solving the Fokker Planck equation in Example
ii for
iii for
and
d the latter being F
measurable
Besides these there are no other ergodic invariant measures since any
other measure would have to satisfy int E
Pas which
is contradicted by the fact that t x
ast
t
for any
x intE for
1

CHAPTER GENERATION
Case N Odd
Now
Dt
R
t
dt dt
t
Rt
Dtt
rt rt
t
R
t
E
dd
fg
The ergodic invariant measures are
i for
the three random Dirac measures
d
and
d
which are all Markov measures for the corre
sponding solutions of the Fokker Planck equation see Example
ii for
iii for
That there are no other ergodic invariant measures is proved as in the
case of an even N
The Lyapunov exponent of for all invariant measures is calculated in
Example
A study of the bifurcation behavior for the cases N
and N is given in Subsect
See Exercise for the real noise
caseOther explicitly solvable SDE are given e g by Horsthemke and Lefever
pp
and Kloeden and Platen
pp
The Manifold Case
There is a similar theory of semimartingale cocycles and their generation
by semimartingale helices via SDE for the state space X M a manifold
which we refrain from developing in detail here see Baxendale Le Jan
and Watanabe
and Kunita Sect for more details
Let us brie y discuss the case of a classical SDE Given a standard
Brownian motion W in Rm and vector elds fj j
m the forward
as well as backward Stratonovich SDE on the manifold M with time T R
formally written with dWt dt as
dxt
mX
jfjxt dWjt
has the following coordinate free meaning A process t x with values
in M is called a solution of
if for each test function h CK M R
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
the space of C functions on M with compact support
htxhxPm
jRtfjh sxdWjs
t
Pm
jRtfjh sxdWjs t
For more details on SDE on manifolds see e g Belopolskaya and Dalecky
Emery
Elworthy
Hackenbroch and Thalmaier
Ikeda and Watanabe
Kunita
and Rogers and Williams
Theorem Local RDS from SDE on Manifold Consider
the Stratonovich SDE on M with time T R
dxt
mX
jfjxt dWjt
andassumethatf Ck andfj Ck j
mk
Theni There is a unique local Ck RDS over the canonical ltered DS
corresponding to W which solves
ii The derivative T TM TM is a local Ck RDS which is the
unique solution of
dvt
mX
jTfjvtdWjt v vTM
where T fj is the natural lift of the vector eld fj to a vector eld on T M
inlocalcoordinatesTfxv xvfx Dfxv
iii If M is a Riemannian manifold and if we use the Riemannian
connection on M to decompose TvT M into horizontal and vertical compo
nents then T fj v has horizontal component fj x and vertical component
rfj x v where rfj x v denotes the covariant derivative of the vector eld
fj in the direction of v TxM and equation
is equivalent to the
two coupled SDE
and the variational equation
Dvt
mX
jrfjxtvtdWjt v vTxM
Here Dvt denotes the absolute stochastic di erential
Furthermore we have Liouville s equation for det T t x For t
Dx
detT tx
expZ t mX
j divfj sx dWjs

CHAPTER GENERATION
expZ t mX
j tracerfj s x dWjs
iv The RDS is global in case M is compact or if by embedding M
into an RN the vector elds fj are restrictions of vector elds fj in RN
for which f Ck
bffmCk
b
and Pm
jPd
i fij xifj Ck
b
Conditions for strict completeness of SDE on general non compact mani
folds which is in general not implied by completeness if dim M
see
Elworthy
p
are given by Li
RDS with Independent Increments
We now return to the situation and notation of Subsect
and dis
cuss the connection between SDE and Markov processes This subject is
classical and is treated in most of the books mentioned in Subsect
For our brief survey following Arnold we choose a more conceptual
and descriptive style often omitting technical conditions under which our
statements hold
The following problem is historically from the beginning of the theory
of SDE Suppose we are given an elliptic di erential operator
LdX
ibixxi
dX
kl aklx xkxl
in Rd with smooth coe cients where a x
aklx isforeachx Rda
non negative de nite symmetric d d matrix we seek a di erential equation
xtfxt
mX
j
jtfj xt
x xRd
also called Langevin equation such that for each initial value xx
x
its solution xx
t t R is a homogeneous Markov process for which the
transition semigroup
Ttgx Egxx
tZRdgyPtxdy
has the in nitesimal generator L and the vector elds f fm are deter
minedbythecoe cientsbandaofL
It turns out that t
t
m
t has to be Gaussian white noise hence
a generalized random process Ito
gave a rigorous meaning

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
to a stochastic generator of the form
by introducing stochastic
integrals and writing
as an Ito SDE
dxt fxtdt mX
jfjxtdWjt x x Rd
The fj in
are derived from b and a as follows Put f b Take
any factorization a x
xxwherex
x
mxisa
d m matrixwithcolumns j x andputfj j j
m
The SDE
indeed generates a Markov family of processes
xz z Rd with time R indexed by their initial conditions all with the
same transition probability
PtxB Pfxz
tsBjxz
s xgPfxx
tBg
The semigroup Tt given by
has generator L given by
and
the coe cients b and a can be obtained from the solutions as in nitesimal
mean
lim
t tExx
txbx
and in nitesimal variance
lim
t tExx
txxx
tx
ax
Note that the semigroup Tt hence the law Px on C R Rd of the solution
xx
ttRof
is in nice cases e g under Oleinik s theorem see
Stroock and Varadhan
p
uniquely determined by L hence by a
and b and is thus independent of the way we factorize a x There are in
general many di erent ways of factorization thus many di erent SDE for
the one and the same in law Markov family
What we of course expect is that the RDS generated by the SDE
for di erent factorizations are di erent which turns out to be the
case After all the RDS describes the simultaneous motion of all initial
points and not just of one
First of all the RDS remembers all one point motions since by the
de nition of t
tx xx
t is the solution of the SDE with initial
value x x but it also describes the simultaneous motion of n points the
forward n point motion
Rt
txn
tx
txn
where x n
x
xn Rd n The Rd n valued stochastic process
t x n is obtained by solving n copies of
with the same Wiener
process but with di erent initial conditions
1

CHAPTER GENERATION
Similarly the two sided RDS also describes the backward n point mo
tions R t
txn
Theorem Solution of SDE as Markovian RDS Assume
the conditions of Theorem for the SDE
so that it generates
a two sided Ck RDS with independent increments Then is Markovian
forwards and backwards in time in the following sense For each n N and
xn Rdn
i the forward n point motion R t
t x n is a homogeneous
Feller Markov process with respect to Ft with transition probability
Pn
t
xnBPftxnBgtR
ii the backward n point motion R t
t x n is a homogeneous
Feller Markov process with respect to Ft with transition probability
Pn
t
xnB
PftxnBgPft
xnBgtR
The proof just uses the fact that has stationary independent increments
see Baxendale
or Kunita
For each xed n N the semigroup of the forward n point motions
Tn
tgxnEgtxntR
uniquely determines the law of the forward n point motions The family
Tn
t
n
is consistent in an obvious way and thus uniquely determines
the distribution of the RDS t t R But the latter can be accomplished
with much less information which we will now explain
The generator of T n
t
Ln gxn lim
t tTn
t
idgxn
is for g C with compact support
Ln gxn
dX
i
nX
pbixpgxn
xip
dX
ij
nX
pq
aijxpxq gxn
xip xjq
wherebx fx and
axy lim
ttEtxxtyy
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
The d d matrix a x y is the in nitesimal covariance of the forward two
point motion t x t y which for the case of the SDE
is
given by
axy mX
jfjxfjy
xy
Now observe that the in nitesimal characteristics a x y and b x can be
reado fromL butnotfromL L
and if this is done the
coe cients of all the L n are determined
In nice cases see above T n
t is uniquely determined by L n The
nal result is quite surprising The law of the RDS as a stochastic process
Rt
t
Di kRd isuniquelydeterminedbyaxy andbx
and thus by the laws of the forward two point motions Baxendale
Remark i The statement that the law of an RDS with sta
tionary independent increments is uniquely determined by the laws of its
forward two point motions is essentially a consequence of the fact that W
is Gaussian The statement is thus generally wrong in the discrete time
caseii With the above ndings we have also basically solved the repre
sentation problem for continuous time Given a generator L of the form
with coe cients a x and b x is there an RDS whose forward
one point motions are homogeneous Markov processes with generator L
We seek a representation by a Markovian RDS in particular by one
generated by an SDE Then the law of is uniquely determined by the law
of its two point motions hence by the functions a x y and b x The one
point motions however only prescribe the diagonal values a x x a x
Suppose we have a factorization a x
xxwherex
x
m x is a d m matrix with columns j x which is regular
enough Then
axy
xy
is an in nitesimal covariance and L can be realized by the SDE
withfx bxandfjx jx jm
The factorization hence the representation is in general not unique
Analogous facts hold for the backward n point motions The generator of
the semigroup
Tn
tgxnEgtxntR
is
Ln
dX
i
nX
pbixp xip
dX
ij
nX
pq
aij xp xq xip xjq
1

CHAPTER GENERATION
where
bxfx
dX
i
mX
jfijxfjx
xi
We stress that both L n and L n describe the n point motions of the
same two sided RDS but L n forwards and L n backwards in time
In general L n L n so the Fokker Planck equations L n
n
and Ln
n
usually have di erent solutions
Based on the concepts of past and future of an RDS developed in Sect
it is now straightforward to carry over the statements on Markov mea
sures formulated in Subsect
for products of i i d random mappings
to the white noise case
Assume the situation of Theorem
and denote by the Ck RDS
with time T R over the canonical DS F P t t R describing Brow
nian motion We choose the past F
tt
and the future
F
tt
which are both exhaustive Also F and F are in
dependent Recall that those IP which are F F measurable are
called forward backward Markov measures of and their sets are denoted
by IP F The restriction of to one sided time is denoted by The
justi cation of this name is given by the next theorem
Theorem One to One Correspondence Between Markov
Measures and Stationary Measures Assume the conditions of Theo
rem for the SDE
so that it generates a two sided Ck RDS
with independent increments Then
i There is a one to one correspondence between the set of invariant
forward Markov measures IP F
the setf IP
Pg
and the set of stationary measures for L the generator of the forward
one point motions given by
E
lim
t
tt
In particular IP F is ergodic under if and only if is an ergodic
stationary measure
ii A completely analogous relation holds between the set of invariant
backward Markov measures IP F
the setf IP
Pg
and the set of stationary measures for L the generator of the backward
one point motions
We stress that typically has more invariant measures than those which can
be seen by the two Fokker Planck equations L
and L
1

CONTINUOUS TIME STOCHASTIC DIFFERENTIAL EQS
Partial statements of Theorem
were obtained by Carverhill
The matter was systematically treated by Crauel
If we start with the Stratonovich SDE dxt Pm
j fj xt dW jt instead
of an Ito SDE the forward backward generators are more symmetric and
have coe cients
bxfx
mX
jfijxfjx
xi
and
axy mX
jfjxfjy
In Hormander form
LfmX
jfj
1

CHAPTER GENERATION

Part II
Multiplicative Ergodic
Theory
1

Chapter
The Multiplicative
Ergodic Theorem in
Euclidean Space
Summary
This chapter is devoted to the presentation of Oseledets s multiplicative
ergodic theorem the most fundamental theorem of the book It provides
a spectral theory for linear cocycles and one can say without exaggeration
that all what follows are just applications of this one theorem
We hence have tried hard to give complete and precise formulations and
clearly structured proofs of the various versions of the multiplicative ergodic
theorem Theorem for one sided time T N and T R Theorem
for two sided time T Z and T R In particular we carefully
describe the invariance properties of the sets on which the statements
hold
As the proofs of the various versions of the multiplicative ergodic theo
rem are based on corresponding versions of the Furstenberg Kesten theorem
Theorem for one sided time T N and T R Theorem
for
twosidedtimeT ZandT R thesamecarehastobetakenforthe
formulations and proofs of the latter
A certain pedantry of our style is motivated by the aim of making this
chapter a source of reference
1

CHAPTER MET IN EUCLIDEAN SPACE
Introduction
It is well known that the dynamics of the autonomous linear system
xt Axt is completely described by linear algebra more precisely by
the spectral theory of A
The local theory of smooth nonlinear dynamical systems such as sta
bility theory invariant manifold theory local bifurcation theory normal
form theory is now based on the following idea Suppose the nonlinear
vector eld f has a singularity at x f x
We can regard the non
linear system xt f xt as a small perturbation of the linearized system
xt Axt A fx x inaneighborhoodofx Wecanthentryto lift
or carry over the dynamics in particular the stability properties of x
of the linearized system to the original nonlinear system
This approach was adopted by Lyapunov in his celebrated classic
Probleme general de la stabilite du mouvement original Russian edi
tion
French translation
also for nonautonomous systems It
can indeed be successfully carried over from a xed point to a periodic or
bit thanks to Floquet theory spectral theory of xt A t xt where A
is a continuous periodic matrix function and then only partly further to
quasi and almost periodic orbits via the spectral theory of Sacker Sell and
Johnson
The key obstacle is the possibility of very com
plicated dynamical behavior of nonautonomous linear systems xt A t xt
It might come as a surprise that an important class of nonautonomous
linear systems namely those driven by a metric DS e g those generated
byxt A t xtorxn A n xn hasabona despectraltheory
with probability one This is the content of the celebrated multiplicative
ergodic theorem of Oseledets
The MET provides us with exactly the
right substitute for deterministic linear algebra i e with the right kind of
spectral objects invariant subspaces exponential growth rates Lyapunov
exponents which allow the lift to nonlinear RDS and hence a local theory
for nonlinear RDS The MET thus occupies a pivotal position in this book
Since the appearance of Oseledets paper
in
there have been
numerous attempts to nd alternative proofs of the MET The original
proof relies on the triangularization of a linear cocycle and the classical
ergodic theorem for the triangular cocycle This technique was also used
in the contemporaneous paper of Millionshchikov
who independently
derived a portion of the MET and was then taken up again by Palmer
Johnson Palmer and Sell assuming a topological setting for
the metric DS and Margulis Appendix A
Multiplicative ergodic theorem is henceforth abbreviated as MET
In August
I counted fteen published proofs besides Oseledets
1

LYAPUNOV EXPONENTS
Another class of proofs uses the singular value decomposition of matri
ces in combination with Kingman s
subadditive ergodic the
orem see Raghunathan
Ruelle
Crauel
Ledrappier
Krengel
Cohen Kesten and Newman
Goldsheid and Margulis
Berger
The subadditive ergodic theorem allows a proof of the
Furstenberg Kesten theorem in a few lines and the Furstenberg Kesten
theorem is then a key ingredient of the proof of the MET
Now begins the hard work consisting of matrix calculations in this
class of proofs It concerns the construction of invariant subspaces in which
the smaller Lyapunov exponents are realized as limits This hard work
can be converted into the direct construction of complementary subspaces
of lower growth rate see Mane
or to the study of the action of
the linear cocycle on projective space also using the subadditive ergodic
theorem see Walters
For extensions of Oseledets theorem to semisimple Lie groups see
Kaimanovich
and Zakharevich
and to local elds see Ragu
nathan
For in nite dimensional versions of the MET see e g Ruelle
Mane
Thieullen
Mohammed
Mohammed and Scheutzow
Flandoli and Schaumlo el Schaumlo el
Lindemann
Here we will follow the established approach via the subadditive er
godic theorem the singular value decomposition and exterior powers and
the Furstenberg Kesten theorem basically following Goldsheid and Mar
gulis
in doing the hard work
Lyapunov Exponents Singular Values
Exterior Powers
Deterministic Theory of Lyapunov Exponents
For an autonomous linear di erential equation xt Axt or di erence equa
tion xn
Axn the origin Rd and thus every point is asymp
totically stable if and only if it is exponentially stable if and only if all
eigenvalues of A have negative real parts or absolute values less than
respectively This can be readily seen by looking at the real Jordan canon
icalformJofAandbyobservingthatifA P JP P GldR then
tetAPetJPornAnPJnP
For stability theory of nonautonomous linear systems xt A t xt where
A is locally integrable or xn An xn it turns out not to be the right

CHAPTER MET IN EUCLIDEAN SPACE
thing to study the eigenvalues of A t or An as they have little or nothing
to do with the asymptotic properties of solutions
We thus need to nd a dynamical formulation of spectral theory of A
i e a formulation which describes spectral objects in terms of the long term
behavior of solutions
Lyapunov
was aware of this problem when he introduced his char
acteristic exponents which today bear his name We will give a brief review
of his theory following the profound monograph of Bylov Vinograd
Grobman and Nemytskii in which proofs and many more facts on the de
terministic theory of Lyapunov exponents for nonautonomous di erential
equations can be found Other good sources of examples and counterexam
ples are Cesari Chap II and Hahn Chap VIII
As a preparation we quote a simple but useful lemma
Lemma Lyapunov Index of a Function Let T R or Z
orNfT Rd andcall
f lim sup
t tlogkftkRf g
the Lyapunov index of f Then
i c ifc const put
ii f fforall Rnfg
iii f g max f g withequalityif f g
iv If T R and f is locally integrable then
Ztfsds f if f
Ztfsds fiff
If f is measurable and locally bounded and g is locally integrable then
Zthfs gsids f Ztkgskds if f
Similarly for T Z or N with integrals replaced by sums
v
hf gi f g if the right hand side makes sense
kfk f for
R put
The proof follows from the de nition of f and is left as an exercise
We now apply this to the function f t
tx
1

LYAPUNOV EXPONENTS
De nition Lyapunov Exponent The forward Lyapunov ex
ponent of the solution t x of a non autonomous linear di erential equa
tion xt A t xt or di erence equation xn An xn starting at time t
atthestatex RdisdenedtobetheLyapunovindexof tx
x
x lim sup
t tlogktxk
and for two sided time the backward Lyapunov exponent of t x is de ned
as the Lyapunov index of t x
x lim sup
t tlogk txk limsup
t jtjlogk t xk
Suppose A t or An is bounded Then and also
satis es
x Rf gforallx Rd and
x
x forall Rnfg
x y max x y withequalityif x
y
A mapping Rd R f g with these three properties is called a
characteristic exponent see e g Pesin
Further the number p of distinct values that x can take on for x
is at most d This can be seen by noting that vectors with di erent
Lyapunov exponents are linearly independent Write
p
p
for the di erent values of x and call them the Lyapunov exponents of
the system Call the top Lyapunov exponent
The sets V fx
x g are linear subspaces of Rd Vi V i
form a ltration ag of subspaces
fgVp Vp
VRd
where all inclusions are proper and
x
i
x VinVi i
p
The integer di dim Vi dim Vi is the multiplicity of i and the set
idii
p is called the forward Lyapunov spectrum
1

CHAPTER MET IN EUCLIDEAN SPACE
If
then xt A t xt is exponentially stable It is however
in general not true that
implies the stability of the origin of the
perturbed nonlinear system
xt Atxt gtxt jgtxjCjxj forjxjhC
For this to hold true one needs a property of the linear system called reg
ularity The equation xt A t xt or xn Anxn is said to be forward
regular if
pX
i di i liminf
t tlogjdet tj
For a forward regular system all lim sup s are in fact limits and
implies exponential stability of
For two sided time analogous facts hold for
x Ithasp dierent
values
p
with multiplicities di the ltration
fg Vp
VRd
and
x
i
x VinVi
The system xt A t xt or xn Anxn is called backward regular if
pX
i di i liminf
t tlogjdet tj
In general the forward and backward spectrum and ltration are not
related We introduce such a relation as follows The two sided linear
system is called regular if
it is forward regular and backward regular
ppdidpii
pii
p
Vi Vp ifgi
p
and imply that dimVi dimVp i d for i
p and
it makes sense for a regular system to de ne the subspaces
EiViVpii
p
1

LYAPUNOV EXPONENTS
where dim Ei di and which form a splitting of Rd
RdE
Ep
The ltrations can be recovered as Vi
pjiEjandVp i
ij Ej
The splitting is dynamically characterized by
lim
t
tlogktxk i
x Einfg
Example Autonomous Case i Two sided continuous time
Consider xt Axt A Rd d t R It follows immediately from the
Jordan canonical form for A and t etA that xt Axt is regular
with
p
the di erent real parts of eigenvalues of A Ei
the sum of the generalized eigenspaces to eigenvalues with real parts equal
toiWehavet i
i t where i is the projection onto Ei
along j iEj equivalently t Ei Ei for all t R i
p Also
Ppdi i traceA
ii Two sided discrete time Consider xn
AxnA GldR
n Z Then n An n Zandthesystemisregularwith i
equal to the di erent values of log j j an eigenvalue of A and Ei the
corresponding sum of generalized eigenspaces Again A i
iA
equivalently AEi Ei and Pp di i log j det Aj
iii One sided discrete time Consider xn Axn A Rd d n Z
The system is forward regular with i as in ii where p
if
is an eigenvalue of A and ltration
Vp
VRdAViVi
where Vi is the sum of the generalized eigenspaces for eigenvalues of
A with log j j i Note that the knowledge of x only allows us to
reconstruct the ltration Vi fx
x
ig and not the eigenspace Ei
corresponding to i as Ei is not dynamically characterizable for one sided
time see
Example Periodic Case Let A R Rd d be continuous
and periodic with period
Then xt A t xt is regular Indeed
see e g Coddington and Levinson
Sect
or Farkas
Sect
write
e R with R Rd d The eigenvalues of R are called
Floquet characteristic exponents and the fundamental matrix is represented
by t PtetR whereP R GldR isC and periodic andP
P and P are bounded on R The new moving coordinates x P t y
transform xt A t xt to yt Ryt Now the Lyapunov exponents i are
the di erent values of log j j where is an eigenvalue of
iethey

CHAPTER MET IN EUCLIDEAN SPACE
are the di erent real parts of the Floquet characteristic exponents which
are uniquely determined by A The Ei are the sums of the generalized
eigenspaces of R corresponding to i Note that
pX
i di iZtraceAtdt
The invariance property of the splitting now reads t i
itt
where i t istheprojectionontoEi t P tEialongFi t
jiEjt
so t Ei Ei t This foreshadows the more general invariance prop
erty of the Ei s in the random case For the embedding into an RDS see
Example
Although it is hard if not impossible to verify regularity for a particular
system the surprise and the key statement of the MET is that linear RDS
including constant periodic quasi periodic and almost periodic A are
regular with probability one
Singular Values
Let Rd be endowed with the standard scalar product let ei be the stan
dardbasisandletOdR fU GldR UU Igbetheorthogonal
group WesayforA Rd dthat
A VDU
is a singular value decomposition of A if U V O d R and D
diag
d with
d The i are called the singular
values of A
Lemma Singular Value Decomposition Any d d matrix A
has a singular value decomposition Moreover the
d
are
necessarily the eigenvalues of pA A also of pAA and the columns of U
are corresponding eigenvectors of pA A In particular kAk where
k k is the operator norm associated with the standard Euclidean norm in
Rd and jdetAj
d
Proof AnyAmaybewrittenasA WpAAwithW OdR polar
decomposition see Gantmacher
IX
NowpAA soitmay
be written aspAA U DUwithD diag
d where the i
are the eigenvalues of pA A Together A W U DU V DU with
UV OdR ConverselyifA VDUthenAA UDUthus
pA AU U D and the i U ei are the eigenvalues and eigenvectors of
pA A respectively

LYAPUNOV EXPONENTS
The geometric meaning of the singular value decomposition is as follows
i is the length and U ei the direction of the i th principal axis of the
ellipsoid A Sd obtained as the image of the unit sphere Sd fx
Rd hx xi g under the linear mapping A
Exterior Powers
Let E be a real vector space of dimension d and for k d let kE
the k fold exterior power of E be the vector space of alternating k linear
forms on the dual space E see e g Temam Chap V or Kowalsky
x Naturally E E E dE R Thespace kEcanbe
identi ed with the set of formal expressions Pm
i ciui
ui
k with
m Nci Randui
j E if we do computations with the following
conventions
u
uj uj
uk
u
uj
uk
u
uj
uk
u
cuj
uk cu
uj
uk
for any permutation of f kg
u
uk sign u
uk
The elements in kE of the form u
uk are called decomposable k
vectors and the set of decomposable k vectors is denoted by kE Clearly
kE span kE
The following facts can be readily checked
Lemma Exterior Powers of Spaces and Operators Let
kE be the kfold exterior power ofE where k d dimE
i Ife
edis abasis ofE thenfei
eik
i
ik
dg is a basis of kE In particular
dim kE dk
iiWehaveu
uk
ifandonlyifu
uk are linearly
dependent
iii Two sets u
ukandv
vk of k linearly independent vectors
in E have the same linear span if and only if
u
uk
v
vk for some R
1

CHAPTER MET IN EUCLIDEAN SPACE
Hence the image of the set kE of decomposable k vectors in the projec
tive space P k E can be identi ed with the Grassmannian Gk E of k
dimensional subspaces of E
iv If E has the scalar product h i then the bilinear extension of
hu
ukv
vki dethui vjik k
de nes a scalar product in kE In particular
ku
ukkqdethui ujik k
is the volume of the k dimensional parallelepiped spanned by u
uk
Further the generalized Hadamard s inequality holds For p k
ku
ukk ku
upk kup
ukk
v If A E E is a linear operator then the linear extension of
kAu
uk Au
Auk
de nes a linear operator k A on kE the k fold exterior power of A with
A A dA detAand kI I kAhas eigenvaluesfi
ik
i
ik dg where
d are the eigenvalues of A Further
kAB
kA kB
kA
kA
kcA ck kA
and
det kA detA d
k
If A is an upper lower triangular matrix in some basis then so is kA in
the corresponding basis of i
vi Let E have a scalar product and let k E be endowed with the scalar
product introduced in part iv If U is orthogonal then kU is orthogonal
Further
hkAu
ukv
vki dethAui vjik k
hence with k k Ak denoting the corresponding operator norm
kkABkkkAkkkBk
kA kA
vii Suppose A is a linear operator on E Then
k etA etAk
1

LYAPUNOV EXPONENTS
where the linear operator Ak kE kE is de ned by the linear extension
of
Aku
uk
kX
i
u
ui Aui ui
uk
and satis es
d
ABkAkBk
RA AAd traceA
and Iku ku Ak has eigenvalues f i
ik
i
ik dg
if
d are the eigenvalues of A In particular
trace Ak dk trace A
If E has a scalar product then Ak
d
Akand
hAk u
uku
uki trace APk ku
ukk
where Pk is the orthogonal projector onto the subspace span u
uk of
E In particular kAkk kkAk
The importance of exterior powers in the study of linear systems stems
mainly from the following two propositions
Proposition Singular Values of Exterior Power Let A be
a d d matrix let A V DU be a singular value decomposition and let
d be the singular values of A Then
i kA kV kD kU is a singular value decomposition of kA
iikDdiagi
ik
i
ik d In particular
the top singular value of kA is
k and the smallest is d k
d
Moreover if A is positive de nite semi de nite then so is k A
iiikkAk
k in particular k d Ak
d jdetAj
andkkmAk kkAkkmAk kmk m dinparticular
k k Ak kAkk Here k k is the corresponding operator norm associated
with the standard Euclidean norm in Rd
For later use we also note
Proposition Metric on Projective Space Let P d be the
projective space of one dimensional subspaces of Rd and denote by x the
class ofx RdnfginPd Then
i
xy
hx yi
kxk kyk
xy Rdnfg
1

CHAPTER MET IN EUCLIDEAN SPACE
is a metric in P d which makes it a complete metric space
ii We have
xykxyk
kxk kyk
Proof i To prove the triangle inequality note that
UxUy
xy
foranyU OdR wherewehaveputU x Ux Ithencesu cesto
prove
xy
xz
zy
forunitvectorsoftheformz e x xe xe y ye ye ye
which is left as an exercise
If denotes the angle between the lines through x and y hx yi cos
then xy jsinjseeFig
Thus has the same set of Cauchy
sequences as the natural Riemannian metric arc length on P d which is
complete
ii By de nition
kxykdethxxihxyi
hyxihyyi kxkkyk hxyi
Figure The metric on projective space
The Furstenberg Kesten Theorem
As a nal preparation for the MET we prove a theorem of Furstenberg
and Kesten
which now bears their names As the MET of which
we present several versions is directly based on the Furstenberg Kesten
theorem we also need several versions of the latter
The Subadditive Ergodic Theorem
We rst introduce a generalization of subadditivity to stochastic processes
De nition Subadditive Stochastic Process Let T N or
R A measurable stochastic process ft
t TwithvaluesinR f g
is called perfectly subadditive over the metric DS F P t t T if
fstfsfts
holds identically We call f superadditive if f is subadditive

FURSTENBERG KESTEN THEOREM
As an example let f
R be measurable and let for T R t
f be locally integrable Then St Ptk f k for T N
and St Rtf s ds for T R are sub as well as superadditive
Theorem Kingman s Subadditive Ergodic Theorem Let
fn be a subadditive sequence of random variables over the metric DS
FPnnN
Assumef L FP where a
max a and put
inf
nNnEfn
Thena There is a forward invariant set F of full measure and a mea
surable function f R f g with f L such that on
lim
n
nfn f
andfk f forallk Nf
in the ergodic case Further
lim
n
nEfn
EfRfg
bIfinadditionfn Lforallnandif
then assertion a
holds with f L and convergence in
also holds in L
We decided to omit the proof as it is easily available in the literature see
e g Krengel or Steele
The Furstenberg Kesten Theorem for One Sided
Time
Let n bealinearRDSwithtimeT NorZ overthemetricDS
FPnnNThen
nAn
An
where A
Rd d is the generator of
We want to study the asymptotic behavior of n in particular of
its singular values k n k
dforn
The cocycle property
nm
mn
n
lifts to kRd k d see Lemma
part v
knm
kmn
kn
1

CHAPTER MET IN EUCLIDEAN SPACE
Theorem Furstenberg
Kesten Theorem One Sided Time Let be a linear cocycle with
onesidedtimeT NorR overthemetricDS FP t t T Then
the following statements hold
A Non invertible case T N If the generator A
Rd d satis es
logkAkL FP
theni For each k
d the sequence
fk
n
logkknknN
is subadditive and f k L
ii There is a forward invariant set F of full measure and mea
surable functions k
R f gwith k L suchthaton
lim
n
nlogkknkk
and
k
k
km
k
m
k
E k in the ergodic case Further
lim
n
nElogkk nkEk inf
nNnElogkk nk
iii The measurable functions k successively de ned by
k
kk
d
where we put k
ifk
have the following properties
on
k
lim
n
nlogk n
where k n
are the singular values of n
and
k
k
d
k
E k in the ergodic case
Further
lim
n
nElogk n
Ek
B Invertible case T N If A
GldR and
logkAkL FP and logkAkL FP
1

FURSTENBERG KESTEN THEOREM
then there exists an invariant set F of full measure on which the wise
statements of part A hold and all k hence all k are nite
Moreover all E k hence all E k are nite and convergence also
holds in L
CInvertiblecaseT R Let t GldR dene
sup
tlogktk
sup
tlogkt k
and assume that
L and
L Then all statements of part B
hold with n and replaced with t and t and F is now invariant with
respectto t t R
Proof Part A
i The cocycle property
forkn
and Lemma vi
imply that f k
n
log k k n k is subadditive Furthermore f
L by assumption This fact and k k Ak kAkk Proposition iii
yieldfk L
ii This is an immediate consequence of Theorem
The forward
invariant set can be taken to be the intersection of the d forward invariant
sets kfork
dThepropertykkmkkkkkmkPropo
sition iii implies k m
k
m
In particular if
k
for some k then m
forallm k
iii By Proposition iii for k
d
nlogkknkn
kX
ilogin
We proceed successively as follows Choose
For k
lim
n
nlog n
exists If
we are nished For
lim
n
nlog n
exists and so on If nally d
d
d
lim
n
nlogd n
d
exists By this construction k
ifandonlyif k
The inequalities
d on follow from
d
1

CHAPTER MET IN EUCLIDEAN SPACE
For the expectations the limit
E
lim
n
nElog n
E
exists If E
we are done as all other expectations E k are also
equal to
If not the limit
E
E
lim
n
nElog n
E
exists and so on
Part B
a We want to apply part b of Theorem
for which we have to
verify that
fk
n
logkk nkL
forallnandkandthatE k
for all k Now
kknkkknkkkAk
kknkkknkkkAk
and
logk kAk klog kAk
result in
fk
n
klogkAkfk
nfk
n
k log kAk
Since P is invariant our assumptions
imply that f k
n Liffk
logk kAk L
b From kAA k kAk kA k it follows that
log kA k logkAk log kAk
and
jlogkAkj log kAk log kA k
Similarly jlogk k Akj k log kAk log kA k
Thereforefk L iflogkA k L
cWecheckthatE k
Indeed
jE kj lim
n
njElogkk n kj
k lim
n
n
nX
i EjlogkA i kj kEjlogkAkj
1

FURSTENBERG KESTEN THEOREM
by step b of the proof
d Dividing equation
bynweseethat
fk
n
n
k
fk
n
n
k
and in that case k
k Hence the set on which the limit
exists and is nite for each k is invariant whenever A
GldR for
all
On this set the limits are invariant We know from Theorem
i that has full measure
Part C
a Since log k
k L part B applies Let be the
invariant set of full measure of those s for which the limits k
exist
and are nite These limits are invariant on
b By the continuity of with respect to t is measurable The
integrability conditions
L and the ergodic theorem assure that the
invariant set on which limn
n n has full measure
Hence
is invariant and has full measure
c We next prove that is t invariant We have for t
t
sup
slogkst
tk
sup
slogksk
sup
slogks
k
similarly for Hence is forward invariant
For the converse use for s t
s
tt
s
tt
which gives
t
t
t
similarly for Hence t
implies
which holds if and
only if
d We now prove that for each
the limit
lim
t tlogkktk
exists and is equal to k
1

CHAPTER MET IN EUCLIDEAN SPACE
The cocycle property gives for t R and n t the integer part of t
kt
ktnn
kn
whence by the invertibility of
kn
ktnn
kt
Hence
logkknkk n
logkk tk
logkknkk n
We conclude from
that for
the limits
exist and are
equal to k
e We nally prove that is t invariant i e t
and that
for
andtRkt
k Consider the case k only
letn t andasssumethats n
t By the cocycle property
logkts n
n klogkn
ttk
logkstk
logkts n
n klogkn
ttk
For
the left hand side and the right hand side of
divided by
s tend to
n
for s
by step iv Consequently
t
i e is forward invariant Moreover
lim
s
slogkstk
lim
n
nlogkntk
t
Rearranging the string of inequalities
we can also conclude that if
t
then
i e is invariant
Corollary Lyapunov Index of Determinant We have
lim
n
nlogjdet n j d
dX
k
k ElogjdetA jjI Pa s
where I F is the algebra of invariant sets and
Ed
dX
kEkElogjdetAjRfg
where the expectations are nite under the assumptions
1

FURSTENBERG KESTEN THEOREM
Proof By Proposition iii
kdnkjdetnjnY
ijdetAij
We know that n log of the left hand side converges on to
d
dX
k
k lim
n
nlogkd nk
The generalized ergodic theorem applies to the right hand side since
log jdetAj f d L Theorem i and yields
lim
n
n
nX
i logjdetAijElogjdetAjjI RfgPas
Remark Furstenberg Kesten Theorem for Other Norms
We have worked with the standard scalar product in Rd to de ne orthog
onal singular value decomposition the corresponding operator matrix
norms in k Rd etc However as all norms in Rd d are equivalent the in
tegrability condition
is satis ed or not for all norms simultaneously
and all limits are independent of the norm chosen
The representation
Ekinf
nNnElogkk nk
is however only valid for so called matrix norms i e norms on Rd d with
the additional property kABk kAk kBk
This remark equally applies to the other versions of the Furstenberg
Kesten theorem that we will present
Remark Furstenberg Kesten Theorem for Backward Co
cycles The Furstenberg Kesten theorem also holds for Rd d valued back
ward cocycles n over satisfying
nm
n
mn
Indeed also in this case for any matrix norm
logkk nmklogkk nklogkk mnk
hence log k k n k is subadditive

CHAPTER MET IN EUCLIDEAN SPACE
Remark i Just as in step c of the last proof we can derive in
any case
E k kE logkAk
soE
implies that E log kAk
ii Under the assumption log kAk L we have
logkAkL
logjdetAj L
allE k are nite
By Theorem B log kA k L implies that E d E logjdetAj
is nite Conversely since kA k d j det Aj kA k
d
d kAkd whence kA k kAkd j det Aj This gives
log kA k d log kAk jlogjdetAjj
This proves the rst equivalence The second one follows from Corollary
De nition Lyapunov Spectrum Suppose is a linear cocy
cle over for which one of the versions of the Furstenberg Kesten theo
rem holds Denote by
p
the di erent numbers in
the sequence
d possibly p
and de
note by di the frequency of appearance of i in this sequence On
the corresponding forward invariant or invariant set of full measure the
functionsp f dgdif p
ig f dgand
ifp
ig R f g are invariant and constant in the
ergodic case The functions i are called the Lyapunov exponents of
and the di their multiplicities The set
S
fidii
pg
is called the Lyapunov spectrum of We also write S A if is generated
by A By Remark
the Lyapunov spectrum is independent of the
choice of the norm in the space Rd d If is ergodic we will agree that the
Lyapunov spectrum consists of the corresponding set of constants where
possibly p
Example Products of
Triangular Matrices Let A
Gl R where
A
a
cb
a
b
These matrices form a subgroup of Gl R and the cocycle on T N over
generated by A is
nAnA
an
aPn
kan
ak ckbk b
bnb
1

FURSTENBERG KESTEN THEOREM
where ak
akbk
b k andck
ck
The following facts are readily veri ed assume for simplicity to be
ergodic
ilogkAkL
log jaj log jbj and log jcj L which we
assume from now on
ii By our assumptions
n
nXlog jakj E log jaj
n
nX log jbkj E log jbj
hence
n logjdet n j
iii The Lyapunov index of n is that of n
is and
that of n
is less than or equal to max
Using the Euclidean
norm in R we obtain by Lemma
nlogk n k
max
hence for
max
min
For
with multiplicity d
iv Generalization Suppose the generator of n in Rd has the
form
ABB
B
a
a
ad
a
ad
add CC
CA
Iflogjaijj Lfor i j dthenlogkAk L Ifthe k
E log jakkj are arranged in decreasing order k
kd then i
kii
d
The Furstenberg Kesten Theorem for Two Sided
Time
We now deal with the asymptotic behavior of the singular values of a cocycle
with two sided time for t
and its relation to the one for t
1

CHAPTER MET IN EUCLIDEAN SPACE
Theorem Furstenberg Kesten
Theorem
Two Sided Time Let be a linear cocycle with two sided time over
themetricDS FP ttT
A Discrete time case T Z Assume that the generator A
GldR of satises
logkAkL FP and logkAkL FP
Then n
n is a cocycle over generated by A
and on an invariant set of full measure we have for k
d
k
lim
n
nlogk k
nkdk
d
and
k
lim
n
nlog k
n
dk
where k and k are the corresponding limits for n In particular
ifS
fidi i
pg is the Lyapunov spectrum of n
as a cocycle over on T N then the Lyapunov spectrum of n
as a cocycle over
onTNis
S
S
fidii
pg
B Continuous time case T R Assume
L FPand
L FP where
sup
tlogkt k
Then all assertions of part A hold with n and N replaced with t and R
Proof Part A
Clearly
is a cocycle over
with generator A
The
generator satis es the integrability conditions
The cocycle property
relates positive and negative times via
n
nn
If
d
are the singular values of A Gl d R then d
are the singular values of A Hence
k
n
knn
dknn
We now apply Theorem B rst to n
which yields an invariant
set of full measure on which the corresponding statements hold and

FURSTENBERG KESTEN THEOREM
then to
n which yields an invariant set of full measure on which
for k
d
gk
n
nlog k
n
k
where k is invariant on
and convergence also holds in L Now gn g
in probability or L and g
gPas impliesgn
n g in probabil
ity or L This applied to equation
gives k
d kPas
consequently k
d k for all in the invariant set of full
measure
fk
dkk
dg
In the ergodic case we could have just taken E log in
to imme
diately arrive at the result
Part B
Theorem C can be applied to both cocycles The integrability
condition for t is satis ed due to the following lemma
Lemma The following three statements are equivalent
suptkt kL
suptktkL
sup tkt kL
The proof is an exercise in the use of the cocycle property
The remaining statements follow as in the proof of part B
Remark Average Lyapunov Exponent Let be a linear
cocycle for which the assumptions of one of the versions of the Furstenberg
Kesten theorem are satis ed Then we have
p
d
pX
idiid
dX
k
k dE logjdetAjjI
where A
in case of T R and the average Lyapunov expo
nent can be calculated by using j det Aj only in contrast the Lyapunov
exponents are quantities about which explicit information is usually hard
to obtain
In the invertible in particular two sided case we can without loss of
generality assume that
Pas since
Sjdetjd S

CHAPTER MET IN EUCLIDEAN SPACE
Example RDE with
Triangular Right Hand Side
Letxt A t xtinR withA
R given by
A
a
cb
Assume for simplicity that is ergodic
iWehaveA L ifandonlyifabc L whichweassumefrom
now on It implies that the RDE generates a linear cocycle t with
time T R see Example
It also implies the integrability conditions
of the Furstenberg Kesten theorem for T R see Example
ii Put Ea
Eb
tZt
asds
tZtbsds ct
ct
The cocycle has the form
t
et etRtes
scsds
et
iii Fromdet t exp t
t we obtain
t logjdet t j
Since the Lyapunov index of t is that of t is and that
oft
is less than or equal to max
we obtain using Lemma
and the Euclidean norm in R
tlogk t k
max
Hence for the case
max
min
while for
with multiplicity d
iv Generalization Consider xt A t xt in Rd with the triangular
matrix
ABB
B
a
a
ad
a
ad
add CC
CA
1

MULTIPLICATIVE ERGODIC THEOREM
ClearlyA L ifandonlyifaij L forall i j d Arranging
the k E akk in decreasing order k
kd we obtain i
ki
i
d See Oseledets
Crauel
and Johnson Palmer and
Sell Lemma
The Multiplicative Ergodic Theorem
This section is devoted to the presentation proof and discussion of the
various versions of Oseledets s multiplicative ergodic theorem MET the
most fundamental and most important theorem in this book See Sect
for an overview of the scope and the various proofs of the MET
We will rst prove the MET for time T N and more generally for
one sided time following Goldsheid and Margulis
and then deal with
two sided time We consider it worth the space to formulate and prove all
versions in detail
The MET for One Sided Time
Theorem MET for One Sided Time Let be a linear
cocycle with one sided time over the metric DS F P t t T Then
the following statements hold
A Non invertible case T N If the generator A
Rd d satis es
logkAkL FP
then there exists a forward invariant set F of full measure such that
for each
i The limit limn
n
n
n
exists
iiLetep
e be the di erent eigenvalues of
possibly p
and let Up
U be the corresponding
eigenspaces with multiplicities di
dim Ui Then
p
p
i
i forallif pg
di
di forallif pg
iii Put Vp
fgandfori
p
Vi
Up
Ui
so that
Vp
Vi
V
Rd

CHAPTER MET IN EUCLIDEAN SPACE
de nes a ltration of Rd Then for each x Rd nf g the Lyapunov exponent
x lim
n
nlogk n xk
exists as a limit and
x
i
xVinVi
equivalently
VifxRd
x
ig
iv Forallx Rdnfg
Ax
x
whence
AVi Viforallifpg
v If FP n n N is ergodic thenthefunctionp is constanton
and the functions i and di are constant on f p
ig
i
d
B Invertible case T N If A
GldR and
logkAkL FP and logkAkL FP
then the set of full measure on which A holds and on which in the
ergodic case p i di are constant can be chosen to be invariant
Further p
on and
AVi Viforallifpg
C Invertible case T R Let t Gl d R Assume
L
and
L where
sup
tlogkt k
Then all statements of part B hold with n and A replaced with t
t and t and the set F of full measure is now invariant with
respectto t t R
D Measurability The function
p f dg measurably
extended from to is measurable The functions
i
R
fgdifdgUi
d
kGkdand Vi
d
k Gk d Gk d the Grassmann manifold of k dimensional subspaces of
Rd measurably extended to f p
ig F are measurable
1

MULTIPLICATIVE ERGODIC THEOREM
In particular
Vi P Rd are random closed sets on their
domain in the sense of De nition
E Lyapunov spectrum The collection f i di i
p gofquan
tities from part A of the theorem coincides with the Lyapunov spectrum
S A of the cocycle
see De nition
The proof of the MET follows in a few lines from the Furstenberg Kesten
theorem and the following deterministic proposition the proof of which
requires the hard work alluded to above
Proposition Deterministic MET Let An n N be a sequence
of d d matrices which satis es the following two conditions
lim sup
n
n log kAnk
Putting n An A
lim
n
nlogkinkiRfg
exists for i
d
Theni The limit limn
nn
n
exists and the eigenvalues
of coincide with e i where the i are successively de ned by
i
ii
dput i
ifi
and satisfy
i lim
n
nlogi n i
d
where i n is the i th singular value of n
iiLetep
e be the di erent eigenvalues of possibly
p
and let Up U be their corresponding eigenspaces with mul
tiplicities di dim Ui Write Vp f g
Vi Up
Uii
p
so that
Vp
Vi
VRd
de nes a ltration of Rd Then for each x Rd nf g the Lyapunov exponent
x lim
n
n logk nxk

CHAPTER MET IN EUCLIDEAN SPACE
exists as a limit and
x
i
x VinVi
equivalently
VifxRd x
ig
To derive the MET from this proposition part of the following elementary
lemma is needed
Lemma Growth of a Stationary Sequence Suppose we are
givenametricDS FP n nN andarandomvariablef R
f gThen
i If f L then the invariant set
f lim sup
n
nfn g
has full measure
ii Let f R Then the invariant set
f lim inf
n
nfn
lim sup
n
nfng
has full measure If in addition f L then the invariant set
f lim sup
n
nfn g
has full measure
If f max f L then the invariant set
f lim inf
n
nfn g
has full measure
iii Let f
RIffLorifonlyf f Lorif
f f L bothimplyingf f L andEf f then
the invariant set
f lim
n
nfn g
has full measure
1

MULTIPLICATIVE ERGODIC THEOREM
Proof The invariance of
and follows from their de nition
i That has full measure is a consequence of the Borel Cantelli
lemma since
X
nPnfn X
nPfnX
nPfnEf
where we have used the elementary fact that for f and any
X
nPfnEfZPftdtX
nPfn
ii If f is nite valued then for any
lim
n Pjnfnj lim
nPjfjn
whence nf n
in probability from which the assertion follows If
f L combine i with what we have just proved
iii If f f L then the representation
fnfnX
iffi
and the extended version of the ergodic theorem say that limn
nf
n z exists and is possibly
withEz Ef f possibly
Byii thislimitisnecessarilyequalto Pa s HenceEz Ef f
in particular f f L Analogously for the case f f L
Proof of Theorem
A Under the asssumption log kAk L condition
of Propo
sition
is true for the sequence An A n for in an invariant
set of full measure by Lemma i applied to f log kA k
Under the same assumption the Furstenberg Kesten theorem A
hence in particular
holds on a forward invariant set of full mea
sureConsequently i ii iii and v are true on
which is
forward invariant and full
For iv just note that n A
n
which immediately
implies
Ax
x
Take x Vi which by iii is equivalent to
x
i Hence
Ax
x
i
i
1

CHAPTER MET IN EUCLIDEAN SPACE
entailing A x Vi
orA Vi Vi
B follows from the fact that the Furstenberg Kesten theorem B
furnishes an invariant set of full measure on which all statements are true
and all limits are nite and constant in the ergodic case
As for the invariance of Vi
note that since A is non singular
dimA Vi
dimVi Hence A iv and dimVi
dim Vi
imply A Vi Vi
C The continuous time case can be reduced to the discrete time case
as follows
i The integrability conditions assure that the corresponding continuous
time Furstenberg Kesten theorem C is valid on a set of full measure
which is t invariant
ii For xed
weintroduce asforT N the agFt corre
sponding to t
t
and show that
Ft tR convergesinFdtoF
limn Fn the
latter limit is known to exist on
the convergence is exponentially fast
lim sup
t tlogFtF
h
from which all other steps of the proof follow
iii The claims in ii would follow from
lim sup
n
nlogFnFns
h uniformly in s
which replaces
Writingn s tn tin
we obtain
lim sup
t tlogFt Ft
h
in particular F t
F and
follows by
and the tri
angle inequality
iv All that remains to be proved is
An inspection of the proof
of Theorem step case and case a shows that we have to assure
that the right hand side of
kPjntPinkktnkin
jnt
ij
ktnkjnt
in
ij
1

MULTIPLICATIVE ERGODIC THEOREM
decays to with exponential speed j i jj uniformly with respect to
t
This is the case provided
lim sup
n
n sup
tlogktn klimsup
n
n
n
note that since
I sup log sup log The integrabil
ity conditions
L and the ergodic theorem assure that in fact
limn
n n on a set of full measure which is invariant with
respect to
In step iii of the proof of part C of Theorem
we
showed that this set is even t invariant and its de ning property was
built into the t invariant set on which we are working
D The measurability of p i and di is a consequence of the measurabil
ity of the i which is assured by the Furstenberg Kesten theorem There
remains to be considered the measurability of Ui and Vi The following
will become clear in the proof of Proposition
On the measurable set
ikf di
kg Ui will be the limit of a sequence Ui n
of measurable functions with values in the Grassmannian Gk d and hence
measurable itself similarly for Vi Vi is a random closed set by Proposition
iandthefactthatforanopensetU Rd
fikViUgfikVi Ukg
whereUk fV Gkd V U gisopeninGkd
E is clear as we have constructed the objects in A by the Furstenberg
Kesten theorem on which De nition
is based
It remains to prove Proposition
For pedagogical reasons we rst
present the proof for dimension d and later for general d
Proof of Proposition
for d
By assumption
the limits
lim
n
n logk nk lim
n
nlog n
and
lim
n
n logk nk
exist and since k nk n
n andif
lim
n
nlog n
This proof was presented by Ilya Goldsheid in a lecture at the University of Bremen
in June
1

CHAPTER MET IN EUCLIDEAN SPACE
exists if
put
Let n VnDnOn be the singular value decomposition of n with
Dn diag n
n Then
nn
nOnDn
nOn
has eigenvalues i n n e i and eigenvectors equal to the columns
One One of On We will in general not be able to prove that On
converges due to the non uniqueness of the singular value decomposition
However it su ces to show that the spaces spanned by the eigenvectors
converge
Lemma Let S be a symmetric d d matrix with spectral represen
tation S P kPk where k are the eigenvalues and Pk are the orthogonal
projections onto the eigenspaces Uk corresponding to k uniquely given by
Pk iZk
zISdz
where k is a circle in the complex plane C around k which excludes all
other eigenvalues of S Let Sn P k n Pk n be a sequence such that for
n
kn
ifork i
a group of indices
Pin Pk iPkn Pi
Then Sn S
Proof By our assumptions
SnSXiX
ki
kn
iPkn
iX
k iPkn Pi
for n
Case
HereD n
n
eIandPn Pn Pn IhencebyLemma
nn
n e I whichprovesi
The ltration in ii is trivial V R We need to prove that for each
x Rnfg
x lim
n
n logk nxk
1

MULTIPLICATIVE ERGODIC THEOREM
But with Onx xn
k nxk kDnxnk n xn
nxn
If
then for every
thereisaC
such that for
i
and all n
Cen
inCen
Hence for every there is a C
independent of x such that for
allnandallx
kxkC en k nxk kxkC en
whence x
forallx If
then the same estimates give
x
for all x
Case
NowD n
n diag e e The eigenvectors of n n
nareu n
One for
nneandunOnefor
nneWenow
prove that for the corresponding projections P n P and P n P
hence by Lemma
nn
n
nPn
nPn
ePeP
This in fact a much sharper result will follow from the next two lemmas
Lemma Let P and Q be orthogonal projections in R with
dimU dimV where U imP and V imQ Then
kPQkUVkxykxUyVkxkkyk
where denotes the distance in the projective space P introduced in Propo
sition
Proof For elementary facts on projections see Kato
pp
i If P Q are two orthogonal projections then always kP Qk
WehavekP Qk ifandonlyifU Vifandonlyif UV
iiIngeneralifx Uy Vkxk kyk then usingx
hxyiyhxyiy
kI QPkjhxyijkx yk UV
The fact that here and at later occasions C is independent of x will be needed in
the proof of a uniform convergence result for two sided time see Theorem v
1

CHAPTER MET IN EUCLIDEAN SPACE
IfkI QPk thenkP Qk kI QPkKato Theorem
and the result follows
Lemma Assume d and the conditions of Proposition
with
Let denote the distance in P introduced in Proposition
andletU n andU n betheeigenspacesof n n
n correspond
ing to the eigenvalues
nn
n n this inequality holds for all
n n Thenfori
lim sup
n
nlogUinUin
which is also true if
In particular the sequence Ui n is a
Cauchy sequence in P and thus converges
lim
nUinUi
and the convergence is exponentially fast
lim sup
n
nlogUinUi
Proof iItsu cestoconsiderthecasei sinceU n TU n
TORthusUnUn
UnUn
ii First observe that we have n u n
n Vne
nun
nVne thusknu nk
nknunk
n and
nun
nu n Bythedenitionof n
knunkkAn
nunkkAnkn
Representu n asu n
nun
nun
and note that
knunkknnun
nnunk
n
n
n
n
jnj n
By Proposition
UnUn kununkjnjkun
unkjnj
so
UnUn kAnk n
n
1

MULTIPLICATIVE ERGODIC THEOREM
note that n
otherwise
The assumptions
and
of Proposition
and Lemma v yield
follows from
UnUX
inUiUi
and the discrete time version of Lemma iv
We now complete the proof of Proposition
for d
The re
sult of Lemma
is that both the eigenvalues and the eigenspaces
ofnn
n converge to limits e i and Ui By Lemma
this is
equivalent to the convergence of the corresponding orthogonal projections
proving part i of the proposition
Toprovepartii takex VnfgwhereV U V R write
xkxkvv
nun
nu n withkvk n
n thus
kxkjnj n knxkkxkn
n
n
n
We have proved in the above lemma that
lim sup
n
n logj nj lim sup
n
nlogUnU
thus n
Lemma iii and v applied to
gives
lim inf
n
n logk nxk
lim sup
n
n logk nxk max
More precisely for each
thereisaC
independent of x such
thatforallnandx Vnfg
kxkC en k nxk kxkC en
whence
x lim
n
nlogknxk ifx Vnfg
Nowtakex RnV andwritex
u v with unit vectors u U
v U Vand hxui Writev
nun
nunu
1

CHAPTER MET IN EUCLIDEAN SPACE
nun
nu n By the above lemma n
n
with exponential
speed
son
n We obtain the estimate
jjjnj n knxk
n
n
n
n
n
n
More precisely by Lemma iii and v for each
there is a
C
independent of x such that for all n and all x R nV
kxkjh x
kxk uijC en k nxk kxkC en
whence
x lim
n
nlogknxk forx RnV
This completes the proof of Proposition for the case d
Remark The crucial fact in the proof of x
forx Vnfg
isthatjnjUUnju unjjhu uniju Udecays
with rate
thus cancelling the rate
of nintheterm
jnj n ontherighthandsideof
It is therefore not just the
convergence U n U but its particular speed that makes the proof
work
Proof of Proposition
for general d
The general idea abstracted from the proof for d is as follows Let
n VnDnOn Dn diag n
d n be the singular value de
composition of n By assumption
lim
nDn
n diage
ed
Denote by
p the di erent numbers among the i Let for
i
p Ui n be spanned by the group i of those eigenvectors of
nn
nOnDn
n On corresponding to eigenvalues k i n n e i
Let Pi n be the corresponding orthogonal projection We prove that
PinPin
for i
p which by Lemma
yields
nn
n Pe iPi assertion i
the convergence in has just the right speed to ensure assertion ii
Step Construction of a sequence F n of ags
We can assume d since the case d is trivial Let
p

MULTIPLICATIVE ERGODIC THEOREM
be the di erent numbers in the sequence
dletdibethe
multiplicity of i If p we can use the proof for d case
verbatim Let now p Assume p
The modi cations necessary
ifp
are either obvious or given in the course of the proof
Put
i
iii
p
min
i
p
i
Choose an
and then an N such that for all i
p
nlogki n
i
foralln N
where k i runs through the group i of di indices for which
lim
n
nlogki n
i
Let
Uin
span On ed
di
Oned
dii
p
and
Vin Upn
Uini
p
The nested sequence of subspaces of Rd given by
Fn Vpn
Vin
Vn
forms a ltration or ag corresponding for all
andalln N to
the vector of dimensions
dpdp dp
dp
dd
which is independent of and n
The set F d of all ags corresponding to the vector of dimensions
can be given the structure of a compact C manifold in a natural way see
Husemoller Chap The ag manifold F d can be endowed with
a complete metric as follows
Take F Vp
V F d anddeneUp Vp Ui orthogonal
complement of Vi in Vi i p
so that
Vi Up
Uii
p
Chooseh d anddeneforanyFF F d
FF
max
ij pij
max
xUiyUjkxkkykjhxyijhji jj
1

CHAPTER MET IN EUCLIDEAN SPACE
Ifp
replacehjp jjandhji pjbyd Then
isa
complete metric on F d compatible with the natural topology For a proof
which mainly consists of verifying the triangle inequality see Goldsheid
and Margulis
pp
Remark i It follows from the formula
kAk sup
kxk kyk jhAx yij
that for two orthogonal projections P Q in a Hilbert space
kPQk kQPk
max
x imPy imQkxkkyk jhxyij
If for F F d Pi denotes the orthogonal projection onto Ui the metric
FFdenedby
can be written as
FF
max
ij pijkPiPjkhji jj
ii Ford weobtainh
which yields exactly the metric in
PF
used above in the proof for d case
Indeed
forFURFUR
FF jhxyijjhyxij
wherex Ux U y Uy U areunitvectors andhencewith
xhxyiyhxyiy
UU kx ykjhxyijky yk FF
Step Reduction to a lemma on the convergence of the sequence
Fn of ags
Proposition
is a consequence of the following lemma
Lemma We have
lim sup
n
nlogFnFn
h
In particular the sequence F n is a Cauchy sequence and thus converges
inF d
lim
nFnF
and this convergence is exponentially fast
lim sup
n
nlogFnF h
1

MULTIPLICATIVE ERGODIC THEOREM
We rst derive Proposition
from this lemma By
and the
denitionofthemetric wehaveFn Vpn
VnF
Vp
V ifandonlyifforeachiUin UiinGdid wherewe
use the complete metric on Gk d given by
UUkIPQkkIQPk
where P and Q are the orthogonal projections onto U and U in Gk d
Indeed denoting by Pi and Pi n the orthogonal projections onto Ui and
Ui n respectively and using the identity I Ppj Pj n
UiUin kI PinPik X
j pjikPjnPik
Recalling the form
ofthemetriconFd Fn FinFd
impliesUi n UiinGdid fori
p
Conversely assume Ui Ui n
for all i A result of Kato
p
applies saying that if P and P are orthogonal projections with
dimimP dimimP andkI P Pk c then
kPPkkIPPkkIPPkc
HencePin Piforalliwhichimplies Fn F
In particular F n F implies n n
n PeiPiiepartiof
the proposition
To derive part ii from the lemma let for xed i
p x VinVi
kxk and write
x
pX
jPjnx
pX
j iPjx
By assumption Pi x
The vectors nPj n x j p are orthogonal
and kPjnxkj n knPjnxkkPjnxkj n
where
jn
min
jkjnjn
max
jkjn
with
jn
jn
j
Hence
pX
jkPjnxkjnknxkpX
jkPjnxkjn
1

CHAPTER MET IN EUCLIDEAN SPACE
SincekPi n xk i n k nxkandkPi n xk kPixk
i lim inf
n
n logk nxk
We now estimate the summands in
fromabove Forj i p
we use the trivial estimate kPj n xk to obtain
kPjnxkj n
i
For j
i weuse
kPjnxkkPjn
pX
k iPkxk pX
k ikPjnPkk
and
to obtain from
kPjnPkkhjj kj hand
kPjnxkj n
j max
kiphjj kj
h
jiji
Hence lim supn
n log k nxk i and altogether
lim
n
nlogknxk i
which is part ii of the proposition
Step Proof of Lemma
Since
FnFX
knFkFk
follows from
by the discrete time version of Lemma
iv It thus su ces to prove
which introducing
ijn kPinPjn kkPjn Pink
is equivalent to
ijn ji jjforij
Caseij
Now i j Foraunitvectorx Uin andy Pjn x Ujn
kn xkkAn
nxkkAnki n
1

MULTIPLICATIVE ERGODIC THEOREM
and
knxkknykknxyk knykjnkyk
so kyk ijnkPjn PinkkAnki n
jn
whence
ijn
ijjijj
For p
we obtain
pjn
for j
p
Caseij
Nowij
a We rst show that we can repeat the argument of case if we have
the additional assumption that An Gl d R and
lim sup
n
n logkAn k
Foraunitvectorx Ujn
andy PinxUin
k nxk kAn
nxkkAnkjn
and
knxk knyk knx yk knyk i nkyk
so
ijnkPinPjn kkAnkj n
in
and hence
ijn
jijijj
b In the general situation we can reduce case i j to case i j
as follows
i Observethat ij n ji jjfori jimplies
lim
nkIPin Pinki
p
hence by Kato s result
lim
nkPinPinki
p
1

CHAPTER MET IN EUCLIDEAN SPACE
To prove
we proceed by induction For i p using I
Ppj Pjn
kIPpn PpnkkpX
jPjn PpnkpX
j pjn
n
Suppose we have proved the claim up to the index i p Then it also holds
for i
For with
IiX
j Pjn
Pin
pX
jiPjn
kPinIPink
iX
jkPi nPjnkpX
jikPi nPjn k
iX
jijn
pX
jikPi nIPjn PjnPjnk
iX
jijn
pX
jikIPjnPjn k
n
We can thus assume from now on that n is chosen so large that
kPinPinki
p
ii If A A and B are bounded operators in a Banach space with A A
kA Ak then kABk kAABk
For
kAABk kAB AAABkkABkkAAABk
kABk kA A k kABk kABk
iii Fori j
Pin Pjn
pX
kPin PknPjn
1

MULTIPLICATIVE ERGODIC THEOREM
thus
Pin PinPjn
X
k
pkiPin PknPjn
For n big enough by
andsincekPi n Pi n k
ijn kPin PinPjn k
X
k
pkikPin PknPjn k
X
k
pki
kin kjn
Since ijp
ij
iX
k
ki kj
jX
ki
ki kj
pX
kjkikjA
where we have omitted the variable n and have marked those terms with a
superscript for which case applies
iv The Lyapunov index of the last term in
can be estimated
as follows using the trivial bound ij n
and the result of case
pX
kjkin kjnApX
kjkinA
max
kjpjkijjjij
v We rst derive the following crude estimate For i j p
ijn
min
i
The proof proceeds by induction First put i and use kj n
The
rst sum in
is not present and
jn
max max
kjjkjjjj
min
kjjkj
Using this for the rst term in
jn
max
max
kjjkjjjj
max
min
1

CHAPTER MET IN EUCLIDEAN SPACE
in general
ijn
min
i
and nally
ppn
min
p
vi We improve this crude estimate as follows For the terms of the
rst sum in
we have k i j and the crude estimate gives
kin kjn
For the terms of the second sum we use the estimate of case for ki and
the crude estimate for kj to obtain
kin kjn jk ij
Altogether
ijn
max
max
kijjkijjijj
max
ijijj
max ji jj
vii We have now two cases
Either j i jj
Then
ijn ji jjandweare
done
Orji jj
In the second case the new estimate
ijn
is strictly better
than the crude estimate
The rst term on the right hand side of
is estimated as
iX
k
ki kj
max
ki
ki
kj
Applying the estimate of step vi and taking into account the de nition of
we arrive at
ki
max
ki
jk ij
for each k
i
I am indebted to Anna Kwiecinska for suggesting an improvement in this step
1

MULTIPLICATIVE ERGODIC THEOREM
On the other hand
jkjjjijj
for each k
i
resulting in
kj
max
ki
jk jj
and nally
iX
k
ki kj
Now the second term on the right hand side of
is estimated as
jX
ki
ki kj
max
kij
ki
kj
ki
kj
for some k fi
j g Applying the estimate of step vi to the
right hand side of the last estimate we obtain
ki
kj
jk
ijmax jk jj
jk
ijjk jjji jjjk jj
jk
ij
jk jj
For the third term on the right hand side of
we use the estimate of
step iv and obtain
pX
kjkikjAji jj
Collecting the three estimates together we nally obtain
ijn
max
iX
k
ki kj
jX
ki
ki kj
pX
kjkikjA
max ji jj
We again have two cases
either j i jj
Then
ijn ji jjandweare
done
1

CHAPTER MET IN EUCLIDEAN SPACE
orji jj
In the second case we apply the estimate
to the right hand side of
to obtain the new estimate
ijn
max ji jj
We continue this procedure until we obtain in nitely many steps
ijn ji jj
which was to be proved
For p
the same procedure gives
ijn
for all i
p
This completes the proof of Lemma
and thus of Proposition
Remark i The ltration and the spectrum of a linear cocycle
are independent of the choice of the vector norm and a corresponding matrix
norm in Rd see also Remark
Moreover let V be a real normed vector space of dimension d with norm
k k and let T L V be a measurable random linear operator with
log kT k L Then the MET Theorem holds for the linear cocycle
Sn
Tn
T for any norm in V and the ltration and
the spectrum are independent of that norm similarly for the other versions
of the MET that follow
To see this choose a basis F in V The coordinate mapping kF V Rd
becomes an isometric isomorphism if we choose the standard scalar product
hiSinRdandhxyiF hkFxkFyiSinV ThenormkkFis
equivalent to any other norm in V
ii The MET is also valid in C d or any nite dimensional normed C
vector space V Our proof now based on the polar and singular value
decomposition in a unitary space see Gantmacher IX holds ver
batimiii Uniqueness of spectrum and ag Suppose the linear cocycle n
satis es the conditions of Theorem
Then its spectrum and ag are
unique This means that whenever there is another ag
fg Wq
WRd
and numbers q
with x
kifandonlyifx WknWk
thenq p k kandWk Vk Theproofisleftasanexercise

MULTIPLICATIVE ERGODIC THEOREM
The MET for Two Sided Time
Theorem MET for Two Sided Time Let be a linear
cocycle with two sided time over the metric DS F P t t T
A Discrete time T Z Let
n
An
A
n
I
n
AnA
n
be generated by A
Gl d R and assume
logkAkL FPandlogkAkL FP
Then there exists an invariant set of full measure on which the statements
of the MET for T N Theorem B hold Moreover for each
there exists a splitting
RdE
Ep
of Rd into random subspaces Ei so called Oseledets spaces depend
ing measurably on hence are random closed sets with dimension
dim Ei di so called Oseledets splitting with the following proper
tiesForif pg
iifPi
Rd Ei denotes the projection onto Ei
along
Fi
j iEj then
APi
PiA
equivalently
AEi
Ei
ii we have
lim
n
nlogkn xki
xEinfg
iii convergence in ii is uniform with respect to x Ei Sd for
each xed
B Continuous time T R Assume that
L and L where
sup
tlogkt k
Then all statements of part A hold with n and A replaced with t
t and t and the set F of full measure is now invariant with
respectto t tR
1

CHAPTER MET IN EUCLIDEAN SPACE
Proof Part A
Theorem A and B applied to the cocycle n yields on an
invariant set of full measure the nite spectrum
S
fidii
pg
and the forward ltration
Vp
V
RdAVi Vi
The same theorem applied to the cocycle
n
with time n N over
yields on an invariant set of full measure the nite spectrum
S
fi di
i
pg
and the backward ltration
Vp
V
RdA
Vi
Vi
The two sided Furstenberg Kesten theorem
assures that on an in
variant set of full measure all statements above are true and
S
S
more precisely p
p andfori
p
di dp
i
i
p
i
We now construct the spaces Ei as intersections of certain spaces
from the forward and the backward ltration We set
Ei
Vi Vp
i
ip
in particular E
Vp Ep Vp As a preparation we prove
Lemma On an invariant set
of full measure
iVi
Vp
ifgip
and ii Vi
Vp
i
Rdip
Proof We show that the statements of the lemma hold on a full set
Passing to
n Z n we have an invariant set of full
measure on which the lemma and the above statements hold This set
also quali es for the invariant set of full measure mentioned in the theorem
1

MULTIPLICATIVE ERGODIC THEOREM
i Selectp f dgarbitrary but xed De ne p f
p
p g This is an invariant set Assume P p
and choose
i f p gWeprovethatPB
where
BfpVi
Vp
i fgg
The idea of the proof is as follows We move for an B the vector
xVi
Vp
i fromtime backtotime Nby N with
rate at most i Then we move the same point from time N forward
totime by N N withrateatmost i
Since i
i
the result will be a small quantity But this would contradict the cocycle
property which gives x N N
N xunlessx
Put
i
i
For each
choose an integer N so
bigthatPC
PpandPD PND
Pp
where
CfpkNxkeNikxk
forallx Vp
i nfgg
and
DfpkNxkeNi
kxk
forallx Vi nfgg
By the proof of Theorem
these estimates hold for all n N
independently of x hence we can choose an integer N such that
PfpN
Ng
Pp
Now
PB
PBCNDPBCNDc
PBCND
WeprovethatB C ND
by the above argument Suppose
BCND
a Since B there exists an x Vi
Vp
i nfgsuch
that without loss of generality
kxkkN N
Nx
kNxkkkNxk
where we have used the cocycle property
1

CHAPTER MET IN EUCLIDEAN SPACE
b Since C
kNxkeNi
c Since
ND foranyw Vi
N nfg
kNNwkeNi
kwk
where we have used the invariance of i and Since N Vi
Vi
Non
w
Nx
kNxkVi
N
Estimating
with
and
gives
eNi
i
e
which is a contradiction Thus P B with arbitrary proving i
ii By i thesumin ii isdirect WehavedimVi Pp
kidk
and by
dim Vp
i
iX
kdk
giving dim Vi
dim Vp
i dandthusii
Continuing the proof of part i of Theorem A we use the elementary
formula
dimU V dimU dimV dimU V UV Rd
andVi Vp i Vi Vp i RdLemma ii toobtain
dim Ei
dim Vi dim Vp
i dimVi Vp
i
pX
k idk
iX
kdk d
di
We now prove that the sum of the Ei is direct This is the case if and only
ifEi Fi fgforalliwhereFi Pj iEj Bydenition
EiFiEiUVUiX
jEjVpX
jiEj
1

MULTIPLICATIVE ERGODIC THEOREM
where for i U is not present and for i p V is not present Note that
UiX
jVjVpjVp
i
and
VpX
jiVjVpjVi
Supposex Ei FiThenx Viandx Vp iandx u vu U
vV
Sincex Viandv Vi Viwehavex v u Vi Vp
i
fgLemma iieu
Sincex Vp iandx v Vi wehavev Vi Vp ifg
Lemma iiev
Hence P Ei is direct in particular dim P Ei P dim Ei d
ii Using the invariance of the forward and backward ltration the
latter one with replaced by the invariance of p and the elementary
relation
AUVAUAVforAGldR
we obtain
AEi
AViAVp
i
Vi
Vp
i
Ei
Since the complementary space Fi
j iEj is also invariant i e
satis es A Fi
Fi
we even have A Pi
PiA
iii By the two one sided versions of the MET
lim
n
nlogkn xki
if and only if
xVinVi
Vp
i nVp
i
Mi
Now
MiViVc
iVpiVp
ic
Since Vi Vp
i fgLemma i Vi Vp
i c Vinfg
similarly V c
i Vp i Vp infgHence
Mi Vinfg Vp infg Einfg
1

CHAPTER MET IN EUCLIDEAN SPACE
iv Uniform convergence in Ei Sd We consider the case n
only
and recall certain facts from the proof of Theorem
and Proposition
Since Vi Vi Ui x Ei Vi can be represented as x
Ri x Qix where Ri and Qi are the corresponding orthogonal projections
By the proof of Proposition and explicitly formulated in our proof
for the case d for each
there exists a random variable C
which is independent of x such that for all n
kn xkkxkeni C inVinVi
knRixkkRixkeni
C forallRi x
Vi
kQixkeniC
knQi xkforallQi x
Ui Vi nVi
Since supx Ei Sd kRi xk and
ci
inf
xEi SdkQi xk
the above estimates yield for x Ei Sd
Ccieni
eni
knxkeniC
where the left hand side and the right hand side are independent of x
Ei Sd
Part B
The continuous time case can be reduced to the discrete time case as
in the proof of Theorem C Note that the integrability conditions
assure that the continuous time Furstenberg Kesten theorem B is
valid on a set of full measure which is t invariant
Remark i The uniformity statement of Theorem iii re
mains valid if Sd is replaced with a set in Rd bounded away from and
ii For all
the cocycle is forward and backward regular in the
sense of Sect
iii The forward and backward ltration can be recovered from the
Oseledets splitting via
Vi
pj iEj
Vp
i
ij Ej
iv Part i of the theorem gives for
nPi
Pin
n foralln Z
1

MULTIPLICATIVE ERGODIC THEOREM
This commutativity of the cocycle with the corresponding projections is a
crucial advantage of two sided time and will play a key role in the theory
v The invariant splitting
RdE
Ep
is in general not orthogonal and the orthogonal splitting Rd U
Up
appearing in Theorem
and its proof is generally not
invariant
vi Represent x
asx
p
iPi xThen
x
kx
where k x
minfiPi xg
vii Uniqueness of spectrum and splitting For a linear cocycle sat
isfying the conditions of Theorem
its spectrum and splitting are
unique This means that if there is another splitting Fq
FRd
with
lim
n
nlogkn xki
ifandonlyifx Finfgthenq p i
i and Fi Ei This follows
from Remark iii applied to Vi and Vi
Remark Measurability Properties of Filtration Let T
ZorRanddeneF
t t for discrete time we have
F
Ann
to be the future and F
tt
for discrete time F
Ann
the past of the cocycle see
Sect
By construction the ltration Vk in particular Ep Vp is
F measurable and Vk in particular E Vp is F measurable If
the sequence A n is i i d or if solves a linear stochastic di erential
equation then F and F hence Vk and Vk in particular Ep and E
are independent
Examples
Example Linear RDE in Rd By Example the equation
xtAtxt
generates a linear cocycle on T R provided A L We now prove
that this condition also implies the integrability conditions
L ofthe
METforT R Takean intheinvariantsetforwhicht A t is
locally integrable Then for t R see Example
kt kZtkAskks kds
1

CHAPTER MET IN EUCLIDEAN SPACE
By Gronwall s lemma B
kt kexpZtkAskdsforalltR
and nally
sup
tlogkt kZkAskds
The right hand side of the last equation is in L by Lemma
The
Furstenberg Kesten theorem and the MET for T R are thus valid In
particular the average Lyapunov exponent is by Liouville s formula and
the ergodic theorem note that jtrace Aj ckAk L
dtraceE AjI
A prototypical and well investigated example see the articles in Arnold
and Wihstutz
or Arnold Crauel and Eckmann
is the damped
linear oscillator with a random restoring force
yt yt
ft yt
fL
Puttingx yx y
xtAtxt
ft
xt
In this case
Example RDE with
Triangular Right Hand Side
Continued We continue Example
and explicitly calculate the
Oseledets splittings see also Crauel Chap
Case
In this case the ltrations and the splitting are trivial E V V
R Assume for example that a b c
and that the
law of the iterated logarithm holds for t Then the orbits t x
exp t
x typically undergo uctuations of the order exp p t log log t
Note in contrast that the deterministic cocycle exp tA has at most
polynomial growth inside an eigenspace corresponding to an eigenvalue with
vanishing real part
Case
E V R e is deterministic and
E
Ru
uZet
tctdt
1

MULTIPLICATIVE ERGODIC THEOREM
which is F measurable The orbit of u is
tu
etRtes
scsds
et
et
ut
That u makes sense and the orbit of u has Lyapunov exponent is a
consequence of the following simple lemma
Lemma Let u L and vt be continuous with vt t
PasThenZevt utdtexistsPas
and
lim sup
t t logZt
evsusds
Proof For each xed t and c
EZt
ectjutjdt
Now for
with c
and
chooseat t
such that
vt
ct for t t at least with probabbility
Since Z t
evtjutjdt
and is arbitrary the rst assertion follows
The second claim follows from the rst by observing that
Zt
evsjusjds e
tZt
e sjusjds
with probability at least
where the second factor on the right hand
side is bounded
Case
Now E V R e is deterministic and
E
Ru
uZet
tctdt
which is F measurable

CHAPTER MET IN EUCLIDEAN SPACE
Exercise Products of Triangular Matrices Continue
Example by explicitly calculating the Oseledets splitting analogous
to the continuous time case of Example see also Berger p
Example Linear SDE in Rd The linear SDE in Rd
dxt A xtdt mX
j Ajxt dWjt
generates a linear cocycle t on T R over the canonical ergodic
DS associated with Brownian motion The integrability conditions of the
MET are automatically satis ed see Theorem
Liouville s formula
Theorem iii yields the scalar cocycle
det t
exp traceA t mX
j traceAj Wjt
so
d trace A
We have det t
forallt RPas ifandonlyiftraceAj for
all j
m The SDE
dxt A dtraceA I xtdt mX
j Aj dtraceAj I xt dWjt
generates the volume preserving cocycle det t
dt SldR
for which
hence p
An important problem is whether
actually p
holds In dimension d we have either
with
multiplicity d or
For criteria for
or for the simplicity of the Lyapunov spectrum
in the white and Markovian noise case see e g Guivarc h and Raugi
Bougerol and Lacroix Arnold and Kliemann Bougerol
and
Goldsheid and Margulis
For genericity results see Knill
and
Arnold and Nguyen

Chapter
The Multiplicative
Ergodic Theorem on
Bundles and Manifolds
Summary
This chapter deals with extensions of the MET from matrix cocycles in
Rd to linear cocycles on bundles in particular to the linearization of a
nonlinear RDS on the tangent bundle by which we prepare the smooth
ergodic theory of Part III
In order to facilitate the transfer of data of the MET from one bundle to
another we need a tempered version of coordinate change called Lyapunov
cohomology Sect
In Sect the MET for the linearization of a nonlinear smooth RDS
on a manifold is derived Theorem
In case the RDS is generated
by an RDE or SDE we also give criteria in terms of the vector elds and
the invariant measure ensuring the validity of the integrability conditions
of the MET Theorem
for the RDE case Theorems
and
for the SDE case
Sect is devoted to one of the most important techniques of smooth
ergodic theory namely the use of random norms which eat up the non
uniformity of the MET Theorem
It is of course crucial that the
random norms do not change the Lyapunov exponents Corollary
We also obtain what is called the strong version of the MET Theorem
1

CHAPTER MET ON BUNDLES
Tempered Random Variables and Lya
punov Cohomology
Tempered Random Variables
We rst introduce a concept which is of fundamental importance for most
parts of the theory of RDS and can even be considered one of its charac
teristic features
In many estimates for RDS we will have random constants whose
values have to be controlled along the orbits of the DS For example the
MET for T Z gives for each
a nite random variable C such that
knkCe
nnZ
There are reasons to consider the norm of n
nnfor
which we obtain from
that
knnkCne
nnZ
In order not to destroy this estimate we have to exclude exponential growth
at other occasions exponential decay of the sequence C n
De nition Tempered Random Variable i A random vari
able R
is called tempered with respect to the DS if for the
associated stationary stochastic process t R t the invariant set for
which
lim
t
tlogR t
t
applies only to two sided time has full P measure
iiR
is called tempered from above if
lim
t jtjlogRt
Pas
while R
is called tempered from below if R is tempered from
above equivalently if with log x max log x
lim
t jtjlogRt
Pas
iii A random variable f
Rd is called tempered from above
or below with respect to the DS if the stationary stochastic process
t kf t k is tempered from above or below

TEMPEREDNESS LYAPUNOV COHOMOLOGY
Since
lim
t jtjlog R t limsup
t jtjlogR t
and
lim
t jtjlogRt
lim inf
t jtjlogR t
R is tempered if and only if it is tempered from above and from below
The following lemma follows immediately from the properties of the
function x log x
Lemma The set of real valued tempered random variables with
respect to a DS is a commutative ring with unit element
There is the following remarkable dichotomy found by Tanny
and
O Brien
Proposition Dichotomy for Linear Growth of Stationary
Process Let FP ttTbeametricDSandletf
Rbe
measurable Then
lim sup
t tf t limsup
tjtjftfgPas
and
lim inf
t tf t liminf
tjtjftfgPas
t
applies only for two sided time
ii For discrete time
f L limsup
n jnjfn
Pas
intheiid case and
f L liminf
n jnjfn
Pas
intheiid case
For continuous time
sup
tft L limsup
tjtjft
Pas
1

CHAPTER MET ON BUNDLES
and sup
tft L liminf
tjtjft
Pas
The P a s statements hold on an invariant set of full P measure
iii If is ergodic the above lim sup s and lim inf s are constant on an
invariant set of full P measure
Proof i We follow O Brien but restrict ourselves to the case where
is ergodic time is discrete and to the consideration of the lim supn
By ergodicity the invariant random variable lim sup f n n is
P a s constant This constant c satis es c
sincef n n
in probability
Assume
c
The event
Un max
kZ cfnk k
is de ned P a s and the sequence Un is stationary ergodic Also Un
maxf n
cUn
so that
cfn UnUn
Pas foralln
Forsomet R PUn t sothatPUn tio io standsfor
in nitely often Let n n be the random positive epochs for which
Un t withni ni foralli LetIn ifUn tandIn
otherwise
Applying the ergodic theorem to the stationary ergodic sequence In we
see that
lim
i
ni
ni
Pas
For ni
n ni we deduce from
and
that
cnfn Un
n Uni
nni
n
tnini
ni
asi
This implies lim sup f n n and hence contradicts our
assumption
ii All statements are easy consequences of the Borel Cantelli lemma
Applied to f log kgk for a measurable g
Rd the lemma yields a
dichotomy for the Lyapunov index of the stationary process t g t
If this index is non zero it is automatically equal to in nity Temperedness
hence prevents the case of superexponential growth of a stationary process
from happening
1

TEMPEREDNESS LYAPUNOV COHOMOLOGY
Remark Existence of Non Tempered Random Variables
Let F P be a standard space and be aperiodic Then there exists a
random variable which is not tempered with respect to The proof relies
on the Rokhlin Halmos lemma see Arnold Nguyen and Oseledets Sect
Lyapunov Cohomology
We have stated the MET in Chap only for linear RDS on the trivial
bundle Rd We need to extend it to nontrivial linear bundle RDS in
the sense of De nition ii to cover e g the linearization T of a C
RDS on a manifold M or the case of random norms k k on Rd
The key notion to facilitate the transfer of spectral objects from one
linear bundle RDS to another is the notion of Lyapunov cohomology which
we will introduce rst
Let FP t t T bea xedmetricDSInthischapterweconsider
linear measurable bundles Y
with typical ber Rd see De nition
ii andalinearbundle RDS t t Tonitie thecocycle
t
tj
t
is linear
AssumethattwolinearRDS it t Ti
over on linear mea
surable bundles Yi
i with typical ber Rd are linearly isomorphic see
De nition ii i e there is a linear measurable isomorphism of the
corresponding bundles Yi
i such that t
t for all
t T For the corresponding linear cocycles de ned by
it
itji
i
it
this is equivalent to being cohomologous with cohomology j
ieforallt T
t
t
t
Cohomology is an equivalence relation in the set of linear RDS on mea
surable linear bundles with typical ber Rd
De nition Normed Linear Bundle A linear measurable bun
dle Y
is said to be normed if there is a measurable function on Y
whichis a normkk on each ber
The norm can in particular be
induced by a scalar product h i in which case we speak of a Euclidean
bundle

CHAPTER MET ON BUNDLES
For linear RDS on normed linear bundles we can de ne the Lyapunov ex
ponent of a vector x Y by
x lim sup
t tlogkt xkt
x
We would like to carry spectral theory i e the objects of the MET
Lyapunov exponents ltrations and splittings from one linear RDS to a
cohomologous one To make this work we have to impose a condition on
the cohomology
De nition Lyapunov Cohomology A cohomology
kk
kk
of two linear cocycles and over on normed linear bundles is called
a Lyapunov cohomology if and are tempered i e if there is a
invariant set of full P measure on which
lim
t
tlogk tkt
lim
t
tlogk t kt
t
applies to two sided time Here k k and k k are
the operator norms of
kk
kk andits
inverse
Lyapunov cohomology is an equivalence relation in the set of linear RDS
on normed linear bundles with typical ber Rd
Proposition Conditions for Lyapunov Cohomology A
su cient condition for a cohomology between two linear RDS and
on normed bundles to be a Lyapunov cohomology is that it satis es
i for discrete time
logkklogkkLP
ii for continuous time
sup
tlogktkt
sup
tlogktktLP
Proof This follows from Proposition ii for f log k k
and f log k k noting that
logkklogkklogkk

TEMPEREDNESS LYAPUNOV COHOMOLOGY
De nition Spectral Theory of a Linear RDS We say of a
linear RDS on a normed linear bundle that it has spectral theory if there
is a forward invariant for the invertible case an invariant set of full
PmeasureonwhichwehaveaspectrumS fi di i
pg
and a ltration Vp
V
for two sided time a
splitting E
Ep
with the properties listed in the
corresponding MET Theorem for one sided time and Theorem
for two sided time
Proposition Lyapunov Cohomology Preserves Spectral
Theory Let and be two linear cocycles on normed bundles which
are cohomologous via
i If has spectral theory and is a Lyapunov cohomology of and
then also has spectral theory with
S
S
and for one sided time the ltrations Vi are related by
Vi
Vi
whereas for two sided time the splittings Ei are related by
Ei
Ei
ii Let T Z or R If both and have spectral theory then is
a Lyapunov cohomology of and
Proof i By assumption
t
t
t
thus
limtlogk t xkt limtlogk t kt
limtlogk t
xkt
limtlogk t kt limtlogk t xkt
Since is Lyapunov on an invariant set of full measure
lim
t
tlogk t
xkt
lim
t
tlogk t xkt
1

CHAPTER MET ON BUNDLES
from which all statements of the MET for follow from those for
ii We stick to the case T Z and use the cohomology relation
in the form
n
n
n
The assumptions and Lemma ii yield
lim sup
n
nlogk n kn
lim
n
nlogk n kn
lim
n
nlogknkn
Hence by Proposition i
lim sup
n
nlogk n kn
The same reasoning yields
lim inf
n
nlogk n kn
Similarly for
and n
Hence is a Lyapunov cohomology
Remark i Proposition i says that each member of an
equivalence class of Lyapunov cohomologous cocycles has spectral theory
provided one member has and that the Lyapunov spectrum S is an
invariant of a cocycle with respect to Lyapunov cohomology
ii Proposition ii says that in the set of linear RDS having spec
tral theory cohomology and Lyapunov cohomology are identical
There remains the basic problem of whether the integrability conditions of
the MET for a linear RDS on a normed linear bundle imply that has
spectral theory i e whether the MET holds The answer is a rmative for
a Euclidean bundle by the following theorem
Theorem MET on Euclidean Bundle i For a Euclidean
bundle the trivializing isomorphism can be chosen to be an isometry on
bersieif
hi fgRdhiS
then h x yiS hxyi forallxy
Integrability conditions of the MET is henceforth abbreviated as IC of the MET
1

TEMPEREDNESS LYAPUNOV COHOMOLOGY
ii Assume that a linear RDS on a Euclidean bundle satis es the IC
of the MET namely
L P where for discrete time
logk k
log k
k
the second condition only applies in the invertible case for continuous
time
sup
tlogktkt
sup
tlogkt kt
Then the MET holds
Proof i De ne a positive de nite operator A of f g Rd to itself
by
hA
x yiShxA
yiS hxyi xy
De ne a new trivializing isomorphism by
A
which is an isometry since
hx
yiS hA
x yiS hxyi
ii The isometric trivialization constructed in i obviously satis es the
integrability conditions of Proposition
hence the cocycle
t
t
t
on the trivial bundle Rd is Lyapunov cohomologous with and satis es
the IC of the MET Thus the MET holds for hence also for
Remark Classi cation of Linear RDS The basic classi
cation problems for linear RDS on normed linear bundles are to decide
whether two linear RDS are Lyapunov cohomologous to characterize the
equivalence classes by invariants and to determine in each equivalence class
the simplest possible element normal or canonical form
If we consider the classi cation problem with respect to Lyapunov coho
mology for linear RDS on Euclidean bundles for which the IC of the MET
hold then the problem can be reduced by Theorem
to considering
matrix cocycles on the trivial bundle Rd Rd endowed with the standard
scalar product for which the MET holds
1

CHAPTER MET ON BUNDLES
Let t be the matrix representation of a linear cocycle in Rd with
respect to random bases F
uiidandFtIft
is the matrix representation in a second basis G and G t then
t
Pt
tP
where P
PF
G Gl d R is the basis transfer matrix from F
to G If we want to simplify by choosing a more appropriate basis
we have to make sure that P de nes a Lyapunov cohomology
Corollary
states that if the two sided MET holds then the change
from the standard basis in Rd to a random basis which draws di orthonor
mal vectors from Ei de nes a Lyapunov cohomology Hence every cocycle
with spectrum f i di i
pg is Lyapunov cohomologous to a block
diagonal cocycle
t
t
t
pt
where i t is di di and has just one Lyapunov exponent i with mul
tiplicity di Classi cation of cocycles with the same spectrum is hence re
duced to classifying the blocks i e cocycles with one point spectrum By
changingfrom it tojdet it j di it seeRemark
we
can assume without loss of generality that the cocycle to be classi ed has
one point spectrum f g For a detailed study see Arnold Nguyen Dinh
Cong and Oseledets
Example Periodic Case We continue Example
and re
consider xt A t xt where A R Rd d is continuous and periodic
with period Let t P t etR be a real Floquet representation with
Lyapunov exponents i and Ei the corresponding splitting
We embed the equation into an RDS by associating the following er
godic metric DS to it
S
exp i
P normal
ized Lebesgue measure t
t mod The cocycle generated by
xAtxAtxis
t
t
PtetRP
Since P and P are bounded t is Lyapunov cohomologous to etR
with Lyapunov exponents i and splitting Ei
PEi

MET ON MANIFOLDS
Example Same Cocycle Under Di erent Norms The use
of appropriate random norms is a fundamental technique in the theory of
continuous and smooth RDS e g for the construction of invariant mani
folds Chap and for normal form theory Chap We will devote Sect
to their construction
Consider for example one linear cocycle on a linear bundle under two
di erent norms k k and k k Since all norms on a nite dimensional
vector space are equivalent there are nite valued random variables ci
with
ckxkkxkckxkforx
The cohomology of under k k with under k k is given by the
identity mapping
id
which together with its inverse needs
to be tempered in order to be Lyapunov The integrability conditions of
Proposition
assuring this will certainly be satis ed if the norms of
and
are bounded away from by a non random constant
see the case of two non random norms discussed in Remark
and the
case of the tangent bundle of a compact Riemannian manifold in Remark
The Multiplicative Ergodic Theorem for
RDS on Manifolds
Linearization of a C RDS
The most important and historically oldest particular case of a nontrivial
linear bundle RDS is the linearization T of a C RDS on a manifold
which will now be considered in detail
First assume that is a global nonlinear C RDS on Rd with invariant
measure The linearization or derivative of t at x Rd for xed
t is the Jacobian d d matrix
Dtx
tx
x
tyi
yj
yx
which is a linear mapping of Rd D t x Rd Rd The chain rule
and the de nition of a C RDS immediately give the following statements
Proposition Let be a C RDS over on Rd with invariant
measure Then

CHAPTER MET ON BUNDLES
iDxv
t xDt xv isacontinuouscocycle
onX Rd Rdover
ii D is a linear cocycle on X Rd over the metric dynamical system
RdF B
ttTwhere t x
t txisthe
skew product associated with
Example i If is not present D is a linear cocycle over the
deterministic ow with invariant measure on Rd a case much studied in
smooth ergodic theory mainly in its manifold version see Remark iv
belowii If the random variable x generates a stationary orbit of i e
iftx
x t then the measure dx
x dxis
invariant In this case t
D t x is a matrix cocycle over
A xedpointx Rdwith t x x isaparticularcase
We now regard Rd as a manifold and identify as usual the tangent bundle
T Rd of Rd with its global trivialization Rd Rd
TRd Rd Rd TxRd fxg Rd
With this identi cation D t x induces a linear mapping of TxRd
fxg RdtoT t xRd ft xg Rdie wealsokeeptrackofthe
motion of the base point x
t x andassuchwealsodenoteitby
TtxTxRdTtxRd
Hence T
D is a cocycle over on TRd
We now assume that we have a C RDS with invariant measure
on a manifold M The linearization or derivative T t x of t at
x M is the linear map
TtxTxMTtxMvwTtxv
for its de nition see Appendix B
As usual T denotes the mapping on the tangent bundle T M M M
which covers and is de ned on bers by
Lemma
T M M id M is a linear measurable
bundle with typical ber Rd
The proof consists in showing that there is a global measurable trivialization
which is linear on bers This follows from part i of the next lemma
1

MET ON MANIFOLDS
Lemma Global Trivialization of the Tangent Bundle Let
M be a smooth manifold of dimension d
i The tangent bundle T M M M is a linear measurable bundle with
base space M and typical ber Rd
ii If M is a Riemannian manifold with Riemannian structure h ix
then T M is a Euclidean measurable bundle and the global trivialization
TM M Rdinicanbechosentobeisometricon bers ie
x TxMhix RdhiS
is an isometry Here h iS denotes the standard scalar product of Rd
Proof We just prove ii as it contains i since every smooth manifold
possesses a smooth Riemannian metric
The tangent bundle T M M M is a smooth vector bundle with typical
ber Rd By de nition for each x M there exists an open set U containing
x and a di eomorphism covering id U which trivializes the bundle locally
ie
U U Rd and which for each x U is a linear isomorphism
as a mapping between bers
x TxM fxg Rd By Theorem
of Klingenberg
this ber mapping can be chosen to preserve
the scalar product in bers i e is an isometry between TxM h ix and
RdhiS
We can now select a countable covering by bundle charts Ui i triv
ializing T M locally and isometrically Disjointify the Ui s by putting
B Uand
Bn Unnn
iBi
to obtain a countable covering Bn of M by disjoint Borel sets Finally
put
TMMRdv
nv forv
Bn
This map is a bimeasurable bijection covering id M and is a linear isometric
isomorphism on bers
Proposition Linearization of RDS is Linear Bundle RDS
Let be a C RDS on a d dimensional manifold M over the metric DS
FP t t T withinvariantmeasure LetT onTMbedenedby
xv
txTtxvThenTis
i a continuous cocycle on T M over
ii a linear bundle RDS on T M over the skew product M F
B
ttTwhere t x
t t x with ber mappings
Ttx
x
TxM
tx
tTtxM
1

CHAPTER MET ON BUNDLES
Proof The cocycle property for T t x with respect to follows by
di erentiating
tsxts
sx
and the chain rule
The MET for RDS on Manifolds
We immediately state the two sided time version
Theorem MET for RDS on Manifolds A Global case
Let be a C RDS on a Riemannian manifold M of dimension d over the
metric DS F P t t T with two sided time and invariant measure
Consider the linear bundle RDS T on T M over the metric DS
MFB
ttTwhere t x
t txisthe
corresponding skew product ow Assume that T satis es
LP
where for T Z
x logkT
xkx
x
x logkT
xkxx
andforT R
x
sup
tlogkTtxkxtx
and
x
sup
tlogkTtxktxx
Then there exists a invariant set
M of full measure on which
the following statements hold
There are p x
p x d real numbers
px
x
x
with multiplicities di x Pp x
i di x dwherepianddiare
invariant random variables which are constant if is ergodic and a
splitting
TxMEx
Epx
x
of the tangent space TxM into random subspaces Ei x depending
measurably on
x hence are random closed sets with dimension
dim Ei x di x the so called Oseledets splitting with the following
properties Fori f p xg

MET ON MANIFOLDS
i if Pi x TxM Ei x denotes the projection onto Ei x
along Fi x
jiEj xthen
TtxPixPitxTtx
equivalently
TtxEixEitx
ii we have
lim
t
tlogkTtxvktx ix
v Ei xnfg
iii Convergence in ii is uniform with respect to v in Ei x Sd
x
The collection S
fidii
pg
is called the Lyapunov spectrum of under
B Local case Let be a local C RDS on a Riemannian manifold
M with time T R Let be an invariant measure for which satis es
L Then all statements of Part A are true and the invariant
set of full measure is a subset of graph E
Proof Part A We use the isometric global trivialization T M
M Rd of Lemma
With x TxM fxg Rd
nx
nxTnx
x
is a linear cocycle on Rd over M
n n Z which is trivially Lya
punov cohomologous to T since k x kh ix h iS Further
knxvkSkTnxxvknxvRd
whencekknxkSkkTnxkxnx
In particular satis es the IC of the MET Theorem
which gives all
statements for and thus for T by Proposition
Part B We know from Theorem
that every invariant measure
of a local C RDS is supported by the strictly invariant set
E
tRDt
tDt
Dtt
of initial values whose orbits do not explode
E
Pas
1

CHAPTER MET ON BUNDLES
Consequently each x E satis es x D t for all t R hence
T t x TxM Tt xMiswelldenedforallt R TheICis
as before where
are well de ned measurable functions on graph E
FB
We saw in Chap that the conditions for generating a local RDS by means
of a generator random or stochastic di erential equation are much milder
than those for a global RDS Therefore part B of the last theorem is a
very useful extension
Remark i The MET remains valid and the spectrum is invariant
if we change the Riemannian metric from h ix to h ix in a way such that
the smooth functions c c M
tting into
c xkvkx kvkx c xkvkx
are tempered In particular if on a compact manifold the MET holds for
one Riemannian metric then it holds for all Riemannian metrics with the
same spectrum and splitting
We later need to change from a given Riemannian metric h ix to a
random Riemannian metric h i x in TxM which depends only measurably
on
x The MET will hold under the new random norm if the ci
M
in
c
xkvkx kvkx c
x kvkx
are tempered see Sect
ii Notethatfor aC RDSwithtwosidedtime txv
T t x v is continuous for each see Theorem ii Hence the
expressions appearing in the integrability conditions of the last theorem
are nite measurable functions in
x which depend continuously on x
Therefore
L P where
supx M log kT
xk
TZ
supxMsuptlogkTtxkTR
would su ce for the validity of the MET for and for any invariant measure
see Subsect
A particular case which has been extensively studied is the deterministic
case where is not present see Katok and Hasselblatt
Mane
Pesin
Ruelle
Here is a C DS on a Riemannian manifold

MET ON MANIFOLDS
M with invariant measure on M B Then the MET holds with the
dependence on ignored If M is compact the IC are always satis ed for
any and any invariant measure since kT t x k is continuous with
respect to t x and thus bounded on a compact set
Remark Total Measure in the MET The invariant set of
full measure
M depends of course on However it has full
total measure in the sense that
for all invariant measures
satisfying the IC This eventually follows from the proof of the MET in
which rst is constructed as a forward invariant invariant set and then
we prove that it has full measure
Remark Sacker Sell Spectrum Assume that is invariant
and ergodic The MET establishes a measurable splitting
TM pM
i Ei as a Whitney sum
into p linear measurable subbundles x f x g Ei x of constant
dimension di and typical ber Rdi which are invariant with respect to
over the invariant set
M of full measure The bundle Ei
is characterized dynamically as containing exactly those tangent vectors
which have Lyapunov exponent i forwards and backwards in time
If t t T happens to consist of homeomorphisms of a compact and
dynamically connected Hausdor space and t is a C cocycle on
a compact manifold M we can consider T as a continuous linear skew
product ow over on M in which case we do not need an invariant
measure See Johnson Palmer and Sell
Sacker and Sell
and
SellThe dichotomy spectrum or Sacker Sell spectrum Sdich of T consists
ofallthose Rforwhich t x
e tT t xfailstohave
an exponential dichotomy Recall that has an exponential dichotomy
if there is a continuous projector P x on TxM and constants K
such that
ktxPx
sxkKetsst
ktxIPx
sxkKestts
for all x
Mandallst T
The spectral theorem of Sacker and Sell states that
Sdich
kj ajbj
1

CHAPTER MET ON BUNDLES
the union of k k d non overlapping compact intervals and that to
each interval there is a continuous invariant subbundle Fj x such that
we have a continuous invariant splitting
TM kM
j Fj as a Whitney sum
Ifv Fj xnfgthenthefour numbers
s xv limsup
t
tlogkT t xvk
i xv liminf
t
tlogkT t xvk
lie in aj bj Sacker and Sell Theorem
LetusdenotebyS fi i
pg the Lyapunov exponents of
T with respect to the ergodic invariant measure the notation empha
sizing the dependence on Johnson Palmer and Sell
have proved
that
Sdich fS
invariantg Sdich
where B denotes the boundary of the set B
More precisely for each a Sdich there exists an ergodic invariant
measure for which a S and for each i S there is exactly one
spectral interval aj bj from Sdich with i
ajbj andEi x
Fj xforall x
where
FjxM
ii
ajbjEi x
For each invariant the measurable invariant splitting of the MET thus
forms a re nement of the continuous invariant splitting of the Sacker Sell
theory The price we have to pay for the re nement of the splitting is
the loss of its continuity
Johnson
constructed an example of an almost periodic linear dif
ferential equation in R for which Sdich a b but S fa a b g with
aab
We refer to Colonius and Kliemann for an extensive treatment of
the various spectral theories of linear nonautonomous systems

MET ON MANIFOLDS
Random Di erential Equations
We now search for conditions on the right hand side of an RDE
xtftxt
on a Riemannian manifold M which su ce for the IC of the MET
ForM Rd
generates a unique global C RDS provided
f LP Cb see Theorem
By de nition and the mean value
theoremf LP Cb ifandonlyifkf k L P and
sup
xRdkDfxkLP
Under this condition we have globally on an invariant set of full P
measure
t x xZtfs
s xds
DtxIZtDfs
s xDs xds
and
Dtx
IZtDs xDfs
s xds
From
and
by Gronwall s lemma
sup
tlogkDt xkZkDfs
s xkds
Suppose is a invariant measure and kDf k L then the right
hand side hence the left hand side of
isinL andtheICofthe
MET are ful lled But by
kDf k sup
xRdkDfxkLP
hence under our conditions for the existence of a global RDS the IC of the
MET are automatically satis ed for any We summarize this as the part
A of the following theorem
Theorem MET for RDE Consider the RDE
AGlobalcaseonRd Ifkf k L PandsupxRdkDf xk
LPthen
uniquely generates a global C RDS which satis es the
IC of the MET for any invariant measure
1

CHAPTER MET ON BUNDLES
B Local case for Riemannian manifold M Assume f LP C
Then
uniquely generates a local C RDS Suppose is a
invariant measure which satis es
krfkL
Then satis es the IC of the MET under
Proof It remains to prove part B
The existence and uniqueness of is guaranteed by Theorem
As
sume that x E Then T solves vt T f t vt which is equivalent
to the pair consisting of the original RDE and the variational equation
vtrfttxvt
Using polar coordinates in T M see Sect
we obtain
logkT t xvk
logkvkZthS u xsrfu
u xSu xsidu
where s
v
kvk and S is the RDS on the unit sphere bundle SM induced
by T Using the elementary fact that for any A Rd d and any basis of
unit vectors vi of Rd kAk d max i d kAvik we obtain from
x
sup
t logkT t xk logdZtkrfu
u xkdu
The same estimate is obtained for by using the same argument for
T
which is generated by vt rf vt
For both the global and the local case Liouville s formula yields for ergodic
d
pX
i di i dEdivf dEtracerf
Recall that for M Rd rf Df
Exercise Continuation of Exercise
The random dif
ferential equation in R
xtatxtbtxN
tN
abLP
generates a local C RDS Verify the validity of the IC for its invariant
measures and determine the corresponding Lyapunov exponents For the
cases N and N see Subsect

MET ON MANIFOLDS
Stochastic Di erential Equations
Now consider the SDE
dxt
mX
jfjxt dWjt
on a Riemannian manifold M We assume that for some
fCffmC
Then
generates a local C RDS over the canonical DS modeling
Brownian motion Theorem
We would like to nd conditions in terms of the vector elds fj and
the invariant measure which imply the IC of the MET The task is com
plicated by the fact that
generally depends on the whole history
F F of the Wiener process
For example if M is a compact manifold
generates a global
C RDS see Theorem iv It was rst proved by Carverhill
Appendix A for M a bounded open domain of Rd and Baxendale
Proposition that automatically satis es the IC of the MET for any
This follows from the stronger result
E sup
xMsup
tkTtxk
The latter result was considerably improved by Kifer
who showed that
for any p
E sup
tktkk sup
tktkk
p
wherek kkisthe norminCkMM providedf Ck f fm
Ck for some k
We hence have the following statement
Theorem MET for SDE on Compact Manifolds Let M
be a compact Riemannian manifold and let the SDE
satisfy
Then the global C RDS it generates satis es the IC of the MET Theo
rem for any invariant measure
Let now M Rd and assume the stronger conditions
fCbffmCb
mX
j Dfjfj Cb
1

CHAPTER MET ON BUNDLES
where Df f Pd
i f i xi f Then also the Ito Stratonovich correction
term
fx fx
mX
j Dfjfj x
where holds for time R and holds for R satis es f Cb
Under
the SDE
generates a global C RDS Further
theJacobianD t x of t atxsolves
dvt
mX
j DfjxtvtdWjt Dfxtvtdt mX
j Dfj xt vtdWjt
Using polar coordinates r jxj
s
x
jxj Sd as in the linear
case see Sect we obtain that the SDE
is equivalent to the two
SDE
dSt
mX
jhjStdWjtSsSd
and
dRt
mX
j qjStRt dWjt QStRtdt mX
j qj St RtdWjt
Rr
Here hj Ajs hAjs sis qj hAjs siand
QhAssi mX
j hBjs si hAjs Ajsi hAjs si
where Aj Dfj x and
Bj DAjxfjx Ajx Dfjxfjx
istheJacobianofgjx Ajxfj x Hencejqj x sj kAjxkand
jQxsjkAxk mX
jkBjxk kAjxk Qx
It su ces to deal with the rst IC
L Integrating
pro
ceeding as in the linear case see Remark
denoting the marginal of
on Rd by and the integral with respect to a measure by E
E
logd EQ
max
i
mX
j EPE sup
tZtqj u xSu xeidWju
1

MET ON MANIFOLDS
where ei is the standard basis of Rd Due to our assumptions
Q x C hence E Q for any The expectation of the stochastic
integrals in
can in general not be estimated by means of classical
stochastic analysis
There is however one important particular case which permits the
use of Burkholder s inequality namely the case where is a forward or
backward Markov measure i e where
is measurable with respect
to the past F or with respect to the future F of the Wiener process W
see Subsect
Theorem MET for SDE Invariant Markov Measure
Consider the SDE
in Rd
A Global case Let
satisfy conditions
so that it gener
ates a global C RDS and let be a invariant Markov measure Then
the IC of the MET are automatically satis ed
B Local case Let
satisfy the conditions
so that it
generates a local C RDS and let be a invariant Markov measure with
marginal Then the IC of the MET are satis ed provided kDf k L
andkD DfjfjkkDfjk L forj
m
C The invariant random variables p di and i from the MET are
independ of
Proof A Restricting ourselves to forward Markov measures they are
by Theorem
in a one to one correspondence with the stationary mea
sures of the one point Markov process generated by
for time R
equivalently with the solutions of the Fokker Planck equation L
LfPm
j fj Further restricting the RDS to one sided time
T R and to the algebra F corresponds to the invariant product
measure
P
Making this restriction for the stochastic integrals in
Burkholder s inequality for any p
and s Sd and then Jensen s
inequality for p yields
E sup
tZtqj u xSu xsdWju
p
CpE EPZkAj u xkdu
p CpEkAjkp
By
kAj x k C hence the right hand side of
is nite for
any p and any Consequently even E
p foranyp
B The estimates
and
using for example p in
show that the IC of the MET are met provided the conditions
listed in the theorem are satis ed
1

CHAPTER MET ON BUNDLES
C If a random variable f Rd R is invariant and P is an
invariant measure for the RDS restricted to time R then f x f x
P a s for some measurable f This is a consequence of the ergodicity
of and the product structure of the invariant measure for details see Kifer
x
Theorem A covers in particular the case of a linear SDE under the
Markov measure see also Theorem
and Remark
We are not able to extend these simple statements from Markov mea
sures to arbitrary invariant measures We have however the following
quite satisfactory general criterion see Arnold and Imkeller
Theorem MET for SDE General Invariant Measure
Let the SDE
in Rd satisfy the conditions
so that it generates
a global C RDS Assume further that Df is globally Lipschitz Let
be a invariant measure with marginal on Rd which satis es
E ZRdqlog kxk dx ZRdqlog kxk dx
Then under meets the IC of the MET
We refrain from giving the proof here as it is based on techniques which
are not used otherwise in this book
Example Continuation of Subsect
In Subsect
we determined all invariant measures necessarily Dirac measures of the
local RDS generated by the SDE
dxt
xt xN
tdt xtdWt
on the state space R where N
R and
We now calculate
the Lyapunov exponent for each of these measures
The linearized RDS vt D t x v satis es the linearized SDE
dvt
NtxN vtdt vtdWt
hence
DtxvvexptWtNZt
s xNds
Thus if
x is a invariant measure its Lyapunov exponent is
lim
t tlogkDt xvk
N lim
t tZt
s xNds
NExN
1

MET ON MANIFOLDS
provided the IC xN L P is satis ed
Case N even
iFor R theICfor
is trivially satis ed and we obtain
so is stable for
and unstable for
ii For
d is F measurable hence the density p
of E satis es the Fokker Planck equation
Lp
x
x xNp
xp
which has the unique probability density solution
pxNx
exp
xN
N
x
see Ariaratnam Equation
Since
ExN EdNZxNpxdx
the IC is satis ed The calculation of the Lyapunov exponent is accom
plished by observing that
dtN
eN
tWt
NRteN
sWsds
t
N
t
where
tZt
eN
sWsds
Hence by the ergodic theorem
EdN N lim
ttlogt
nally
N
iii For
d is F measurable Since L d
Ld andNiseven
EdN
EdN
1

CHAPTER MET ON BUNDLES
thus
N
Case N odd
iFor R
and
ii For
d
and
d
are F measurable
The solution p of the Fokker Planck equation corresonding to is given
by the same expression as for the case N even and p x p x The
IC can be veri ed directly using the expression for p and
EdNEdN
thus
N
A detailed analysis of the bifurcation behavior of this RDS for the cases
N and N is given in Subsect
Random Lyapunov Metrics and Norms
One of the key technical steps in the construction of invariant manifolds of a
deterministic di eomorphism at the hyperbolic xed point is the change
from the standard Euclidean norm in Rd to a norm under which D is
contracting on its stable space and expanding on its unstable space See
e g Irwin
p
or Ruelle
p These new norms are called
adapted to or Lyapunov norms
This is also possible in fact essential in the random case For the trans
fer of the results of the MET from the linearized RDS t
Dt
to the original nonlinear RDS t in a neighborhood of we need control
of the non uniformity in the MET which is due to the fact that Lyapunov
exponents are an asymptotic concept Since the non uniformity in t
that has to be suppressed by the new norm depends on the new norm
will inevitably depend on too
The construction of the new random norms has to be done carefully
since we want our linearized cocycle under the new norm to be Lyapunov
cohomologous to itself under the original norm in order not to change the
Lyapunov spectrum
This section is based on the now classical work of Pesin
followed
by that of Fathi Herman and Yoccoz
and of Pugh and Shub
who all treat the case of the iteration of a deterministic di eomorphism on

RANDOM LYAPUNOV METRICS AND NORMS
a manifold M which leaves a measure invariant smooth ergodic theory
We go beyond these papers and follow Boxler
and Dahlke
who
develop random versions of the construction in which
every subspace from the Oseledets splitting is equipped with a new
scalar product
the dynamics in both time directions t
and t
is taken
into account
time can also be continuous
In this section we make the following standing assumptions on our un
derlyingDS FP ttT
timeistwosidedT ZorT R
is ergodic this assumption is only made to simplify presentation
The Control of Non Uniformity in the MET
We treat the cases T Z and T R as one Assume that a matrix cocycle
t is given which satis es the IC of the MET Theorem
We will now introduce an improved temperedness property
De nition Slowly Varying Random Variable Given
and a DS A random variable R
is called slowly varying
withrespectto ifPas
ejtjR Rt
ejtjR forall t T
Lemma i Each of the following conditions is equivalent with R
being slowly varying
R t R ejtjforalltTPas
e jtjR
R t foralltTPas
R is slowly varying
ii If R is slowly varying then it is slowly varying for
and
cR is j j slowly varying for each c
and R
iii If R and R are slowly varying then so are R R
maxR R andminR R FurtherR R is slowlyvary
ing Also the pointwise limit of slowly varying random variables is
slowly varying
1

CHAPTER MET ON BUNDLES
The proof is elementary
Proposition Tempered Versus Slowly Varying i If R
is slowly varying for some
then it is tempered
ii Conversely if f
is tempered and in case T R
ift logf t iscontinuousPas thenfor any
there is an
slowly varying random variable R for which
RfR
Proof i We have
lim inf
t
tlogR t limsup
t
tlogR t
whence the result follows from Proposition i
ii By de nition and utilizing the continuity of t log f t in the
case T R there is for each
a C in general not slowly varying
with
Cejtjft Cejtj
for all t T Part i of the proof of the next theorem shows that there is
then a slowly varying random variable R for which
R
fR
Theorem Non Uniformity of MET is Slowly Varying As
sume that the linear cocycle with two sided time satis es the IC of the
MET Then for each
there exists an slowly varying random variable
R
such that on the invariant set of the MET the cocycle
has the following properties
iForeachi
pxEitT
R eitjtjkxkkt xkR eitjtjkxk
ii IfMandNaredisjointsubsetsofRandif EM EN
is the angle between EM
i MEi
and EN
i NEi denedby
cos EM EN
max
xEM yEN
xy jhxyij
kxk kyk
1

RANDOM LYAPUNOV METRICS AND NORMS
then
R
EM EN
Proof We work on the invariant set throughout
AssumewehavefoundR foriandR foriithenR
max R R will satisfy both i and ii
i Let
The MET implies that there is a random variable N
such that for each t T with jtj N
i
pxEi
eitjtjkxkkt xkeitjtjkxk
N
does not depend on x due to the uniform convergence of
limt
tlogkt xk ionthesetEi Sd Denetheran
dom variables
A
min
i
p min
tT min
xEikt xk
e it jtjkxk
A
max
i
p max
tT max
xEikt xk
e it jtjkxk
We have A
and A
since mint T
and maxt T
outside of a nite interval of times and since for T R it is part of the
de nition of a linear cocycle that t
t is continuous
NowtakeB min A
B maxA
and nally
C
max B
B
Then for all i
pxEitT
C eitjtjkxkkt xkC eitjtjkxk
C is in general not yet slowly varying Rather by the cocycle property
foranyst T
max
xEisktsxk
e it jtjkxk
max
yEikts
syk
eitjtjks yk
max
yEiktsyk
eitjtjks yk
1

CHAPTER MET ON BUNDLES
max
yEiktsyk
eits jtsjkykeisjsjkyk
ks ykejsjjtsjjtj
e jsj max
yEiktsyk
eitsjtsjkyk min
yEiksyk
e is jsjkyk
whence
AsAAejsj
With a similar estimate for A we obtain
CtCejtj
Working with t instead of we arrive at
pCejtjCt CejtjforalltT
It will turn out to be crucially important that for continuous time t
C t is continuous hence sup s and inf s are measurable with no
need of completion of F as follows from
and the continuity of
t
t We now prove that there exists a slowly varying random
variable R with
R
C
R
Indeed put
R
inf
tTCtejtj
R
sup
tTCtejtj
R and R are measurable and R
R
by
Hence
RejtjCt RejtjforalltT
Nowforallts Tby
Cts Rejsjejtj
and also
CtsCts
Rsejtj
By de nition R is the smallest number satisfying the right hand side
inequality of
for all t and all
Comparing
with
R s R ejsjforallsT
1

RANDOM LYAPUNOV METRICS AND NORMS
i e R is a slowly varying random variable Similarly
RejsjRsforallsT
Finally
RmaxRR
is slowly varying and satis es
and thus by
the assertion
i of the theorem
ii By Corollary
the angle is tempered on For each
there are thus random variables M M for which for all t T
MejtjEMtENt
M ejtj
By the same procedure as in the proof of i we construct now a slowly
varying random variable RM N satisfying for all t T
RMNejtjEMtENt
RMN e jtj
Taking nally
R maxfRMNMNg
we have found a slowly varying random variable which satis es ii
Corollary Choose and then x a constant
Then for each
there exists an slowly varying random variable G
such that on the invariant set for all i
pxEitT
G eitjtjkxkkt xkG eitjtjkxk
Proof Take
min
and choose a slowly varying random vari
able R according to Theorem
Then G R is slowly varying
since
and satis es equation
since
Random Scalar Products
The following is the main theorem of Sect
Theorem Random Scalar Product Assume that the linear
cocycle satis es the IC of the MET with two sided time Choose and

CHAPTER MET ON BUNDLES
then x a constant
Introduce on the invariant set of the MET
and for any x
p
i xiandy p
i yi withxi yi Ei
hxyi pX
i hxi yii
where for ui vi Ei
huivii Rhtuitvii
eitjtjdtTR
PnZhn ui n vii
einjnj
TZ
Puthxyi hxyifor
Then
i h i is a random scalar product in Rd which depends measurably
on and under which the Ei are orthogonal
ii For each
there exists an slowly varying random variable
B
such that
BkkkkBkk
where
kxk hxxi pX
i kxik
and
kxik Rktxik
eitjtjdtTR
PnZkn xik
einjnjTZ
de nes the square of the random norm corresponding to h i
iii The function
xyt hxyit
is continuous with respect to the Euclidean metric in Rd Rd T and
R Inparticular xyt ht x t yit andxt
k t xk t are continuous
iv Foralli
pxEitT
eitjtjkxkkt xkt
e it jtjkxk
Proof i For T Z note rst that the series in
converges for
each
since
jhuvij X
nZkn uk
einjnjkn vk
einjnj

RANDOM LYAPUNOV METRICS AND NORMS
GX
nZ
e jnj kuk kvk
where we have applied Corollary
with replaced with and the
fact that G is an slowly varying random variable Clearly hx yi is
measurable with respect to as a series of measurable random variables
For each
it is a scalar product in Rd and by de nition hx yi
if x Ei and y Ej i j For T R the arguments are essentially
the same
ii First take T Z We rst verify the estimate of kxk from below
The series de ning hx yi in Ei contains in particular the summand
hxyifor n hencekxk kxk inEi Forageneralx
p
ixi
this yields
kxk pX
i kxik pX
i kxik ppkxk
The estimate of kxk from above is more complicated Denote by
iRdEi
the projection onto Ei
along Fi
j iEj
Then for x
Pp
iix
kxk pX
ikixkkixkX
nZkn
ixk
einjnj
Choose an slowly varying random variable G corresponding to ac
cording to Corollary
Then
ki xkcG kikkxk
wherec c PnZe jnj
Consequently
kxk pcG kxk pX
ikik
To estimate the norm of i which is an orthogonal projection in the
new norm so k i k we use the following lemma
Lemma Norm of Projection Suppose that U h i is a vec
tor space with scalar product and U U is a projection i e
Let im ker
be the angle between im and ker Then
kk sin
1

CHAPTER MET ON BUNDLES
For a proof see Pugh and Shub p
By Lemma
kiksinEiFi
By Theorem
there is an slowly varying random variable R
for which for all i
psinEi Fi
Ei Fi
R
thus
kikR
Altogether
kxkpcpG RkxkBkxk
where the random variable B
is slowly varying by Lemma
We now treat the case T R The proof of the upper estimate is the
same as for T Z Denote the resulting slowly varying random variable
by B For the lower estimate take the slowly varying random variable
G from Corollary
and use the lower estimate in
to obtain
kixkZktixk
eitjtjdt
Gckixk
and hence
kxk
pX
iki xk
pc
G
pX
iki xk
pc
Gpp
pX
iki xkpc
ppG kxk
B kxk
The nal slowly varying random variable is B max B B
iii This follows readily from the de nition
the cocycle property
and the crucial fact that by our de nition of a linear cocycle x t
t x is continuous A detailed proof is given by Nguyen Dinh Cong
Theorem
1

RANDOM LYAPUNOV METRICS AND NORMS
ivForTZxEi andsT
ksxksX
tZkts
sxk
eitjtj
X
tZkts xk
eitsjtsjeisjtsjjtj
e is jsjkxk
analogously for the lower bound and for T R
Remark Lyapunov Metric Lyapunov Norm The quanti
ties hx yi and kxk and their relatives on the tangent bundle are called
the Lyapunov metric and the Lyapunov norm e g by Pesin
Rd
withh i onthe berfg Rdis aEuclideanbundleinthe sense of
De nition
We stress that the Lyapunov metrics hx yi also depend
on the xed parameter
Remark Uniformity Properties of in the Random
Norm i The key property of k k is that the random factors which are
still present in equation
and which account for the non uniformity
of the cocycle in the original norm have disappeared in equation
An equivalent formulation of
is
eitjtjktjEikt
eitjtj
ForT Zandn
and n
we e g obtain the uniform bounds
eikAjEik ei
eikAjEikei
Ifi
i
and is so small that i
i
then
t is contracting expanding on Ei for t
ii More generally if M R and EM
i MEi then for each
x
i Mxi EM k t xk t iscontinuouswithrespecttotby
Theorem iii and
X
iM
eitjtjkxikktxktX
iM
e it jtj kxik
Withmin miniMi max maxiMiweobtain
for t
emin tkxkktxkt
e max tkxk
1

CHAPTER MET ON BUNDLES
for t
emax tkxkktxkt
e min tkxk
iii For M
M
M f g we obtain the stable
unstable and center subspaces
Es
iEi
Eu
iEi
Ec
iEi
For
small enough the cocycle in the new norm is thus uniformly
contracting on Es and uniformly expanding on Eu hence uniformly
hyperbolic if Ec is not present
More speci cally if is any number in minf s
u g where
s max
ii
u min
ii
then ktjEskt
e t forallt
ktjEutkt
e t forallt
and on Ec if present at all
ejtjktjEckt
ejtj forall t T
iv ForM R
eptktkt
e
tfort
e
tktkt
ep tfort
and so together with max
p
e
jtjktkt
e jtjforalltT
e
jtjktkt
e jtjforalltT
i e the cocycle and its inverse are uniformly bounded in the random norm
Corollary The cocycle t with respect to the Lyapunov
normkk for any
xed is Lyapunov cohomologous to itself
with respect to any xed norm in Rd
Proof By Remark
we can assume without loss of generality that
the xed norm is the standard Euclidean norm k kS By Example
we have to check whether we have suppressing
lim
t
t logkidkS t
lim
t
t logkidk t S
1

RANDOM LYAPUNOV METRICS AND NORMS
where kid kS
sup
x kxk
kxkS and kidk S sup
x kxkS
kxk
By Theorem ii for each
B kidkS B
B kidkS B
which implies
by Proposition i
Corollary Projections to Oseledets Spaces Let M and N
be disjoint subsets of R whose union is R and let M N be the projection
onto EM
i MEi along EN
i NEi Then for each
there exists an slowly varying random variable R such that
kMNkR
In particular M N is tempered
Proof Combine Theorem ii with Lemma
to obtain the rst
assertion Then use Proposition i
Corollary Strong Version of the MET Assume that
the matrix cocycle t is written in the standard basis S ui of
Rd endowed with the standard scalar product and its corresponding Eu
clidean norm Construct a new random basis G
vi
adapted
to the Oseledets splitting Rd E
Ep
as follows Pick
measurable unit vectors in the standard Euclidean norm in such a
way that the v
vd
are orthogonal and taken from E the
vd
vdd
are orthogonal and taken from E
and the
vd
dp
vd
are orthogonal and taken from Ep The
matrix representation of the cocycle with respect to the bases G and
Gtis
t
PttP
where P
PS
G is the basis transfer matrix from S to G Then the
cocycle t is block diagonal
tdiagt
pt
where the blocks i t are cocycles of size di and one point spectrum
f i di g Moreover the cocycles and are Lyapunov cohomologous
1

CHAPTER MET ON BUNDLES
Proof The existence of a measurable selection of a basis G is guaran
teed by the Measurable Selection Theorem Proposition
since by the
MET the Oseledets spaces Ei are random closed sets the procedure is
carefully carried out by Crauel and Walters
The columns of Q P
are the coordinates kS vj
where
qij is obtained as the orthogonal projection of the unit vector vj onto
the subspace Rui thus jqij j and hence kP k C
Further the columns of P are the coordinates kG uj obtained
by rst applying one of the projections i Rd Ei and then by
orthogonal projection onto one of the vk in Ei Thus
jpijjkiksini
where i is the angle between Ei and Fi
j iEj see Lemma
By Theorem ii there exists for each
an slowly varying
random variable R
for which sin i
iRthus
CkPkCR
Analogously
CRkPkC
Summarizing we have the following lemma
Lemma Let P be the basis transfer matrix from a xed basis
to a basis adapted to the Oseledets splitting Then there exists a constant
C and for each
there exists an slowly varying random variable
R such that
CkPkCR
CRkPkC
Continuing the proof of Corollary
the lemma implies in particular
that P and P are tempered hence P is a Lyapunov cohomology between
the two cocycles
Random Riemannian Metrics on Manifolds
Consider now a C RDS t on a Riemannian manifold M h ix over
an ergodic DS F P t t T with two sided time Assume that is an
ergodic invariant measure The linearized cocycle T is a linear bundle
RDS over the skew product t x
ttx
1

RANDOM LYAPUNOV METRICS AND NORMS
Assume that T satis es the IC of the MET Theorem
with respect
to Using the global trivialization T M M Rd such that on bers
x TxMhix RdhiS isanisometryseeLemma
all
results of the last subsection and their proofs carry over verbatim since they
can be transferred from the cocycle t x
txTtx
x onRdtoT t x where
hTtxuTtxvitxhtxxutxxviS
Theorem Non Uniformity of MET is Slowly Varying
Let beaC RDSonaRiemannianmanifold Mh ix Let bean
ergodic invariant measure and assume that T satis es the IC of the
MET with respect to Theorem
Then for each
there is an
slowly varying random variable R
M
over t such that
on the invariant set
M of full measure the cocycle T t x
has the following properties
iForeachi
puEixtT
R xeitjtjkukxkTt xukt xR xeitjtjkukx
ii IfMandNaredisjointsubsetsofRandif EM x EN x
is the angle between EM x
iMEi xandEN x
i NEi xdenedby
cosEM xEN x
max
uEM xvEN xuvjhuvixj
kukx kvkx
then
Rx
EMxENx
Corollary Choose and then x a constant
Then for
each
there exists an slowly varying random variable G
M
such that on the invariant set
Mforalli
p
uEixtT
G xeitjtjkukxkTt xukt xG xeitjtjkukx
Theorem Random Riemannian Metric Let be a C
RDS on a Riemannian manifold M h ix Let be an ergodic invariant
measure and assume that T satis es the IC of the MET with respect to
1

CHAPTER MET ON BUNDLES
Choose and then x a constant
and introduce for any x
and u
p
iuiv
p
iviTxMuiviEi x
huvix
pX
ihuiviix
whereforui vi Ei x
huiviixRhTtxuiTtxviitx
eitjtj
dtTR
PnZhTn xuiTn xviin
x
einjnj
TZ
Puthu vi x hu vixfor
x
Then
i h i x is a random scalar product on T M which depends measur
ably on
x i e is a measurable Riemannian metric and under which
the Ei x are orthogonal
ii For each
there exists an slowly varying random variable
B
M
such that for the corresponding measurable Lyapunov
normkk x
B xkkxkkxB xkkx
iii For all i
puEixtT
eitjtjkukxkTtxuktx eitjtjkukx
Corollary The cocycle T t x with respect to the Lyapunov
normkuk x huui xisforany
Lyapunov cohomologous
to itself with respect to the norm kukx hu uix associated with the Rie
mannian metric on M
Remark i Remark
carries over to the manifold case in
an obvious way In particular kT t x k x t x is uniformly
bounded under the Lyapunov metric and in case S
and for
small enough T t x is uniformly hyperbolic
ii Pesin
Fathi Herman and Yoccoz
and Pugh and Shub
considered the case T Z and independent of the classical case
of smooth ergodic theory See also Katok and Strelcyn for an account
and generalizations of Pesin s theory and Liu and Qian
for Pesin s
theory for RDS

Chapter
The MET for Related
Linear and A ne RDS
Summary
Let V be a nite dimensional vector space and A V V be a linear
operator Then the standard constructions of linear algebra yield A on V
A on the dual space V kA on kV the k th exterior power of V where
k d in particular dA detA on dV If U V is A invariant
A induces operators AjU on U A on the quotient space U V U and
A on the orthogonal complement U Further if B W W is another
linearoperator wehaveA BonthedirectsumV WandA Bonthe
tensor product V W
Assume now that and are linear cocycles on V and W respectively
In Sects to
we will systematically study the multiplicative ergodic
theory of those cocycles obtained from the given ones by means of
i the above constructions i e of
k in particular d
det
and
ii time reversal i e by considering
iii by studying the above objects over and
The results of our study will be used in subsequent chapters of the book
e g exterior powers in the theory of rotation numbers Sect
tensor
products and a ne RDS in normal form theory Chap
In Sect we apply multiplicative ergodic theory to the simplest possi
ble nonlinear cocycles namely those taking their values in the a ne group
in the discrete time case also known as iterated function systems The main
result is Theorem
on the existence of a unique invariant measure for
hyperbolic a ne RDS
1

CHAPTER MET FOR RELATED LINEAR RDS
Inverse and Adjoint
Suppose we are given a linear cocycle on Rd over an ergodic DS
FP t tT withtwosidedtimeT ForT ZseeSect
and
with A
n
An
A
n
I
n
AnA
n
andtheICoftheMETarelogkA k L P Wesayasusualthat
is generated by A
For T R the two ways of generating a linear cocycle are by solving a
linear RDE real noise case see Sect
xtAtxtA
Rdd
or a linear SDE white noise case see Sect
dxt A xtdt mX
jAjxtdWjtAA AmRdd
In case
theICfor areimpliedbyA L P andwesaythat
is generated by A In case
the canonical underlying DS the shift
on Wiener space is ergodic and the IC of the MET are always satis ed see
Theorem
Wesaythat isgeneratedby A A AmW
Once is generated by one of the above mechanisms we can consider
it as a Gl d R valued cocycle see Sect
to which we can apply an
arbitrary combination of
i the operation of time reversal
ii matrix inversion
iii matrix transposition
and by considering the result over or
This yields di erent ob
jects
of which four turn out
to be cocycles namely
An
other four are Gl d R valued backward cocycles namely
while the remaining eight objects
etc are neither cocycles nor backward cocycles By the canonical left ac
tion g x gx of Gl d R on Rd the four cocycles induce linear cocycles
In the whole Chap we assume that the underlying DS is ergodic but only
to simplify presentation All statements hold with obvious modi cations in the non
ergodic case too
1

INVERSE AND ADJOINT
while the four backward cocycles induce linear backward cocycles on Rd
see Remark
We have the following facts about generators spectrum and splitting of
the four resulting linear cocycles
Theorem MET for Inverse and Adjoint Let
be the
linear cocycle generated by A with log kA k L P in case T Z
by AwithA LPincaseofanRDEandby AA AmW
in case of an SDE Let
S
be its Lyapunov spectrum and
E
Ep its splitting Then the following holds
i The cocycle
has generator A in case T Z
A intheRDEcaseand A A
AmW intheSDEcase
Its spectrum is
and its splitting is
Fp
F
where Fi
jiEj i
p
ii The cocycle
has generator
A
in case
TZ
A incaseofanRDEand
AA AmW
in the SDE case Its spectrum is
and its splitting is Ep
E
iii The cocycle
has generator A
in case
TZ
A intheRDEcaseand A A
AmW in
the SDE case Its spectrum is and its splitting is F
Fp
Proof iii follows from i applied to the data in ii so we only need
to prove i and ii
i The generator of
is obtained for T Z by applying
to
For the RDE case we di erentiate
I with respect to time
t to obtain
use A and transposition to arrive
at
A
Formally the same calculation goes through for a
Stratonovich SDE
The cocycle t
t
satis es
t
t
t
by the MET but
t
t
t
t
t
t
hence
In this chapter all splittings are written in the order of decreasing Lyapunov
exponents
1

CHAPTER MET FOR RELATED LINEAR RDS
In particular following the proof of the MET
ppdkdpk
k
p kand
Vk
Vpk
this is a one sided result we only need one sided time and t
Gl d R for this to hold Now everything applied to the cocycle t
over following now the proof of the two sided MET
gives the
spectrumpppdpkdkdpkpk
k
pk
and the backward ltration
VpkVk
Bydenitionfor i p
Ei
ViVpiVpi
Vi
Remembering that
Vpi
pjp iEj
Vi
pi
jEj
and noting that for subspaces U V we have U V
UVwe
obtain
Ei
jp iEj
Fpi
ii The calculation of the generators of t
tforTZ
andtheRDEcaseareasabove IntheSDEcaseforallx Rd andt R
txxZtAsxds
mX
jZtAj s x dWjs
with the interpretation of the two sided stochastic integral given in Sect
By the calculus developed there
Ztfs dWsZtf s dWs
Using this and inverting time in the Lebesgue integral gives
txxZtAsxds
mX
jZtAj s x dWjs
i e has generator
AA AmW
The rest of the statement follows readily from
lim
t
tlogk t xk lim
t
tlogkt xk i
ifandonlyifx Ei

INVERSE AND ADJOINT
The cocycle being generated by A means in more detail in the RDE
case that it solves xt A t xt in particular xt A t
xt if
the original system has the form xt A t
xt etc
Remark More Cocycles Inspecting the generators of the four
linear cocycles above e g for the RDE case shows that some generators are
missing in the list of Theorem
namely the family A
A
A
A These are the generators of the four cocycles obtained
from the four Gl d R valued backward cocycles through the right action
g x g x of Gl d R on Rd Spectrum and splitting of the members
of this family are again closely related with each other the statement anal
ogous to Theorem
is left as an exercise but have in general nothing
to do with the spectrum and splitting of the rst family hence there is no
general relation between S A and S A S A However the
following condition ensures equality
Proposition i Let for T Z or in the RDE case the generating
process A of a linear cocycle be time reversible i e
LAttTLAttT
where L denotes the probability law of the random variable Then
LALA
henceSASA
ii In case of an SDE we always have
LAAAmWL
AAAmW
henceSASA
Proof iForT Zandn
LAn
LAn
A
LAn
A
LAn
Similarly for n
and any nite dimensional distribution of A
ForT Rthesolutionof t At t
I is a certain
measurablefunctionofA fA IfLA LB thenLfA
LfB
Relation
implies
L lim
t
At
At
t Llim
t
At
At
t
1

CHAPTER MET FOR RELATED LINEAR RDS
andhenceLS A
LS A intheergodiccaseS A
SA
ii The solution of dxt A xtdt PAjxt dWj and dyt A ytdt
PAjyt dWjwithW W havethesamelawsinceLW LW
The hypothesis of time reversibility cannot be dropped for an example see
KeyFor T Z time reversibility is implied by exchangeability and exchange
abilityisimpliedby A n n Zbeingiid
For the RDE case assume that xt A t
xt where t t RisaFeller
Markov process on a locally compact state space E with a countable base
see Appendix A
Then time reversibility follows from reversibility
or detailed balance of the transition probability of which in turn is
equivalent to the generator being self adjoint in L E where L
see e g Ikeda and Watanabe
p
MET on Linear Subbundles and Quo
tient Spaces
Let in this subsection be again a linear cocycle in Rd over an ergodic DS
with two sided time
Recall from Sect that a mapping
U taking values in the
set of linear subspaces of Rd is called a random linear subspace if
dxU ismeasurableforeachx Rd wheredxC inffjx yj y
Cg
Lemma If U and V are random linear subspaces then so are U
U VandU V Further
dim U is measurable
TheproofisleftasanexercisenoteegthatU V U V
A random linear subspace U
Rd is called invariant with respect
to if
tU
U t forallt
Since m
dim U is a invariant random variable and thus a con
stant m U is a measurable linear bundle over F P t t T with
typical ber Rm An important particular case is a deterministic invariant
subspace U Rd
The restriction of on U
Ut
tjUU
Ut
1

MET ON LINEAR SUBBUNDLES
is a linear cocycle on the bundle U
We de ne a second linear cocycle
on the quotient space U
Rd U of equivalence classes with respect to the equivalence relation
xy
xyUby
U
x
tx
txtUt
where x is the equivalence class containing x Rd over the ber
Finally let U be the orthogonal complement of U in Rd with
respect to the standard scalar product and let the cocycle
on the
linear bundle U be de ned by
U
x
tx
txtUt
where y y
y is the orthogonal projection onto U over
The restriction of the canonical projection
RdU
x
x toU denesanisomorphismofU andU
which is an
isometryifwede neascalarproductonU byhx y i hx yi
With this done we can identify the cocycles
onU and onU
for our purposes
Choose an adapted measurable random basis
F
v
vm
vm
vd
inRd ie suchthatF
v
vm is an orthogonal basis of
U andF
vm
vd is an orthogonal basis of U
Then represented in the bases F F t takes the Lyapunov coho
mologous exercise form
t
t
tt
where the cocycle t on Rm represents U t in the bases F
inU andF t inU t whilethecocycle t onRd m
represents t in the bases F
vm
vd
of
UandFtofUt
and represents t in the bases
F
vm
vd
ofUandFtofUt
We will now determine the spectrum and splitting of U and and
their relation to the spectrum and splitting of As far as the spectrum is
concerned we can basically follow Crauel Appendix A who did the
case of a deterministic invariant subspace and who generalized earlier work
by Furstenberg and Kifer Lemma Hennion Proposition
and Key Theorem
Note that U is in general not invariant with respect to
1

CHAPTER MET FOR RELATED LINEAR RDS
Theorem MET for Subbundle and Quotient Let be a
linear cocycle in Rd over the ergodic DS F P t t T with two sided
time which satis es the IC of the MET Let have spectrum S
fidii
pg and splitting E Ep Assume that the random linear
subspace U
Rd is non trivial and invariant Then
i The cocycle U jU satis es the integrability conditions of the
MET and has spectrum
S UfkidU
kii
rg
and splitting
EkUEkrU
where
kk
kr p are the indices for which dU
k
dimEk U
In particular U has the form
U
rjUEkj
p
iUEi
ii The cocycle
on U equivalently the cocycle
on U satis es
the IC of the MET and has spectrum
S
S
fmidmii
qg
and splitting
EmU
Emq U
where
Emi
kiFkU
and m
mq p are the indices for which dk dim Fk
U dimEk U
iii We have S
SUS
where the multiplicities of i with respect to U and add up to di
di dU
i di In particular denoting by
the top Lyapunov exponent
of a cocyle
max
U
Proof i The IC for U follow from
kUtkt ktjUkt
sup
xUkxkkt
xkktk
1

MET ON LINEAR SUBBUNDLES
The bundle U can be isometrically trivialized by picking a measurable
orthonormal basis F in U h iS and taking the coordinate mapping
kF
UhiS
Rmh iS implyingk k
By Proposition
the MET holds for U with spectrum S U
fjdjj qgsplittingE
Eq and
U
qj Ej
Let k
kr p be those indices k for which
dU
k dimU Ek
note that dU
k is invariant and thus constant
Now E
UEk
k andd dU
k
Eq
U Ekq
q
kq anddq dU
kq Henceq rand
U
qj Ej
p
iUEi
U
consequently
dimU qX
idi
pX
idU
i
rX
idU
ki dimU
andhenceq rdi dU
kiEi U Ei and
U
p
iU Ei
ii It su ces to study the cocycle
on U The integrability condi
tions for follow from
kkand
kksup
xUkxkk xkkk
The IC of the MET then follow as in part i
Further one readily checks that U is invariant for the cocycle over
if and only if U is invariant for the cocycle
over Since also
jU
The cocycle
over has by Theorem i the spectrum
S
fidii
pg and ltration E
Ep where i
p idi
dp iandEi
Fp i where Fi
j iEj
i
p
1

CHAPTER MET FOR RELATED LINEAR RDS
The cocycle
jU has by part i of this theorem spec
trum
S
fidii
qg
where i
li di dli and the indices l
lq pare
those for which
dkdimEkU dimFpkU
while the ltration is
EiEliUFpliU
Now the original cocycle
on U is obtained as
jU jU
By applying Theorem i a second time its spectrum is
S
S
fidii
qg
where
i
qi
lq
i
plq
i
and
didqidlq
i dimFp lq
iU
and its ltration is orthogonal complements to be taken in U
EiGqiUGijiEj
jiFp ljU
We now introduce the new index
mip
lqii
q
with m
mq p Then
i
mi di dimFmi U dmi
and
EiGqiU
kiFmkU
UEmiU
Note that
dim Emi U
dimU dimkiFkU
dimU dimU dimFmi U
dmi
1

EXTERIOR POWERS VOLUME ANGLE
as it should be
iii We now prove that
dk dimEk dU
kdk
where dU
k dimEk U anddk dimFk U Since
FkU
FkU
jkEj jEjU
EkU jkEj
having used relation
below we have
dkddimEkUddkdkdU
k
Corollary Structure of Invariant Subbundle Every
invariant measurable linear bundle U is subordinate to the Oseledets
splitting i e has the form
U
p
iUEi
p
iEi
Further if U p
iUEiandVp
i V Ei are invariant
bundles then so are
UVp
iUEiVEi
and
UVp
iUVEi
One can repeatedly apply Theorem
to an invariant subbundle V of U
etc See the non random MET of Kifer Chap III and Carverhill
Exercise Triangular
Matrices Reconsider Example
and Exercise
in the light of this subsection
Exterior Powers Volume Angle
Exterior Powers
Thekthexteriorpower k t in kRd k dofacocycle on
Rd was already used in our proof of the MET see Sect for basic facts
on exterior powers see Lemma
We now determine the spectrum and
splitting of k t
1

CHAPTER MET FOR RELATED LINEAR RDS
Theorem MET for Exterior Power Let be a linear co
cycleinRdovertheergodicDS FP ttT withone ortwo
sided time satisfying the IC of the respective MET Let
be the
forward invariant for two sided time invariant set of full measure on
which all statements of the MET for hold where the Lyapunov spectrum
S fidii
pg is constant and
lim
t
t
t
t
exists Let
d
be the list of the i where i appears di times Then
iThecocycle k t on kRd
k d satisestheICofthe
respective MET and the MET holds on the same forward invariant resp
invariant set In particular
lim
t
kt
kt
t
k
andthespectrumS k f k
idk
ii
p k g is determined as fol
lows The k
i are the di erent numbers in the list of d
k numbers
i
i
ik
i
ikd
where this list starts with k
k and ends with k
pk
dk
d anddk
i is the number of times the value k
i appears
in this list
iiTheagfg Vk
pk
Vk
kRdofk can
be obtained from the ag f g Vp
V
Rd of asfol
lows Extract a measurable basis F
v
vd
of Rd which is
adapted to the ag of i e such that the rst d vectors are taken from
VnV
the last dp vectors are taken from Vp n Vp
Then
Vk
i
span fvi
vik i
ik
k
i
i
ik dg
iii If time is two sided then the splitting E k
i
ipk
of k t can be obtained from the splitting Ei of t as follows
Choose the measurable basis F fv
vd g adapted to the split
tingof ie suchthattherstd vectorsareinE
the last dp
vectors are in Ep Then
Ek
i
span fvi
vik i
ik
k
i
i
ik dg
1

EXTERIOR POWERS VOLUME ANGLE
iv If has a generator then so has k In the discrete time case
the generator of k is
kA In the continuous time case T R the
generators are Ak for the RDE case and Ak L if A L and
Ak Ak
m W for the SDE case where
Aku
uk
kX
i
u
ui Aui ui
uk
Proof i We assume knowledge of Lemma
and of the proofs of the
respective MET s in Sect The IC for k t follow from
kk
tkkktkktkk
Elementary manipulations with exterior powers yield
kt
kt
t
kt
t
t
k
where the eigenvalues of k
are exp i
ik exp i being the
eigenvalues of
This proves the claim about the spectrum of k t
ii Forthe agVk
i remember that
Vk
i
Uk
pk
Uk
i
where U k
i is the eigenspace of k
corresponding to the eigenvalue
exp k
i with dimU k
i
dk
i Now the right hand side of the last
equation is equal to the right hand side of equation
iii If time is two sided the splitting of k t is de ned by
Ek
i
Vk
i
Vk
pki
ipk
But again the right hand side of the last equation is equal to the right hand
side of equation
iv This follows immediately from Lemma vii
Remark i The bases F appearing in part ii and iii of the
theorem are so called normal bases in the sense of Lyapunov i e bases vi
for which P vi is minimal
ii The order of the exponents k
i is in general not determined by
the order of the i It is only clear that the top exponent of k t is
k
k and the smallest exponent is k
pk
dk
d
iii Extending Remark
the subspace E k is F measurable
while the subspace E k
p k is F measurable for each k
d
Theorem has several important applications to the distorsion of vol
umes and the dynamics of angles under a cocycle see in particular the
theory of rotation numbers in Sect
1

CHAPTER MET FOR RELATED LINEAR RDS
Volume and Determinant
Recall that if V h i is a Euclidean vector space then
ku
ukk vol u
uk dethui ujik k
is the volume of the k dimensional parallelepiped
u
uk fx
u
kuk
ii
kgV
spanned by u
uk Since
voltu
tukkktu
ukk
Theorem
immediately gives the following result
Corollary Lyapunov Index of Volume Assume the situa
tion of Theorem
Then on for all u
ukRdnfgkd
lim
t tlogvol t u
tuk
k
i
where i i
minfj u
uk Vk
j nVk
jg
De nition Determinant of a Linear Map on Subspace Let
T Vh iV Wh iW bealinearmapofEuclideanspacesandlet
U V be a k dimensional subspace k with basis u
uk Wede ne
the determinant of T jU by
detTjU volW Tu Tuk
volV u
uk kTu
T ukkW
ku
ukkV
Equivalently
detTjU k kT u
uk kW
ku
ukkV kkTjkUk kkTjUk
anddetTjU is wellde nedsince kU R u
uk is one dimensional
det T jU thus measures the distortion of volumes in U under T We have
the usual properties of the determinant
det T jU is independent of the basis of U and if AU is a matrix repre
sentation of T jU with respect to orthonormal bases of U and T U
then detTjU jdetAUj
detTjU
and detTjU
T U TU isbijective
1

EXTERIOR POWERS VOLUME ANGLE
ifTV WandSW Gthen
detS T jU detSjT U detTjU
For U V W we recover the absolute value of the classical determinant
detT k dTk
Corollary Lyapunov Index of Determinant on Invariant
Subspace Assume the situation of Theorem
with two sided time
Let
U
p
iU Ei
dU
i dimU Ei
be a non trivial invariant bundle Then det t jU is a positive
scalar cocycle satisfying the IC of the MET and
lim
t
t logdet t jU
pX
idU
ii
For discrete time T Z and A
lim
n
nlogdet n jU ElogdetA jU
Proof By Theorem i the cocycle t jU has spectrum
fkidU
kii
rg where the indices k
kr p are those
for which dU
i
The cocycle m t jmU
m dimU
Pp
idU
i
is a scalar cocycle on the one dimensional m invariant
subbundle mU
mRd and by Theorem
has the single the
Lyapunov exponent Prj dU
kjkjPp
idU
i i The result follows from
dettjUkmtjmUk
For discrete time e g for n
logdet n jU
nX
klogdetAkjUk
so that the ergodic theorem can be applied
For continuous time formulas for the Lyapunov index of volumes and de
terminants see Sect We list the following important particular cases
Example Lyapunov Index of Determinant on Oseledets
Space Assume the situation of Theorem
with two sided time Then
1

CHAPTER MET FOR RELATED LINEAR RDS
i If Ei is an Oseledets space then
lim
t
tlogdet t jEi di i
ii Let Es
iEi Ec
iEi Eu
i Ei be the stable center and unstable Oseledets space respec
tively Then
lim
t
t logdet t jE
Pidii
s
c
Pidii
u
if E is non trivial
Angles
Recall that the angle x y
between two vectors x y Rdnf g
isdenedby
sin xy kxkkyk hxyi
kxk kyk
kx yk
kxk kyk
Corollary Lyapunov Index of Angle Between Vectors
Assume the situation of Theorem
with two sided time Then for all
i For all linearly independent x y Rd
lim
ttlogsintxty
lim
ttlogtxty
xy
xy
ii Speci cally let x
p
ixiyp
i yi be linearly indepen
dentandi x
minfi xi g Then
lim
ttlogtxty
ifandonlyifxi x yi y ie xi x andyi y arelinearly
dependent Otherwise
lim
ttlogtxty
1

EXTERIOR POWERS VOLUME ANGLE
Proof i Equality of the two limits follows from
sin
while
existence and the formula follow from Theorem
since
sintxtyktxtyk
ktxkktykktxyk
kt xkkt yk
ii With the representations x
p
i xiandy pj yjwehave
xy max
xi yj
ij
x max
xii
ix
y max
yjjiy
Incaseix iy wealwayshavexi x yi y
and
xy
xyix
iy
ix
iy
Now assume i i x i y Then either xi yi implying
di
hence
xy
xy
i
i
i
orxi yi
Then by linear independence of x and y p and
ixyminfiixi
oryigf pg
is well de ned whence
xy
xyi
i
i
i
i
i
Example Ifeitherix Ei nfgandy Ej nfgi j
or ii x y Ei are linearly independent then
lim
t
tlogtxty
Remark i Since by Proposition
xy sin xyisa
metric on projective space P d Corollary
makes a statement about
exponentially fast clustering of orbits of the cocycle induced by on P d
see also Sect
ii In particular inside an Oseledets space there is never exponentially
fast clustering The clustering is exponentially fast if and only if the domi
nating components of x and y are linearly dependent This is e g the case

CHAPTER MET FOR RELATED LINEAR RDS
with probability if the top Lyapunov exponent is simple and i x
Pasforanyx
ThelatterholdsintheT Ziid caseifthe
distribution of An Gl d R satis es certain conditions Furstenberg
Theorem Guivarc h and Raugi
Bougerol and Lacroix Sect
VI and in the case T R for SDE and RDE with Markovian noise
under a Lie algebra condition on P d Arnold et al
Imkeller
Recall that the angle between subspaces U V Rd is de ned by
sin UV
inf
xUyVxy sin xy
inf
xUyVxykxyk
kxk kyk
Corollary Angle Between Oseledets Spaces is Tem
pered Assume the situation of Theorem
with two sided time Then
foralliFori j
lim
t
tlogEitEjt
i e the angle between Oseledets spaces is tempered
ii More generally if M and N are disjoint subsets of R and
EM
i MEi
EN
i NEi
then
lim
t
tlogEMtENt
Proof i It su ces again to prove the statement for replaced by sin
With t Ei
Eit
sinEitEjt
inf
xEiyEjkxkkykktxyk
kt xkkt yk
Now
xyijx
iyjandxyE
where
E
is the space in the splitting of corresponding to
ij
By the uniformity statement of the two sided MET
the convergence
in
lim
t tlogkt xki
is uniform with respect to x Ei Sd and convergence in
lim
ttlogktxykij

TENSOR PRODUCT
is uniform with respect to x y such that x Ei Sd and y
Ej Sd thesex yformacompactsubsetofE nfg sothat
for each
thereisaT
such that for all t T
sinEitEjt
et
similarly for t
iiForx
i Mxi EM andy
jNyjENxyPxiyj
where xi yj if and only if xi
or yj Therefore
xy max
xi yj
ijmax
xiimax
yjjxy
Similarly for t
with max replaced by min The result fol
lows again by uniformity of convergence in the MET and the fact that
t EMN
EMN t
Tensor Product
Let V Vk be nite dimensional real vector spaces of dimensions
dim Vi
nii
kandletV V
Vk be their tensor
product a vector space of dimension n n n
nk Linear cocycles i
on Vi uniquely induce a linear cocycle
k the tensor product of
k on V whose spectrum splitting and generator can be obtained
from the spectra splittings and generators of the i
As we will not need the full machinery and since the extension to general
k is obvious we will restrict ourselves to the case k The matrix version
of
in Rn Rn Rn n is called Kronecker product and will turn
out to be basic for normal form theory of RDS see Chap
ThetensorproductV V V ofV andV isde nedtobethevector
space L V V R of bilinear forms on V V contravariant tensors
Deneforx Vy Vtheelementx y V Vby
xy
xyiViLViR
Then V V can be identi ed with the set of linear combinations
Pm
i cixi yi m Nci Rxi Vyi Vwherethefollowing
rules of computation hold
xxyxyxyxyyxyxy
xyxy
xy
The following facts which we also need in Chap can be found in
many textbooks on multi linear algebra see e g Kowalsky Chap
or readily obtained from the de nitions
1

CHAPTER MET FOR RELATED LINEAR RDS
Lemma Tensor Product of Spaces and Operators Let
V V V be the tensor product of the nite dimensional vector spaces
V and V with dimensions n dimV n dimV Then we have the
following properties
iIfeiisabasisofVandfjisabasisofV then eifjisa
basis of V V In particular
dimV V dimVdimV nn
ii The splittings V p
i Vi V pj Vj induce the splitting
VV
pp
ijViVj
iii If Ti Vi Vi are linear operators then the linear extension of
TTxyTxTy
de nes a linear operator T T on V V the tensor product of T and
TWehaveTaTaTTaTT
TSTTTST
TTSTTTS
TTSSTSTS
In particular
TIITTTITTI
where Ii are the identity operators on Vi
IfT andT areinvertible thensoisT T and
TT
TT
iv If T has eigenvalues i and eigenvectors xi and if T has eigen
values j and eigenvectors yj then T T has eigenvalues i j and
eigenvectors xi yj with corresponding multiplicities In particular
detT T detT n detT n traceT T traceT traceT
v Let in the bases of part i the matrix representations of T T and
T T beABandA B respectively Then
AB
aB
anB
anB
an nB
1

TENSOR PRODUCT
ie then n n n matrixA BistheKroneckerproductofthen n
matrix A and the n n matrix B
vi We have
etTetTetTIIT
vii If the spaces Vi have scalar products h ii then the bilinear exten
sion of
hxyxyihxxihyyi
de nes a scalar product in V V In particular
akx yk kxkkyk
bkT Tk kTkkTk
cTTTT
dIfUiareorthogonalthensoisU U and U U U U
The following theorem is a more or less immediate consequence of the
lemma
Theorem MET for Tensor Product Let in Rn and
in Rn be linear cocycles over the ergodic DS F P t t T with two
sided time satisfying the IC of the MET Let
be the invariant
set of full measure on which the statements of the MET for and hold
simultaneously Let S i f i
jdi
jj
pig be the spectrum
and Ei
j j pi the splitting of i respectively Then
it
t
t is a linear cocycle on Rn Rn which
satis es the IC of the MET
ii The MET holds on and
lim
t
t
t
t
where i
limt
it
it
t The spectrum of
is
S
S
S
fi
jiS
jSg
and the multiplicity of S
is
dX
ij
di dj
The Oseledets spaces of
are
E
ijEiEj
S
1

CHAPTER MET FOR RELATED LINEAR RDS
and dimE d
iii If the i have generators then so does
In the discrete time
caseT Z A A isthegeneratorof
ForT Randthe
RDEcase it Ai t it
d
dt
t
AtIIAt
t
ForT RandtheSDEcased it Pm
jAi
j it dWjt
d
tmX
jAjIIAj
t dWjt
Proof i The cocycle property follows from equation
of Lemma
the integrability condition follows from part vii b of the same
lemma
ii The claim about the matrix follows from
t
t
t
t
t
t
t
t
t
t
t
The assertion about spectrum and splitting follows from
kt
txykktxkktyk
and the dynamical characterization of the splitting
iiiIfiAit ithen
d
dtt
t
At
t
t
tAt
t
AtIIAt
t
t
Analogously for the Stratonovich SDE case
Manifold Versions
All standard constructions of linear algebra by which we derived the co
cycles investigated in sections to
also make sense on the level of
vector bundles Hence if is a linear bundle RDS on a Euclidean bundle
for which the MET holds see Theorem
we can derive similar state
ments which we refrain from formulating as they should be obvious for the
reader by now
1

AFFINE RDS
We brie y recall the basic situation for the tangent bundle of a Rie
mannian manifold If is a C RDS on a Riemannian manifold M h ix
with invariant measure then T t x TxM T t xM is a linear
bundleRDSon TMovertheDS MF B
ttTwhere
tx
ttx
Now if the MET holds for T with spectrum and splitting Ei x
then for example T t x is a linear bundle RDS over with spec
trum
and splitting j p iEj x
etc Here TxM is as usual
canonically identi ed with TxM
In Chap we will use the manifold versions of the MET for k T on
k T M and its o springs for the volume the determinant and the angle
whose de nitions in TxM h ix remain unchanged
A neRDS
Representation
The group A d R of invertible a ne transformations of Rd has the struc
ture of a semidirect product of Gl d R and Rd More speci cally every
elementg AdR isrepresentedbyg Ab wheregx Ax b with
A GldRandb Rd and
gg AAAbb gi Aibi
gAAbeI
A d R is a Lie group of di eomorphisms of Rd with dimension d d and
with Lie algebra a d R gl d R Rd semidirect sum
Let T be two sided A mapping T
AdR t
t
t
t which is measurable and for which t
t is continu
ous is called an A d R valued cocycle if
ts
ts
s
It follows that
e
is equivalent to
ts
ts
s
ie isaGldRvaluedcocycle and
ts
ts
s
ts
EveryAdR cocycle t inducesbytheleftactionofAdR onRd
gxgxAxbananecocycletx tx tin

CHAPTER MET FOR RELATED LINEAR RDS
Rd We now determine its generator and a more explicit form of the cocycle
by means of the variation of constants formula
i T Z The time one mapping is
xAxbandthe
cocycle is generated by the a ne di erence equation
xn
nxnAnxnbn
x xRd
the solution of which is see exercise
nx
n
xPn
jj
bj
n
x
n
n
xPjnj
bj
n
where is the linear cocycle generated by xn A n xn
ii T R RDE case Every A d R valued cocycle which is abso
lutely continuous with respect to t induces an a ne cocycle in Rd which is
generated by an a ne RDE
xtAtxtbt
If A b L P the solution is see Example
txtxZt
s bsdsZttssbsds
where is the linear cocycle generated by the RDE xt A t xt
iii T R SDE case Here the cocycle is generated by the a ne SDE
dxt
mX
j Ajxt bj dWjt
Its solution is see Example
txtx
mX
jZt
s bj dWjsA
Invariant Measure in the Hyperbolic Case
If a linear RDS is perturbed by additive noise then the xed point x
should survive as a stationary solution We will prove that this is indeed
the case provided the xed point is hyperbolic
1

AFFINE RDS
This subsection is mainly based on Arnold and Crauel Arnold and
Xu
and for the white noise case on Arnold and Imkeller We rst
prove a general theorem for a ne cocycles in Rd which do not necessarily
have a generator The notion of temperedness see De nition
will
prove to be crucial here
Theorem Invariant Measure for Hyperbolic A ne RDS
iLettx
t x t beananeRDSinRdwithtwo
sided time such that the linear part satis es the IC of the MET Assume
that is hyperbolic i e that all Lyapunov exponents are non zero Further
assume that
s
sup
tTks t tk Pas
and
u
sup
tTku t tk Pas
where su Rd Esu istheprojectionontoEsu alongEus
Es u being the stable unstable space of
Then there exists a unique invariant measure This measure is a
random Dirac measure
where is given by
s
u
limt inpr s
tt
limt
inpr u
tt
and su
su
Wehaveforallx Rd
s
lim
tinprs
t
x
and
u
lim
tinprut
x
If the limits in the de nition of are P a s then the limits in
and
arealsoPas
ii If moreover the random variables s u are tempered from above
the limit in probability in the de nition of is a P a s limit and is also
tempered Further the stationary stochastic process t has continuous
trajectories
We henceforth write tempered for tempered from above
1

CHAPTER MET FOR RELATED LINEAR RDS
Proof i a We rst prove that the expressions in the de nition of
make sense by proving that the sequences in question are Cauchy in
probability
We use the random norms of Sect and recall that for all
there
exists an slowly varying function B
i e satisfying
e jtjB
Bt
e jtjB
such that
BkkkkBkk
Choose so that
s max
ii
min
ii
u
then choose
min s
u
and nally choose
such
thatBy
t
t
tt
tt
t
hence
sut
t
su
tt
ttsut
t
Thus for t
and the stable part
ks
t
t
s
ttk
kttst
tk
kt tjEs tkks t
tk
BtkttjEstkt
st
Btetst
e
tB
st
in probability as t
where wehaveusedthatk t jEs k t
et
Consequently s
t t is Cauchy in probability hence con
verges in probability as t
Similarly for the unstable component
b Invariance We now show that
is invariant i e that for
allt T
sutt
sut
t
1

AFFINE RDS
For the stable part
stt
ts
stt
ts
s t t in probability as
stt
On the other hand putting
stt
t
s
st
t
st
t
stt
t
t
st
t in probability as
Hence
stt
st
t
similarly for the unstable part
c Uniqueness We show that if is a invariant measure then neces
sarily
Pas
Since a measure on a product space is a Dirac measure if and only if its
marginals are Dirac measures it su ces to prove that
su
su
Let be invariant and let f Cb Rd Then
sut
tfsut
tf su
ft
and
sut
t
fsut
tfsu
ft
Consequently
su
ft
su
ft
fsuttZfsuttxdx
Zfsutt
fsuttxdx
1

CHAPTER MET FOR RELATED LINEAR RDS
Due to the a ne structure of
sutt
suttxtsu
x
Since is hyperbolic the stable unstable part of the right hand side of the
last equation goes to zero for all x Rd for t
t
exponentially
fastPas
Suppose f is uniformly continuous Then by the dominated conver
gence theorem the last line of
converges to zero as t
t
while the rst line is a stationary stochastic process Thus necessarily
su
f
su
f
Pas
Since uniformly continuous f Cb Rd su ce to identify a measure see
Ganssler and Stute
p
we obtain
Pas
d We now prove
Observe that
s
t
x
s
ttx
t
ttstxs
tt
ttstxs
s
tt
s
Now using random norms and the fact that k s k
and
ktjEskt
exp t
kttstxkBetkstxkt
BetBtkxk
Be
tkxk
exponentially fast as t
Pas
Hence
follows with mode of convergence the same as in the de nition
of Similarly for the unstable component
ii Step i a of the proof shows that under our assumptions
su
t t is Cauchy P a s exponentially fast hence converges
P a s exponentially fast Moreover for each
and the stable part
kskks
skks
k
e
B
s
s
Hence s similarly u and thus is tempered
Finally t
t
has continuous trajectories since t
t x is continuous by de nition

AFFINE RDS
Remark Invariant Measures for Nonlinear RDS Theorem
in combination with the Banach xed point theorem can be used to
establish the existence and uniqueness of an invariant measure
for the RDS
txtxtNtx
where is hyperbolic satis es assumptions similar to
and
but now with respect to the random norm and the nonlinear term N ful lls
a Lipschitz condition
sup
tkNtxNtyktLkxyk
with L small enough see Arnold and Xu Theorem
and for a local
version Arnold and Boxler
Corollary Invariant Measure for Linear RDS Let be a
linear cocycle with two sided time satisfying the IC of the MET If is
hyperbolic then its unique invariant measure is the Dirac measure at
Corollary Spectrum Splitting and Invariant Manifolds
The splitting and spectrum of T t under
are the same as for
hencePas
Ei
Ei
Thus for the stable and unstable manifolds of under
Msu
Esu
and the invariant foliation of Rd is Ms u x x Es u
In particular for
Ms
Rd Mu
fg
ProofFortxtxt
Ttxvtv
hence spectrum and splitting of T under are the same as for so for
almostallx Rd ieforx
Eix
xEi
Ifwedeneforanyx Rd
MsuxfyRdktytxkastg
1

CHAPTER MET FOR RELATED LINEAR RDS
thenyMsuxifandonlyiftytx tyx
t
ifandonlyify x Esu ory x Esu Forx
we obtain the stable unstable manifold of under and for arbitrary x we
obtain the invariant foliation of the state space see subsection
We now deal with the case where has a generator
Theorem Invariant Measure for A ne RDS with Gener
ator i T Z Suppose is generated by the a ne random di erence
equation
xnAnxnbn
in Rd Assume
logkAkLP logkbkLP
the integrability condition for b can be replaced by the assumption that b
is tempered If A generates a hyperbolic linear cocycle then the unique
invariant measure is concentrated at
s
u
Pn
n
snbn
Pn
n
unbn
Moreover is tempered
ii T R real noise case Suppose is generated by the a ne RDE
xtAtxtbt
in Rd where A b L P the integrability condition for b can be replaced
by the assumption that b is tempered If A generates a hyperbolic linear
cocycle then the unique invariant measure is concentrated at
s
u
Rt
stbtdt
Rt
utbtdt
Moreover is tempered and the stationary stochastic process t has
absolutely continuous trajectories
iii T R white noise case Suppose is generated by the a ne SDE
dxt
mX
j Ajxt bj dWjt

AFFINE RDS
inRd noICneeded Ifdxt Pm
j Aj xt dW j generates a hyperbolic
linear cocycle then the unique invariant measure is concentrated at
s
u
Pm
jRt
stbjdWjt
Pm
jRt
utbjdWjt
The integrals exist as limits P a s of the non adapted Stratonovich inte
gralsRT andRT asT
Moreover is tempered and the stationary
stochastic process t has continuous trajectories
Proof iT ZBy
and the cocycle property we obtain for
n
s
nn
nX
kkkskbk
Using random norms as in the proof of Theorem
we obtain
s
sup
nks
nnk
X
kkkkskbkk
BX
k
e
kkbkk
since b is tempered or by Lemma i if log kbk L P Hence the
expression for s makes sense Further s is tempered hence s is also
tempered and
s
nn
s P a s exponentially fast
as nSimilarly for u and u
ii T R real noise case By
t tZt
uubudu
Using again random norms
s
sup
tksttk
Zkt t
tbtkdt
BZe
tkb t kdt
1

CHAPTER MET FOR RELATED LINEAR RDS
since b is tempered or by Lemma
if b L P Hence the expression
for s makes sense Further s is tempered since
stBe
tZt e
ukb u kdu
and b is tempered or again by Lemma
if b L Thus s is tempered
and
s
tt
s P a s exponentially fast
as tSimilarly for u and u
iii For the SDE case the problems steming from the non adaptedness of
require new techniques which will be developed in Sect See Theorem
Corollary Stable Case If then
i
Pn
n
bn
TZ
Rtbtdt
T RRDE
Pm
jRt bjdWjt
T RSDE
In particular
is a Markov measure
iiForanyx Rdandt
t
x
tx
t
Pas
both exponentially fast
iii For any probability measure on Rd
lim
t
t
weakly Pas
andforallx Rd
lim
tLtxL
where L denotes the law of the random variable
Proof As i and ii are clear we only prove iii For f Cb Rd and
by the fact that t
x
Pas
tfZft
xdx
Zf
dx
f
1

AFFINE RDS
Furthermore
LtxLttxLtxL
by the above for
x
Remark Invariant Measures in the Non Hyperbolic Case
Cohomological Equations and Coboundaries If an a ne cocycle
has a linear part which is not hyperbolic then has an invariant measure
only under rare circumstances For a complete treatment of the case of a
non random A for T Z and T R RDE see Arnold and Wihstutz
To nd a invariant Dirac measure
amounts to looking for a
random variable for which t
t or solving the equation
t
t
t
is then called a coboundary with respect to the cocycle or twisted by
see Katok and Hasselblatt
p
if there exists a random variable
such that the cohomological equation
is satis ed
For I we obtain the de nition of a classical additive coboundary
i e t is a helix which is cohomologous to the trivial helix and two
solutions di er only by a constant if is ergodic
For the generator cases the cohomological equation can be equivalently
written in the di erential form
Lb
where L is the cohomological operator More precisely
iforT ZL
A
b ie withUf
fLUA
iiforTRRDELt
d
dttAt
tbtie
Ld
dtAt
iii forT RSDE
dLdmX
jAjdWjmX
jbjdWj
ie
dLdmX
jAjdWj
We will make use of this in Chap

CHAPTER MET FOR RELATED LINEAR RDS
Time Reversibility and Iterated Function Sys
tems
We now compare our results on invariant measures for a ne cocycles with
those of Barnsley and Elton
and Elton
on stationary measures for
iterated function systems
Let be an a ne RDS with discrete time T Z and time one mapping
x
xAxb
Under the assumptions log kA k L P log kbk L P and
we have constructed in Corollary
the unique invariant Markov
measure
andforanyx Rd Pas
lim
n
n
x lim
n
nx
i e we recover by moving any x from time n forward to time and
letting n
We also have
L
lim
nLnxlim
nLn
x
i e we recover the law of by moving any x from time forward to time
n and letting n
The orbit n x itself does not converge but
approaches the stationary process n exponentially fast
Barnsley and Elton
constructed in the i i d case and for
byshowingthat L whereforallx RdandPas
lim
n
n
x
i e by somehow going from time n backward to time and letting
n Since obviously
is measurable with respect to the past while
is measurable with respect to the future of in the i i d case and
are independent the question arises why L L
The general answer is given in the next theorem For the discrete time
case and the continuous time RDE case we write A b for the a ne
cocycle generated by A and b
Theorem Invariant Measure for Time Reversible Case
Assume the situation of Theorem
andletforT ZorT RRDE
case A t b t tTbetimereversibleie
LAtbttTLAtbttT
1

AFFINE RDS
Then
LAbt
t
L
Abttt
For T R SDE case the analogue of
always holds
In particular the P a s limit
lim
t
Abt
x
exists if and only if the P a s limit
lim
t
Abttx
exists in case of which L L
If then both and exist and are independent of x
Proof We may and will restrict ourselves to the case T Z
We have for n
Abn
Abn n
n
Time reversibility implies
LAbn
nL
nn
Now
n
Abn n
from which
follows In the SDE case note that L W
L WIf all nite dimensional distributions of two stochastic processes are the
same then one of them converges P a s if and only if the other one does
so Further the laws of the limiting random variables coincide
Remark iNotethatif A t b t t Tistimereversible
thensois A t t ThenceS A S A byProposition
The discrete time i i d case is always time reversible
ii Assume
ForthecaseT Ziid orforthecaseT R
SDEL E
is the unique stationary measure of the cor
responding one point motions t x t see Subsect
for discrete
time and Subsect
for continuous time i e the unique solution of
ZRdPtxdxPtxBPftxBg
1

CHAPTER MET FOR RELATED LINEAR RDS
For T R this is equivalent to being the unique solution of the Fokker
Planck equation L
L being the generator of t x
For p
we have a similar situation with now being the unique
stationary measure of the one point motions t x t i e the unique
solution of
ZRdPt x dx
wherePtxBPftxBgPft xBg
For T R this is equivalent to being the unique solution of the Fokker
Planck equation L
L being the generator of t x

Chapter
RDS on Homogeneous
Spaces of the General
Linear Group
Summary
Following the pioneering work of Furstenberg
the study of the asymp
totic behavior law of large numbers positivity of top Lyapunov exponent
alias Furstenberg constant simplicity of the Lyapunov spectrum central
limit theorem large deviations principle etc of products of i i d random
matrices more generally products of i i d random variables in a Lie group
became an important research area This development was highlighted in
by Guivarc h and Raugi s profound paper
and the book by
Bougerol and Lacroix
Furstenberg made the following basic observation In order to deter
mine the asymptotic behavior of the product n An A of group
valued i i d random variables An one has to let the group act on certain
manifolds M Furstenberg boundaries in the matrix case Grassmannian
and ag manifolds g x gx and study e g the stationary measures
of the transition probability P x B Pfg gx Bg of the Markov chain
xn
nx on M Furstenberg s approach proved to be extremely fertile
and practically every author investigating products of random matrices has
made use of it
One of the lasting outcomes was Furstenberg s formula for the top Lya
punov exponent as an integral over projective space
Formula
This formula was independently found by Khasminskii
for linear
1

CHAPTER RDS ON HOMOGENEOUS SPACES
SDEAs explained in the Preface we decided to omit the subject except for
the study of the action of linear cocycles on certain homogeneous spaces
thus permitting us in particular to derive Furstenberg Khasminskii formulas
for all Lyapunov exponents
More speci cally in this chapter we will study some nonlinear RDS
which are induced by a Gl d R valued cocycle on homogeneous spaces
ofGldR
The most interesting of these homogeneous spaces for us are the unit
sphere Sd the projective space P d the Grassmannian manifold Gk d
of k planes the Grassmannian manifold Gk d of oriented k planes the
Stiefel manifold Stk d of orthonormal k frames the action of Gl d R
is followed by orthonormalization and the ag manifolds F d where
ddr
d
dr d is a multi index of dimensions di
of subspaces Vi and f V V
Vr is a generic element of F d
We also need to study RDS on a principal bundle and investigate under
which conditions it induces an RDS on the base manifold In case that the
original RDS has a generator we determine the generator of the induced
system The cases of particular interest for us are the principal bundles
StdZF d StkdOkRGkdandStkdSOkRGkdwhichare
the basis of our theory of rotation numbers in Sect
In Sect we collect some abstract nonsense about group valued
cocycles and cocycles on principal bundles
Sect is devoted to the dynamics and ergodic theory of the RDS
induced on Sd and P d by a linear RDS We determine all invari
ant measures Theorem
and their spectrum and splitting Theorem
On this basis we obtain Furstenberg Khasminskii formulas i e
phase averages over Sd for all Lyapunov exponents of Theorem
for the real noise case and Theorem
for the white noise case The
white noise formulas are derived by anticipative calculus
In order to obtain Furstenberg Khasminskii formulas for sums of Lya
punov exponents Theorem
we have to study the RDS induced by
on Grassmannian manifolds see Sect
Manifold versions of the above are treated in Sect
Finally Sect is devoted to our concept of rotation numbers a si
multaneous generalization of the two dimensional concept and the notion
of imaginary part of eigenvalue of a matrix to d dimensions and to non
linear RDS We prove an MET for rotation numbers which states that if the
Lyapunov spectrum is simple then every two plane has a rotation number
and this is taken from a nite collection of basic rotation numbers ij
realized in the canonical planes Ei Ej Theorem
for the real noise

COCYCLES ON LIE GROUPS
case and Theorem
for the white noise case
Cocycles on Lie Groups and Their Ho
mogeneous Spaces
A continuous or Ck cocycle with two sided time on a state space X de nes a
group valued cocycle on Homeo X or Di k X where
means
composition and where those groups act naturally on X on the left see
SectConversely if G is a group then a G valued cocycle induces a cocycle
on any space X on which G acts on the left We are particularly interested
in the case where G is a Lie group and X G H is a homogeneous space
for which case we will determine the generators of the induced cocycles
Group Valued Cocycles and Their Generators
De nition Group Valued Cocycle Let F P t t T
be a metric DS with two sided time Let G be a measurable group A
measurable function
T
G
is called a G valued cocycle if it satis es
ts
ts
s
is called continuous if G is a topological group and t
t is contin
uous
It follows that
e the unit element of G and t
tt
The study of through its G valued cocycles and their cohomology
invariants has become known as Mackey s program in algebraic ergodic
theory see Schmidt
Generators
As all G valued cocycles with discrete time are obviously generated by their
time one mapping we consider the case T R
On the group level it would be equally natural to write the factors in
the
other way around Only when G acts on a space X on the left or on the right is
the symmetry broken In the deterministic case no and in the random case if G is
Abelian the two notions coincide
1

CHAPTER RDS ON HOMOGENEOUS SPACES
LetGbeaLiegroupandletgbeitsLiealgebra Avector eldXon
G is called right invariant if Rg X X for Rg see De nition
The following reasoning is completely analogous to the one in Sect
Let
A
TeG g
be a g valued random variable Then there exists a unique right invariant
random vector eld X de ned by
XgRgATgG
giving the corresponding right invariant RDE on G
t
Xtt
RtAt
e
IfA
Lloc R g then the RDE
uniquely generates a global
G valued cocycle use for example Kobayashi and Nomizu
p
Conversely for a G valued cocycle which is absolutely continuous with
respect to t there exists A
if is C such that solves the
right invariant RDE
We have thus obtained the following statements
Theorem Cocycles Through Right Invariant RDE There
is a basically one to one correspondence between
i G valued cocycles which are absolutely continuous with respect to t
ii right invariant RDE on G of the form
ExampleiG GldR g Rd d HereRgA Aghence
G valued cocycles are generated by right invariant RDE
tAtt
I
ii G Rd g Rd GisAbelian AGcocyclealsocalledahelixin
the additive case satis es
ts
ts
s
A right invariant RDE on Rd has the form
t
tbt
where b
Rd satis es b
Lloc R Rd with solution
tZtbsds
1

COCYCLES ON LIE GROUPS
This is the most general absolutely continuous Rd valued cocycle
iii G A d R the group of a ne transformations of Rd This case
is treated in detail in Sect Examples i and ii above are particular
cases of iii
In the white noise case choose A Am g Consider the helix
Ft
tA mX
jWjtAj
which is Brownian motion in g Then the right invariant SDE generating a
G valued cocycle reads
dtRtdFtmX
jRtAjdWjt
e
Cocycles Induced by Actions
De nition Action of a Group on a Space Let G be a
measurable group and X B be a measurable space A measurable map
pingGXXgx
g x suchthatx
g x is a bimeasurable
bijection of X is called a left action G acts on X on the left if
gg
g
g
and a right action G acts on X on the right if
gg
g
g
The group G acts naturally on itself i e X G
iontheleftby gx
gx gx LgxsinceLgg LgLg
ii ontherightby gx
gx
xg Rgx sinceRgg
Rg Rg
WewriteX GLorGRforthestatespaceX Gwiththeleftor
right action of G
Combining a cocycle with a left or right action the symmetry is broken
as follows
Lemma Let be a G valued cocycle and let be a left or right
action ofGonX Dene
tx
tx
Then is measurable and is a cocycle or backward cocycle on X respec
tively
1

CHAPTER RDS ON HOMOGENEOUS SPACES
The proof follows immediately from the de nitions
We now investigate the question whether a cocycle induced by the
left action of the Lie group G on a manifold M by a cocycle on G
via
has a generator if does The answer is given by the non
autonomous version of a well known fact from di erential geometry see
Kobayashi and Nomizu
p
Theorem Generator of Cocycle Induced by Action Let
be a G valued cocycle generated by the right invariant RDE t
RtAt
e and let be a left action of G on the manifold
M Then the cocycle de ned by
is generated by the RDE
xtAtxt
xxM
where the random vector eld A
XM isgivenby
Axd
dttxjt
ExampleLetG GldR andM Rd Then t
At t andgx gxinducetx
t x alinearco
cycle in Rd which solves xt A t xt
In the white noise case if solves
and is a left action on a manifold
M then the induced cocycle de ned by
solves the SDE
dxt
mX
j Ajxt dWjt
where Aj X M is the vector eld corresponding to the ow j t x
exptAj x onM
Theorem becomes even nicer if M is a homogeneous space of G
ieifGactstransitivelyonMmeaningthatfgx g Gg Mfor
all x M Without loss of generality we can assume that M G H
gH g G is the set of left cosets of a closed subgroup H G
ThereisanaturalleftactionofGonGHdenedby
gxH gxH
If G G H g gH is the canonical projection then
Lg
g
1

COCYCLES ON LIE GROUPS
Corollary Generator on Homogeneous Space Let be a
G valued cocycle with generator t
RtAt LetMGH
be any homogeneous space of G with the natural left action of G Then
the cocycle
L hasageneratorxt A t xt onM where
A
A
Proof The existence of a generator follows from Theorem
For
xG
Ax
d
dtt xjt d
dtt
xjt
d
dttxjt
t xRtxAtjt
xRxA
Corollary Generator on Homogeneous Space of Gl d R
Let be a Gl d R valued cocycle generated by t
Att
I Then on any homogeneous space M of Gl d R induces a
cocycle
L with generator A A Here is the natural left
action of Gl d R on M and Gl d R M is the canonical projection
For the white noise case the right invariant SDE
on G induces an
SDE
on M which for a homogeneous space M G H becomes
dxt
mX
j AjxtdWjt
Cocycles on Principal Bundles
LetPMG beaprincipal berbundle iePthetotalspace isaman
ifold G is a Lie group structure group acting freely on the right on P
written p g
gp pg M PGisthe orbitspacebase space
P M the canonical projection with g for all g G such
that P is locally trivial i e for each x M there exists a neighborhood
Usuchthat U U Gbyadieomorphism
which inter
twines the right action g of G on P and the right action Rg of G on itself
gRg
It follows that all bers
x are closed submanifolds of P on which
the right action of G is transitive and free hence
xG
isafreeactionofGonPif gx xforsomex Pimpliesg e
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Theorem Cocycle on Principal Bundle Let be a cocycle
on the principal bundle P Then the following is equivalent
i projectstoacocycle onMvia ie
ii is right invariant with respect to G i e
gt
t
g forallg G
In case the cocycle has a generator xt X t xt on P ii is equivalent
to either of the following statements
iii X is a right invariant generator on P i e
gX
X
ivX canbeprojectedtoMieX
X is a random
vector eld on M that is related to X in which case X is the
generator of
The proof follows again immediately from well known facts of di erential
geometry
RDS Induced on Sd and P d
We now follow Furstenberg s program
and study the action of a lin
ear cocycle in Rd induced on the manifolds Sd and P d more pre
cisely the left action of a Gl d R valued cocycle on the above homogeneous
spaces see Sect
While working on Sd is very convenient for calculations the situation
on P d is less redundant since P d is smaller than Sd hence
certain trivial ambiguities are removed
Only if necessary we will distinguish between a quantity associated
with Sd and associated with P d We will however mainly elaborate
the case Sd and then omit the bar in
and leave the formulation of
the completely parallel statements for P d to the reader
We assume throughout that is ergodic and that time T is two sided
Invariant Measures
The Manifolds Sd and P d
Although both Sd and P d are homogeneous spaces of Gl d R we give
a more concrete and elementary description here
The unit sphere of Rd h i is the manifold
Sd fxRdhxxigRd
1

RDS INDUCED ON SD AND P D
It inherits its natural Riemannian structure from Rd by the inclusion map
ping i Sd Rd which is an isometric imbedding
Denote by
Rdnfg Sd
x
x
x
kxksSd
the canonical projection onto Sd Note that Rd n f g Sd
is
a at principal bundle over Sd with bers structure group
s
The coordinates s
x
kxk r kxkofapointx rs Rdnfgin
this bundle are called polar coordinates
The tangent bundle T Rdnfg of Rdnfgis globally trivial and is
identi ed with Rd n f g Rd We also identify the tangent bundle T Sd
of Sd with the subbundle of Sd Rd de ned by identifying s TsSd
withsfvRdhvsig ss
TSdfsvsSdhvsigSd RdRdRd
wecanalsothinkofTsSd asthea nespaces s inthespaceRdin
which Sd is imbedded
NowT TRdnfg TSd sends xv x Rdnfgv TxRdn
fgRdto xTxv where
Txvkxkvhv xix
x TxSd
and kxkTx is the orthogonal projection of Rd onto x
A Ck di eomorphismf of Rdnf ginduces a Ck di eomorphismf on
Sd ief
f if f is positively homogeneous or just homogeneous
or even linear In this case the unique f is
fs fs
kfsk s Sd
As a consequence a linear mapping A Gl d R induces naturally a
smooth di eomorphism A on Sd via
Sd
sAs
kAsk As Sd
which satis es A
A hence
ABABABGldR
FurtherbyhomogeneityA g g Aforg G fidSd g
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Every linear mapping x Ax can be described equivalently in polar
coordinates by s r As kAskr
ACkvector eldXonRdnfginducesaCkvector eldX T X on
Sd if X is positively homogeneous or just homogeneous or even linear
In this case the unique X is
XsXshXssissSd
and the corresponding ows and are related by
t
t
To see this note that X induces the vector eld X if and only if X and X
are related in the sense that for all x y
sTXxTXy
Boothby p
Now use
Hence every linear vector eld XA x Ax A Rd d induces a
smooth vector eld
XAs AshAssis
on Sd
The real projective space P d of one dimensional subspaces of Rd is the
quotient space Rd n f g where the equivalence relation is de ned by
x yifandonlyifthereexistsa Rwithx yLet
Rdnfg Pd
x
xx
be the canonical projection Rd n f g is thus a non at principal bundle
over P d with typical ber structure group R R n f g
There is yet another way of looking at Sd and P d Note that the
group
GfidSdgZ
is the cyclic group of order with generator g id Sd and that
Pd SdG
i e P d is obtained from Sd by identifying antipodal points s Sd
is thus a two fold covering of P d with covering projection
Sd Pd
satisfying g for g G It is also a principal bundle with typical
ber G We endow P d with the Riemannian metric which makes a
local isometry In particular
his ghigshis
WehaveS P
1

RDS INDUCED ON SD AND P D
whereforg idSd g TSd TSd sendsv TsSd tog v
v TsSd andT TSd
TPd identi es v TsSd and
v T sSd Further
onRdnfg
ACk di eomorphismf onRdnf ginduces aCk di eomorphismf onPd
ief
f if and only if f induces a di eomorphism f on Sd
withf s fs
If this holds the unique f is given by
fx
d
fx fs
The conditions are in particular satis ed if f is homogeneous or even linear
As a consequence a linear mapping x Ax A Gl d R induces
naturally a smooth di eomorphism A PGl d R Gl d R R I the
projective general linear group on P d via
Pd
xc
AxAxPd
which satis es A
AonRdnfgand A A onSd hence
d
ABABABGldR
Further a Ck vector eld X on Rd n f g induces a Ck vector eld
X T X onPd ifandonlyifXinducesavector eldXonSd
withX s Xs
If this holds the unique X is given by
XTXTXTTX
and the corresponding ows
and are related by
t
t
t
t
t
t
The conditions are particularly satis ed if X is homogeneous or even linear
Hence every linear vector eld XA x Ax A Rd d induces a
smooth vector eld on P d given by
XAx TXAs TAshAssis
All of the above relations are summarized for the mapping case in the fol
lowing commutative diagram of Fig
Figure Actions induced by a matrix
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Invariant Measures for the Induced RDS on Sd and P d
With the help of the MET we will determine the invariant measures of the
nonlinear RDS
onSd and on Pd induced by a linearRDS
in Rd but only spell out the results for Sd
Proposition Let be a linear RDS in Rd over the ergodic metric
DS with two sided time and let
tsts
ts
ktsksSd
Theni is a C RDS on Sd over satisfying
t
t
ii The function
qt s logkt sk
isahelixover t s
ttsie
qtusqtsquts
identically on T T
Sd
iii The set IP of invariant measures is nonvoid convex compact
and is invariant with respect to G f id Sd g i e if is invariant
then so is g the measure with factorization g
iv Suppose satis es the IC of the MET let be the largest and p
the smallest Lyapunov exponent of Then the helix q de ned by
satis es the IC of the subadditive ergodic theorem for any invariant mea
sure If is such a measure
p
s lim
t
tlogk t sk
where the limits exist on an invariant set of full measure and also in L
and on this invariant set is an invariant random variable
In case is ergodic
E logk sk
Proof i ii The cocycle property of follows from that of and AB
A B The helix property of q over follows from the cocycle property of
and kABsk kABsk kBsk
1

RDS INDUCED ON SD AND P D
iii Since Sd is compact metric PP Sd and IP are nonvoid
compact convex sets see Theorem
If IP theng IP
sinceAggA
iv We have
logkt k logkt k
logkt sklogktklogktk
Thusthehelixqt s
log k t sk over is bounded above
by log k t k which is subadditive over
and bounded below by
log k t k which is superadditive over both satisfying the IC of
the Furstenberg Kesten theorem Theorem
invertible cases Hence
for any invariant the subadditive ergodic theorem holds First assume
T Z By additivity and the technique used to prove Theorem B
lim
n
nlogk n sk
s
exists on an invariant set of full measure and also in L and the limit
is invariant on this set Further
p
s
for the left hand side inequality note that n
is a backward cocycle
over with spectrum
For T R we use additivity the discrete time
result and the technique used to prove Theorem C to obtain the same
statements
Now q t s has analogous properties with respect to
and
hence
s lim
ttlogktskp
Since by additivity q t s
q t t s the reasoning used in
the proof of Theorem
yields
Since additivity implies E q t t E q
Eq
in the ergodic case
Remark Lift of Measure from P d to Sd In this remark
we write and for and
The set IP of invariant measures is also nonvoid convex and com
pact Further the mapping
IP
IP
g
1

CHAPTER RDS ON HOMOGENEOUS SPACES
is continuous a ne and surjective the latter meaning that if is
invariant on P d then there exists a invariant on Sd which is a
liftof ie
inthiscaseg g Gisalsoaliftof
This can be seen as follows The continuous a ne mapping
PP Sd
PP Pd
satis es IP
IP since A
A and
g since
gThe existence of a lift of IP is equivalent to IP
IP
being surjective This follows from the surjectivity of
as follows
Assume the compact convex set
PP Sd is nonvoid For
IP
t
implying that
IP
see Corollary
It remains to be proved that is surjective By the Krein Milman
theorem PP P d is the closed convex hull of its extremal points
which are the random Dirac measures
xIfPdSdis
a measurable section of Sd P d i e
idPd then
x
s is an extremal point of PP Sd with
Now let
PP P d be arbitrary and assume n
n Ppknxkn
Then n Ppkn xkn Ppkn skn hastheproperty n
n
since is a ne
Now pick a converging subsequence of n
n which is possible
by compactness By the continuity of
limnlimnlimn
thus
Theorem Invariant Measures on Sd Let be a linear
RDS in Rd over the ergodic metric DS with two sided time satisfying the
ICoftheMET LetS
fidii
pg be its Lyapunov spectrum
and Rd
p
i Ei its Oseledets splitting Then
i On the invariant set of full P measure of the MET
Ei
Ei
EiSdi
p
are disjoint strictly invariant random compact submanifolds of Sd of
dimension di
ii For each ergodic invariant measure on Sd there exists a
unique i f pg such that realizes i in the sense that
lim
t
tlogkt ski
as
1

RDS INDUCED ON SD AND P D
Equivalently is supported by the random compact set Ei i e
Ei
Pas
Conversely for each i f pg there is a invariant measure on Sd
which realizes i
iii Each invariant measure on Sd has the form
pX
i
ii
i
pX
i
i
i realizes i
Proof i The strict invariance of Ei follows from the strict
invariance of Ei and A
A
ii Since is ergodic both limits in
are equal to the same con
stant
But the MET for and the characterization of the Ei assert
that for
the two limits
lim
t
tlogk t sk
s
are equal if and only if
s
i for some i equivalently if and only if
s Ei for some i Since
s
a s this i is independent of
s
as and Ei
Pas
Conversely let IP jEi be the weakly compact set of invariant mea
sures supported by the strictly invariant hence forward invariant random
compact set Ei By Theorem
IP jEi is nonvoid and equal to the
closed convex hull of its ergodic elements see Remark
Note that
any IP jEi and not just the ergodic ones ful lls
iv Assume that is invariant By Proposition iv
lim
t
tlogk t sk
s
as
By the MET the possible values of the invariant random variable
are
the i with
fs
s
igfssEigi
and
fss
p
iEigpX
i
i
Denefor i
i
i
Ei
1

CHAPTER RDS ON HOMOGENEOUS SPACES
and for i
choose any i realizing i see ii Then i is invariant
though not necessarily ergodic and realizes i and we have
pX
i
ii
Remark Markov Measures on Sd The Oseledets space
E is F measurable and Ep is F measurable see Remark
By
Theorem
the sets IP F jE and IP F jEp are nonvoid compact
convex sets In particular there is always an ergodic invariant forward
Markov measure realizing and an ergodic invariant backward Markov
measure realizing p
Random Eigenvectors
The block diagonalization of a linear cocycle with two sided time satisfy
ing the IC of the MET can be easily accomplished as follows see Corollary
We choose a random basis G
vi
which is adapted to the
Oseledets splitting If P
PS
G is the basis transfer matrix from the
standard basis S to G then
t
PttP
diag t
pt
is block diagonal and Lyapunov cohomologous to For the proof one
uses the fact that the property t Ei
Ei t implies that if
vEithenbothvtEitandtvEit
However in general t v
v t Can we nd random vectors
for which equality holds i e which are invariant
De nition Random Eigenvector Eigenvalue Let be a
linear RDS in Rd A random variable u
Sd is called random
eigenvector of if it satis es
tu
ut
equivalently
tuktukutQtut
and the cocycle Q t k t u k is called the corresponding
random eigenvalue of

RDS INDUCED ON SD AND P D
u being a random eigenvector is equivalent to
u being
invariant and necessarily ergodic
For example if di then Ei
Ei
Ei is one point
and there is exactly one ergodic invariant measure realizing i namely
Ei Now Ei consists of two points
can be either lifted to
the two ergodic invariant Dirac measures
si andg
si
g idSdieg
or to the single ergodic invariant measure
si
si
the weight coming from g
In the second case there is no mea
surable way of distinguishing between the two points in Ei
However if we are willing to accept a possible extension of the underly
ing DS we can have the following random analogue of the deterministic
diagonalization of a matrix by means of a basis of eigenvectors
Theorem Existence of Random Eigenvectors Let be a
linear RDS with two sided time over the ergodic DS satisfying the IC of
the MET Then the following holds possibly over an extension of
i In each Oseledets space Ei there exists at least one random eigen
vector ui and the corresponding eigenvalue is Qi t k t ui k
with
iElogk uik
ii If the spectrum of is simple then there exists a basis of random
eigenvectors
G
u
ud
where ui
Ei P a s with random eigenvalues Qi t
k t ui k Thus t the cocycle written in the bases G and
G t is Lyapunov cohomologous to and diagonal
t diagktukktudk
withElogk ui k iforalli
d
Proof We follow Arnold and Xu Appendix B
i Assume without loss of generality that the MET holds without an
exceptional set We consider the random compact set
CE
Ep
Sdp
which is strictly invariant with respect to Hence IP jC is nonvoid com
pact convex thus there exists an ergodic invariant measure supported
by C If happens to be a Dirac measure u
up we are done
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Otherwise we consider the extended DS P where C P
with t
t trivially extended and de ne random
vectors by ui
uis
sp
sii
p
Then
tui
tsiuit
ii This follows from i for p d
If di the random eigenvector in Ei just constructed might be the
only possible one see the deterministic case For more examples see Crauel
Note that all diagonal elements in
are positive meaning that any
ip behavior is built into the basis of random eigenvectors
Furstenberg Khasminskii Formulas
The study of the action induced by on Sd in Subsect
gave rise
to the helix
qt s logkt sk
Proposition
and for each i there is a invariant measure i real
izing i
lim
t
tqt s
i ias
Theorem
When has a generator the helix q has a generator too i e it can be
represented as an integral as a sum if T Z of a stationary stochastic
process Hence the left hand side of
becomes a time average so
that the classical ergodic theorem applies and gives an expression for i as a
phase average expectation with respect to i We call these expressions for
i the Furstenberg Khasminskii formulas The prototypes of these formulas
for the top exponent were discovered independently by Furstenberg
for discrete time and by Khasminskii for linear stochastic di erential
equations This was well before Oseledets
proved the MET in
We only treat the continuous time case here and leave the case T Z
as an exercise
Exercise Furstenberg Khasminskii Formulas T Z Let
A with log kA k L P be the generator of a linear cocycle and let
be the corresponding cocycle on Sd If i is invariant and realizes
i then
iZSd logkA skdi s
1

RDS INDUCED ON SD AND P D
Real Noise Case
Let be a linear cocycle in Rd generated by the RDE
xtAtxt
In polar coordinates s
x
kxkSd rkxk
this equation is
equivalent to
sthtst
rtQt strt
where
hsAshAssis
is the projection of the linear vector eld A x and
QshAssi
Equations
and
follow easily from
by di erentiating
s xhx xi andr hx xi exercise
Equation
is decoupled from
and generates the cocycle
while
with inserted is equivalent to
d
dtlogrt Q t s
and yields with initial condition r
the helix
qt s logkt skZtQu
sdu
Theorem
and the classical ergodic theorem give
Theorem Furstenberg Khasminskii Formulas Real Noise
Case Suppose A L P so
generates a linear cocycle for
which the MET holds Let be the corresponding cocycle on Sd generated
by
andQ s hA ssiThen
i For any invariant measure Q L
ii If i is invariant and realizes i then
i EiQ
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Proof The integrability of Q follows from that of A by
kAkminAA hAssimaxAA kAk
A very satisfactory theory based on Theorem
can be developed in the
Markovian context for the case where t is a nice ergodic di usion process
and xt A t xt See Arnold Kliemann and Oeljeklaus Arnold and
Kliemann
and Bougerol
We will nally use the MET to decouple a linear RDE While in the de
terministic case the block diagonalization of a linear ow t exp tA
and of its generator A are the same problem since T etAT et T AT
this is not true in the random case Suppose we block diagonalize the
linear cocycle by means of a basis adapted to the Oseledets split
ting as in
Then the resulting block diagonal cocycle t
PttP
is in general not absolutely continuous with respect
to t hence has no generator This means that the basis which block
diagonalizes in general does not block diagonalize the RDE xt
A t xt What we need to accomplish the latter is a basis of random
eigenvectors see Theorem for a condition
Corollary Diagonalization of a Linear RDE Suppose the
linear cocycle generated by xt A t xt with A L P has a basis
G
u
ud
of random eigenvectors Then t the cocy
cle t written in the bases G and G t is diagonal and generated
by the d uncoupled scalar RDE
ytdiagQt Qdtyt
where
QiQuihAui
uii
withiEQii
d Hence has the form
t diag expZtQ r dr
expZtQd r dr
Proof We start with equation
Since has generator A
it kt uikexpZtQr
ruidr
Since ui is a random eigenvector r ui
ui r hence
itQit it

RDS INDUCED ON SD AND P D
We will need this corollary in Sect We can write
as
ytdiagiQityt
where Qi Qi
i has mean This is the closest one can in general
get to an equation with constant coe cients by means of a linear random
coordinate transformation
White Noise Case
Now let the linear cocycle in Rd be generated by the SDE
dxt A xtdt mX
j Ajxt dWjt
mX
j Ajxt dWjt
Since the real noise calculations remain formally valid for Stratonovich in
tegrals
written in polar coordinates is
dst
mX
j hjst dWjt
drt
mX
j qjstrtdWjt
where for Aj Rd d
hjs AjshAjssis qjs hAjssi
Again equation
is decoupled from
and generates the
cocycle on Sd while
yields with initial condition r
the
helix
qtslogktskmX
jZtqj u s dWju
The following integrability condition assures the applicability of the MET
to and of the classical ergodic theorem to q in
it also assures
the substitution of non adapted initial values into the stochastic integrals
of
in place of s
Proposition For any j m and p
E sup
sSd sup
tZtqj u s dWju
p
The analogous statement holds for sup t
1

CHAPTER RDS ON HOMOGENEOUS SPACES
We omit the rather technical proof which consists of verifying the conditions
of Proposition of the application of which yields
Furstenberg Khasminskii formulas in the white noise case were derived
by Khasminskii for the largest exponent but under a rather restrictive
condition for the largest and smallest exponent by Arnold Oeljeklaus and
Pardoux
and for the general case by Arnold and Imkeller the latter
paper being based on a non adapted substitution formula derived by Arnold
and Imkeller in
Proposition Preliminary Furstenberg Khasminskii For
mulas White Noise Case Let be the solution of the SDE
Theni Lp P for any p where
sup
tlogkt k
In particular always satis es the IC of the MET
ii For any invariant measure i realizing i we have
i
mX
jEiZqjtsdWjt
EiQ mX
jEiZqj tsdWjt
and
i
mX
jEiZqjtsdWjt
EiQ mX
jEiZqj tsdWjt
where
Qs qs
mX
jhjqjs
and
hjqjs hAj AjsAjsi hAjssi
iii If i is F measurable realizing i with marginal i on Sd then
iEiQZSdQsids
1

RDS INDUCED ON SD AND P D
If i is F measurable realizing i with marginal i on Sd then
iEiQZSdQsids
Proof i Sincelogisincreasingk k andjQ sj kAk
relation
gives
sup
tlogktksup
sSd sup
tlogktsk
sup
sSd sup
t
mX
jZtqj u s dWju
mX
j sup
sSd sup
tZtqj u sdWju
kAk mX
j sup
sSd sup
tZtqj u sdWju
By Proposition
LpPforanyp
As for
sup t logk t krecallthat t
t
is generated by
dyt
mX
j AjytdWjt
see Theorem i to which the above reasoning applies
ii The rst equality signs in the formulas for i follow immediately
from Theorem
The second equality signs are a consequence of the
following form of the Stratonovich Ito correction term We have
Ztqj u s dWjuZtqj u sdWju
signtZthjqj u sdu
where sign t takes care of the cases t forward integral and t
backward integral and hj qj the function obtained by applying the
vector eld hj to the function qj is given by
hjqjs hhjsgradSd qjsihAj AjsAjsi hAjssi
since
gradSdqjs Aj AjshAj Ajssis
1

CHAPTER RDS ON HOMOGENEOUS SPACES
iii If i is F measurable then
FtZSdqjtsids
is adapted hence using g s hj qj s and recalling that the expecta
tion of an Ito integral with respect to an adapted integrand vanishes
EZFtdWjtEZZSdgtsidsdt
Since i is invariant
ZSdgtsidsZSdgsitdsGt
thus
EiZqj t s dWjt
EZZSdqjtsidsdWjt
EGZSdgsE dsZSdgsids
Similarly for an F measurable i
Remark IC of the MET The fact that
LpPforany
p hence the IC for can also be directly obtained as follows First
note that for any A Rd d and for any basis of unit vectors ui in Rd
kAk d max
i d kAuik
Hence
ktkdmax
iexpmX
jZtqj u ui dWjuA
and
logd kAk max
i
mX
j sup
tZtqj u ui dWju
1

RDS INDUCED ON SD AND P D
For proving
LpP forallp itthussu cestoprovethatforeach
xeds Sd
sup
tZtqj u s dWju LpP
Converting again the Stratonovich integral into an Ito integral it remains
toshowthatM sup t jMtj LpP where
MtZtqj u sdWju
is a continuous square integrable martingale
By the Burkholder Davis Gundy inequality Ikeda and Watanabe
p
for any p
EM p CpEhMMipCpEZqj u sdu
p
CpkAjk p
For showing
Lp P consider the cocycle t
t
which
solves dyt Pm
j AjytdWjt
Remark Stationary Measures for Extreme Exponents
By Remark
there is always a invariant forward Markov measure
realizing and an invariant backward Markov measure p realizing p
By Theorem
the invariant forward Markov measures are in a
one to one correspondence with the stationary measures of the di usion
process given by
for time R i e with the solutions of the Fokker
Planck equation
L
Lh
mX
jhj
Likewise there is a one to one correspondence between invariant back
ward Markov measures and solutions to the Fokker Planck equation of the
di usion process given by
for time R
L
L
h
mX
jhj

CHAPTER RDS ON HOMOGENEOUS SPACES
For obtaining true formulas for i it remains to calculate the expectations
of the stochastic integrals with respect to the measures i for the general
case as explicitly as possible a highly nontrivial problem due to the non
adaptedness of these measures We only elaborate the case t
Theorem Furstenberg Khasminskii Formulas White
Noise Case Let be generated by the linear SDE
and assume
that has simple Lyapunov spectrum Then for i d
iEQEi
mX
j E trace Aj Aj NiAjRi
where Q is given by
Ni is a random matrix de ned by
Ni
ISi
PiPdiIPi
Si
PiPdiPi
Ri
and Pi Pd i and Ri are the orthogonal projections on Vi
Vd i and on the Oseledets spaces Ei
Vi
Vdi
re
spectively
Proof Under the integrability condition
the following anticipa
tive substitution formula holds For any random variable X
Sd
ZqjtsdWjtsXZqjtX dWjt
where both sides make sense see Arnold and Imkeller Corollary
Now suppose for simplicity that i
si
si
Ei
realizes
iBy
and Proposition
iEZqsitdtmX
jEZqjsi t dWjt
Since hAj si sii trace Aj Ri
EZqsi tdt Eqsi EtraceARi
and
Cij EZqjsi t dWjt EZtraceAjRi t dWjt
for i dand j m All wehavetocalculateisCijwhichis
a technically formidable task but can be accomplished by using Malliavin
calculus For details see

RDS INDUCED ON SD AND P D
Particular cases
Case i Here N so the correction term in
vanishes
identically and we recover the classical Furstenberg Khasminskii formula
for
EQEZSdQs ds
where is the distribution of E which solves the Fokker Planck equation
L
since E is F measurable
Casei d HereNd I Rdhence
trace Aj Aj NdAjRd hj qj Ed
thus
dEQEdZSdQsdds
where d is the distribution of Ed which solves the Fokker Planck equation
Ld
since Ed is F measurable
Remark Simple Lyapunov Spectrum Assume that the
subgroup G of Gl d R generated by the matrices A Am equals Sl d R
or Gl d R Then the Lyapunov spectrum of is simple see Bougerol
and Lacroix Example
WehaveG Gl dR foranopenand
dense set of A Am Rd d m Thus for matrices from this set
in particular generically the corresponding SDE dxt Pm
j Ajxt dWjt
has simple Lyapunov spectrum
We would like to draw the reader s attention to a line of research with many
applications which is due to our chosen restrictions beyond the scope of
this book by quoting the following theorem
Theorem Given the linear SDE
Assume that the vec
tor elds of the corresponding SDE
interpreted on P d satisfy
the following hypoellipticity condition
dimLAh hm x d forallx Pd
where LA Z denotes the Lie algebra generated by the set Z of vector elds
Then the following holds
i The di usion process on P d generated by
with generator
LhPm
j hj admits a unique stationary measure The measure
has a C density with respect to Lebesgue measure on P d
iiWehave ZPdQxdx
1

CHAPTER RDS ON HOMOGENEOUS SPACES
In particular the invariant Markov measure corresponding to is sup
ported by E and is the only such Markov measure
iii Law of large numbers For each xed x more generally for
each random x F
x lim
t tlogkt xk Pas
In other words the projection of any xed vector x onto the random sub
space E does not vanish P a s
The only hard part is to prove the uniqueness of Arnold Oeljeklaus and
Pardoux
use a transfer mechanism between di usion processes and
geometric control theory known as support theorems and the character
ization of C supp as an invariant control set by Kliemann
It
turns out that under
C is unique and has nonvoid interior For an
up to date account see Colonius and Kliemann
There is also a central
limit theorem and large deviation theory associated with
etc see
Arnold and Kliemann Baxendale Arnold and Khasminskii
and the references therein
The formula
has been and still is the starting point of numerous
asymptotic studies of for small or large white noise to prove stabilization
or destabilization by noise the existence of stochastic bifurcation see Chap
etc From the vast literature we just mention Ariaratnam and Xie
Arnold Eizenberg and Wihstutz Kao and Wihstutz
Pardoux
and Wihstutz
Pinsky
and Pinsky and Wihstutz
In Sect we need the following white noise analogue of Corollary
Corollary Diagonalization of Linear SDE Suppose the
linear RDS generated by
has a basis G
u
ud
of
random eigenvectors Then t the cocycle t written in the bases
G and G t is diagonal and generated by the d uncoupled scalar
SDE
dyt
mX
j diagqju t
qjud t ytdWjt
with stationary anticipative coe cients
See Arnold and Imkeller Theorem
Spectrum and Splitting
In Theorem
we have determined all invariant measures of the C
RDS on Sd induced by the linear RDS on Rd We now apply the

RDS INDUCED ON SD AND P D
manifold version of the MET Theorem
to T and a invariant
measure and explicitly determine its Lyapunov spectrum and Oseledets
splitting
LemmaLet kd v
vk Rdands Sd
Further let
wiTsvi vihvisisi
k
be the orthogonal projection of vi to s Then
kw
wkkk kv
vk skk
Proof By de nition see Lemma iv using s Sd and wi s
kw
wk skk
det hwi wjik k
kw
wkkk
Further using the elementary rules of Subsect
w
wks
vhvsis
vkhvksis s
v
vks
kX
i hvi siw
wi
swi
wks
v
vks
Proposition Let A Gl d R and A Sd
Sd be the
di eomorphism induced by A Then for any s Sd the linear operator
TA s TsSd
s TAsSd As
is given by
TA s v kAsk Av kAsk hAv AsiAs
vsRd
Moreover
kTAs vk kAv Ask
kAsk
hence for all s Sd
kTA sk supkvk v skAv Ask
kAsk
kAkkAk
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Proof Since A
A
TA
TATTATA
more precisely since s s
TAsvTAsAv
so that
gives
We now use Lemma
fork v Av w AvhAvAsiAs
and As instead of s which gives
kTA s v k kAskkAv hAv AsiAsk
kAskkAv As
kAskk k Av sk
kAsk
Since by Proposition iii
kAvskkAkkAk
and
kAksup
ksk kAsk
we obtain
kTA sk supkvk v skAv Ask
kAsk
kAkkAk
Theorem Spectrum and Splitting of Induced RDS on
Sd Let be a linear RDS on Rd over an ergodic DS with two sided
time satisfying the IC of the MET with spectrum S f i di i
pg
and splitting Rd
p
i Ei Denote by the RDS on Sd induced by
Leti f pgandlet
i be a invariant measure realizing
i Then
iT and i satisfytheICoftheMET
ii The Lyapunov spectrum
S
i fjdjj pg
is given by
j
jiji
ji dj
djji
di ji
1

RDS INDUCED ON SD AND P D
where for di
the case j i is not present in which case we only have
p Lyapunov exponents
iii On the invariant set of full i measure which is contained in
Ei
Ei Sd the splitting TsSd
pjEj sis
Ejs spanfvhvsisvEjgs TsSd j
p
fordi j i only andEi
s is not present
Proof i By Proposition
logkTAsk logkAk logkA k
ii and iii We prove our assertion by exhibiting a basis of vectors
which span Ej s and have growth rate j i for t
By
uniqueness of spectrum and splitting we are done
Pick s in the invariant set of full i measure in particular s
Ei
Now choose a random basis which is adapted to the splitting Ei
ie d orthogonalunitvectorsinE d inE etc suchthatsisthe
rst vector chosen in Ei Denote the basis vectors without s by F
The images w v hv sis of the vectors v F under Ts where Ts
is the orthogonal projection of Rd onto s de ned by Ts v v hv sis
form a basis of TsSd where Ej s Ts Ej is spanned by the
images of the basis vectors in Ej
We have for w Ej s by Proposition
kTtswkktwtskktsk
ktvtskktk
ktvskktk
Since by construction t s has growth rate i and t v s has
growth rate j i Theorem
lim
t
tlogkTtswkji
iji
The spectrum of under i is thus obtained by subtracting i from the
exponents of and by reducing the multiplicity of the exponent by
multiplicity means removal
If realizes a Lyapunov exponent which is simple then is hyperbolic
under
1

CHAPTER RDS ON HOMOGENEOUS SPACES
If has simple spectrum then is hyperbolic under any invariant
measure The only stable measures are the ones realizing a measure
realizing d has exponents j d j
d
We refrain from treating the RDS induced on P d for which com
pletely analogous statements hold
RDS Induced on Grassmannian Mani
folds
We now aim at Furstenberg Khasminskii formulas for sums of k Lyapunov
exponents of for which we have to study the RDS k induced by on
the Grassmannian Gk d
Invariant Measures
In this subsection we basically follow Crauel
The Manifolds Gk d and Gk d
Let k d The Grassmannian manifold Gk d is the manifold of k
dimensional linear subspaces of Rd It is a compact smooth manifold of
dimension dim Gk d k d k which can be easily seen by considering
Gkd asahomogeneousspaceGHofG OdR where
H OkROdkR
NotethatG d Pd Gd d Pd andGdd isaonepointset
However we here prefer to represent Gk d as a subset of the projective
space P kRd obtained as the projection k k Rd of the set of decom
posable vectors k Rd
kRd for details see Subsect
where
kkRdPkRd
is the canonical projection Hence
GkdkkRdPkRd
is considered as a k d k dimensional proper if k d compact
connected submanifold of P kRd
A linear mapping A Gl d R induces naturally a smooth di eomor
phismAkonGkd via
Gkd x Akx Gkd
1

RDS ON GRASSMANNIANS
where Ak x is the image of the subspace x under A The mapping A also
lifts naturally to kRd by linear extension of
kAu u
ukAuAu
Auk
obviously leaving the set kRd of decomposable vectors invariant kA in
turn projects down to P kRd as k k A and the latter coincides with
Ak from above on the invariant manifold Gk d
A linear RDS in Rd hence induces a smooth RDS k on Gk d by
k kkjGkd
For computational reasons we prefer to work on the oriented Grassmannian
in which every subspace appears twice but with opposite orientation
Gkd kkRdSkRd
which is a submanifold of the unit sphere S k Rd of kRd and a two fold
covering of Gk d Gk d Z As is well known from linear algebra two
k frames x
xkandy
yk spanning the same element of Gk d
hence y
yk
x
xk for some
have the same
orientation if and only if
Gkd andGkd areinexactlythesamerelationasG d Sd
and G d P d in Sect hence we will spare most details and abuse
notation by using the same symbol for corresponding objects of Gk d and
Gk d If we write Gk d then the statement holds for both Gk d and
Gkd
Invariant Measures for k on Gk d
As in the particular case k see Subsect
the invariant mea
sures of k can be classi ed by the MET but now by the MET for
the linear cocycle k on k Rd see Theorem
In fact Theorem
holds verbatim for k k now based on the spectrum S k
fk
idk
ii
p k g and the splitting kRd
pk
iEk
i
But here we are not so much interested in the full system k k but
in its restriction k to Gk d
Note rst that since E k
i is the span of certain decomposable vectors
Gk
i
Gkd kEk
i
i
pk
are nonvoid disjoint k invariant random compact submanifolds of Gk d
hence IP kjG k
i is nonvoid We can thus but will not repeat the argu
ments leading to Theorem for G k
i and obtain the following theorem
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Theorem Invariant Measures on Gk d Let be a linear
RDS on Rd over the ergodic metric DS with two sided time satisfying the
IC of the MET Assume the situation described above Then
iForanyi
pk thesetIP kjGk
i of k invariant measures
supported by G k
i is nonvoid compact and convex
ii Each ergodic k invariant measure realizes one of the k
iie
for each thereis a uniquei f pkgsuchthat
lim
t
tlogkk t uk
kuk
k
i
as
equivalently IP kjG k
i
iii Conversely for any k
i there exists a k invariant measure which
realizes k
i
The spectrum of the induced RDS k on Gk d under can also be deter
mined see Crauel
Furstenberg Khasminskii Formulas
We now generalize the procedure of Subsect
and obtain representa
tions of the sum k
i
i
ikofanyofkexponentsof asa
phase average of a function Q k on Gk d with respect to a k invariant
measure k
i realizing k
i For k these formulas were rst derived by
Baxendale for the white noise and manifold case see Sect
Let be a linear cocycle in Rd satisfying the IC of the MET Let
k
k k jGk d be the cocycle induced on the oriented Grassmannian
Gk d Then
qktu logkktuklogvoltu
tuk
uu
uk Gkd isahelixover
k andforeach k
i there is
a k invariant measure k
i realizing k
iie
lim
t
tqkt u
k
i
k
ias
We now assume that has a generator Then k has a generator see
Theorem iv hence k has a generator see Subsect
and the
helix qk has a generator whose explicit form will now be determined
We will however restrict ourselves to the continuous time white noise
case from which the real noise case can be recovered by ignoring the sum
Pm
j
1

RDS ON GRASSMANNIANS
Let be generated by the linear SDE
Then hence k
satisfy the IC of the MET and k is generated by the linear SDE
dut
mX
j Akjut dWjt
where
Akj u
uk
kX
i
u
ui Ajui ui
uk
Hence k on Gk d is generated by
dut
mX
jhk
j ut dWjt
where with
qk
j u hAkju ui traceAjPk u
Pk u being the orthogonal projection onto U span u
uk
hk
ju TkAkju Akjuqk
juuAjkqk
juIku
Finally the SDE for
rtkktukvoltu
tuk
is
drt
mX
jqk
j utrtdWjt
Hence we obtain Liouville s formula
logdet t jU
mX
jZtqk
jkrudWjr
Converting Stratonovich to Ito integrals for j m and staying with
the case t yields
Ztqk
jkrudWjr
Ztqk
j kr udWjrZthk
jqk
jkrudr
For the calculation of the correction terms we use the following lemma
whose part i generalizes Lemma vii note that I k u k u
1

CHAPTER RDS ON HOMOGENEOUS SPACES
LemmaiForAB Rdd k dandu u
uk
kRd
hAku Bkui traceA BPk u traceAPk u traceBPk u
traceA Pk u BPk u kuk
traceAI PkuBPku
trace APk u trace BPk u kuk
where Pk u is the orthogonal projection onto span u
uk
ii The correction terms h k
jqk
j have the form
hk
jqk
j u traceAjAjIPkuAjPku
Proof i after notes of Schmalfu
We rst assume that
u
uk is an orthogonal basis of u By the de nition of Ak
hAku Bkui X
l mhu
Aul
uku
Bum
uki
For each summand we use the de nition of the scalar product in k Rd as
a determinant which we expand along its nontrivial row column
Ifu
uk is not orthogonal note that both sides of the claimed equa
tion remain unchanged if u
uk are replaced with the orthogonal basis
v
vk obtained by the well known Gram Schmidt procedure i e the
vj are orthogonal with vj uj span u
ujj
k
ii By
hk
jqk
j u hAkj Akj uAkjui hAkju ui
Since by Lemma vii Akj Akj Aj Aj k
hk
jqk
j u hAj Aj kuAkjui hAkjuui
We now use part i of this lemma for both terms on the right hand side of
the last equation and the fact that trace Aj Pk u trace Aj Pk u
Theorem Furstenberg Khasminskii Formula for Sum of
Lyapunov Exponents White Noise Case Let be generated by
the linear SDE
andlet kbetheRDSinducedby onGk d
Ifk
i is a k invariant measure on Gk d realizing k
i then
k
iEk
iQk
mX
jEk
iZqk
j kt dWjt
1

MANIFOLD VERSIONS
where
Qk u traceAPku
mX
j traceAjAjI PkuAjPku
and Pk u is the orthogonal projection onto span u
ukuu
ukIfk
i is a forward Markov measure with marginal k
i onGk d then
k
iZGkdQkuk
idu
In particular there is always a forward Markov measure realizing the
top exponent k
k
Proof Just use Theorem
and Lemma
We could also derive a more explicit expression of the expectations of the
second term in formula
using Malliavin calculus
Fork dwehavePdu I qd
j trace Aj hence
d
pX
i di i dtraceA
Exercise Flag Manifolds Let d dr
d
dr d be a multi index of dimensions and let
FdffV
Vr Vi Rd dimVi dig
be the ag manifold corresponding to Determine the generator of the
RDS induced on F d by Develop Furstenberg Khasminskii formulas
for the r vector whose i th component consists of the sum of di Lyapunov
exponents In particular for
d this gives a formula for the
whole Lyapunov spectrum
Manifold Versions
For further use and ease of reference we will now brie y develop the manifold
versions of Sect and For a C RDS with two sided time on a
Riemannian manifold M we study the RDS induced by the linearization
T on the unit sphere bundle SM the projective bundle P M and the

CHAPTER RDS ON HOMOGENEOUS SPACES
Grassmann bundles GkM The induced systems are denoted by S P
and Gk respectively
Suppose is an invariant measure for for which T satis es the IC
of the MET Then we are able to describe all invariant measures S of S
on SM etc which are lifts of i e which project down to under the
canonical projection of SM M S
If has a generator then all the above induced RDS have generators
which we will determine By introducing polar coordinates in T M respec
tively in kT M x M k TxM we can derive Furstenberg Khasminskii
formulas for the Lyapunov exponents respectively for the sum of Lyapunov
exponents of under
Sphere Bundle and Projective Bundle
Let M be a Riemannian manifold of dimension d with tangent bundle
T M x M TxM unit sphere bundle SM x M SxM and projective
bundle P M x M PxM SxM the unit sphere and PxM the projective
space of TxM To avoid redundancy we only treat the sphere bundle case
hereThe following proposition is the nonlinear analogue of parts of Propo
sition
and Theorem
with analogous proofs use M and
in place of and and recall that SM has compact bers
Proposition Let be a C RDS on M with two sided time let
T be the linearization of on T M and let
Stxs
Ttxs
kTtxsktx sSxM
betheRDSS inducedbyT onSM
Suppose is an ergodic invariant measure Denote by S an invari
ant measure of S which is a lift of i e the image of which under the
projection SM M is and let
I S fS S isS invariantandaliftofg
Then the following hold
i I S is a nonvoid convex compact set
ii Let T under satisfy the IC of the MET Let
M
be the invariant set of full measure on which the MET holds and de
notebyS fidii
pg the Lyapunov spectrum and by TxM
p
i Ei x the Oseledets splitting Then
SEi
xEixSxMi
p
1

MANIFOLD VERSIONS
are disjoint strictly S invariant subbundles of SM and
ISjSEifS IS SxSEi x
asg
is nonvoid
The measures S I S jSEi realize i in the sense that they are
those for which
lim
t
tlogkTtxskiSas
Further for each ergodic S I S there exists a unique i f pg
such that S realizes i
Furstenberg Khasminskii Formulas for RDS on Manifolds
We restrict ourselves to the white noise case
Carverhill was the rst to carry over the classical Furstenberg
Khasminskii formula for the top Lyapunov exponent of a linear SDE
to the case of a nonlinear SDE on a manifold Baxendale
and Arnold
and San Martin
made further contributions There is a wealth of deep
and beautiful results about the geometry of stochastic ows based on
the Lyapunov spectrum of which we mention just Baxendale
Baxendale and Stroock Carverhill Carverhill and Elworthy
Elworthy
and the references therein
and
are
surveys
Let
dxt
mX
jfjxt dWjt
be an SDE on a Riemannian manifold M of dimension d where f
C andfj C forj
m
so that
generates a local
C RDS see Theorem
Further T is generated by
dvt
mX
j Tfj vt dWjt
where Tfj is the natural lift of fj from M to TM Tfj v has horizontal
component fj x and vertical component rfj x v Hence T is generated
by
and
Dvt
mX
j rfjxtvtdWjt
1

CHAPTER RDS ON HOMOGENEOUS SPACES
We introduce polar coordinates in T M x M Tx M Tx M TxM n
fgviaTxM v
rsrkvk
s
v
kvk SxM Then the
local Ck RDS S on SM induced by T is the angular part of T and
is generated by the SDE
dst
mX
j Sfj st dWjt
where Sfj is the natural lift of fj to SM or the projection of T fj from
T M to SM The decomposition of Sfj s into the horizontal component
fj x and vertical component hj x s gives that
is equivalent to
the two coupled SDE
and
Dst
mX
j hjxt st dWjt hjxs rfjxshrfjxssis
The radial part rt kvtk of T is generated by
drt
mX
j qjstrtdWjt qjs hrfjxssi
and
is decoupled from
Therefore for any v T M
log kvtk log kvk mX
jZtqj su dWju
where st solves
with s
v
kvk
Converting the Stratonovich integrals in
into Ito integrals yields
logkvtk logkvk ZtQ su du
mX
jZtqj su dWju
where
Qs qs
mX
j Sfjqjs
and Sfj qj is the function obtained by applying the vector eld Sfj to
the function qj Carverhill and Baxendale have calculated the
correction terms in
more explicitly to which we refer for the purely
geometric proof
1

MANIFOLD VERSIONS
Lemma Let f be a C vector eld on M Sf be the natural lift
offtotheunitspherebundleSMandletqs hrfxs si Thenfor
s SxM
Sfqxs hrrfxfxssikrfxsk
hrfxssi hRfx sfx si
where R is the Riemannian curvature tensor on M
Forthe atspaceM RdTM Rd RdSM Rd Sd R
rfxs AxsAx Dfx theJacobianoffatxrrfxfx s
BxsBx DAxfx
Dfxfx Ax theJacobianof
gx Axfx atxand
Sfqxs hBxssihAxsAxsi hAxssi
Fortheparticularcasefx AxwehaveAx AandBx A
reproducing the correction term of Theorem
Theorem Furstenberg Khasminskii Formulas for SDE on
Manifold Assume that the SDE
satis es the conditions of Theo
rem fork sothatitgeneratesalocalC RDS onM Let be
an ergodic invariant measure ful lling the IC of the MET Let S denote
the local continuous RDS on SM canonically induced by T on T M and
let I S jSEi be the nonvoid set of S invariant measures which realize
i and are lifts of
iIfS ISjSEi then
iZSMQsS ds
mX
jZSMZt qjS t dWjt dS
whereS ds ES ds isthe marginalofS onSMandQisgivenby
ii If S I S jSEi is F measurable then
i ESQZSMQsSds
iii In particular if has an invariant forward Markov measure
whose marginal hence solves the Fokker Planck equation L
whereL f Pm
j fj is the generator of the di usion process cor
responding to the SDE
then S has a forward Markov measure
S I S jSE It hence realizes the top Lyapunov exponent and the

CHAPTER RDS ON HOMOGENEOUS SPACES
marginal S lifts and solves the Fokker Planck equation SL S
where
SLSfmX
j Sfj
is the generator of the di usion process corresponding to the SDE
Thus
ZSMQsS ds
The statement under iii follows from Theorem
since SE is an invari
ant F measurable random compact set Formula
is the nonlinear
analogue of the classical formula
If S is not a Markov measure the expectation of the second term in
can only be treated further by anticipative calculus as in Theorem
For the nonlinear analogue of Theorem
see Arnold and San Martin
Grassmannian Bundles
We now very brie y develop the nonlinear version of Sect in the SDE
case For the geometry we basically follow Baxendale
Let M be a Riemannian manifold of dimension d
Wedenethe
k th exterior power of the tangent bundle by
kTM xMkTxMk
d
and the k th Grassmannian bundle by
GkM xMGkTxM k
d
WehaveGM PM andGdM M
As in Subsect
we consider GkM as a submanifold of the projective
bundle P kT M
GkM P kTM
by representing the ber Gk TxM as the submanifold of those elements of
P k TxM which are projections of decomposable elements in k TxM
Assume that the SDE
on M satis es the conditions of Theorem
for k which ensures the existence and uniqueness of a local C
RDS on M Then the local continuous RDS T generated by
acts

MANIFOLD VERSIONS
naturally on kT M and induces a local continuous RDS kT generated
by
dvk
t
mX
j
kTfjvk
t dWjt
where the vector eld kTfj on kTM is the lift of fj
Similarly T acts naturally on GkM and induces a local continous RDS
Gk generated by
dsk
t
mX
j Gkfjsk
t dWjt
Gkfj being the vector eld on GkM which is the lift of fj
The vertical components of the vector elds kT fj and Gkfj can be
explicitly written down by combining the expressions in
and
with what was done in Subsect
for kRd and Gk d which we leave
as an exercise
By introducing polar coordinates in kT M a decomposable vector
uu
uk can be represented by the pair s r GkM
where s
u
kuk and
r kuk dethui ujik k
vol u
uk
is the volume of the k dimensional parallelepiped spanned by u
uk
see Subsect
The SDE for
rk
tkkTtxukvolTtxu Ttxuk
coupled to
is
drk
t
mX
jqk
jsk
trk
t dWjt
where
qk
j u hrfjx kuuitracerfjxPku
Pk u being the orthogonal projection onto U span u
uku
u
uk
Therefore for any decomposable u
kTMands
u
kuk we obtain
Liouville s equation for det T t x jU
logdetT t xjU
logvolTtxu Ttxuk
vol u
uk
mX
jZtqk
jskdWj
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Converting the Stratonovich integrals in
into Ito integrals yields
logdetT t xjUZtQk skd mX
jZtqk
jskdWj
where
Qksqk
mX
j Gkfjqk
The following lemma is a simultaneous generalization of Lemmas
and
and was proved by Baxendale to whom we refer for the proof
Lemma Let f be a C vector eld on M Gkf the natural lift of
f to the Grassmann bundle GkM and q k u trace rf x Pk u where
Pk u denotes the orthogonal projection onto U span u
uk and
uu
uk Then
Gkfqk x u
tracerrfxfx Pku tracerfxPku
tracerfx I PkurfxPku
traceRfx Pkufx
where R is the Riemannian curvature tensor on M
Finally assume that is an ergodic invariant measure ful lling the IC
of the MET for T Hence the IC of the MET for kT are also satis ed
LetthespectrumbeS kT f k
idk
ii
p k g and the splitting
kTxM pk
iEk
i
x PutGkEk
i
xEk
i
x GkTxM
We hence have
Theorem Furstenberg Khasminskii Formula for the Sum
of Lyapunov Exponents SDE on Manifold Assume that the SDE
satis es the conditions of Theorem for k which ensures
the existence and uniqueness of a local C RDS on M
Let be an ergodic invariant measure ful lling the IC of the MET Let
Gk denote the local continuous RDS on Gk M induced by T on T M and
letI GkjGkEk
i be the nonvoid set of Gk invariant measures which
realize
k
i
i
ik
and are lifts of
iIfGk I GkjGkEk
i then
k
i ZGkMQk sGkds
1

MANIFOLD VERSIONS
mX
jZGkMZt qk
jGktdWjtdGk
whereGk ds EGk ds isthe marginalofGk onGkMandQk is
given by
ii IfGk I GkjGkEk
i is F measurable then
k
i EGkQkZGkMQk sGkds
iii In particular if has an invariant F measurable measure
whose marginal hence solves the Fokker Planck equation L
where
LfPm
j fj is the generator of the di usion process corresponding to
the SDE
then Gk has an F measurable Gk I Gk jGkE k
It hence realizes the top Lyapunov exponent k
k and the
marginal Gk lifts and solves the Fokker Planck equation Gk L Gk
where
GkL Gkf mX
j Gkfj
is the generator of the di usion process corresponding to the SDE
Thus
k
kZGkMQk sGk ds
For k we recover Theorem for P M instead of SM
We nally have a closer look at the case k d where
qd x tracerfx divfx
Qdx divfx
mX
j fjdivfj x
and
becomes Liouville s equation for det T t x
detT t x expmX
j Ztdivfj s x dWjs
see Theorem
This leads to the following formula for the average
Lyapunov exponent
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Corollary Average Lyapunov Exponent Under the as
sumptions of the above theorem
d
pX
i
idi
dZM divf x
mX
j fjdivfj x dx
mX
jZMZt divfj t dWjt d
If is F measurable with marginal on M then
dZMdivf x
mX
j fjdivfj x dx
Further if M is a compact manifold and if has a density p x
then
dZM
mX
j pxdivpxfjx pxdx
and
ifandonlyif t pxdx pxdx
Formula
can be obtained from
using integration by parts
and the divergence theorem see Baxendale
The formula for hence does not require the invariant measure to
be lifted so quantitative information on is easier to obtain than for
See the literature quoted at the beginning of this section and Chappell
Rotation Numbers
The Concept of Rotation Number of a Plane
Motivation
The MET furnishes through its Lyapunov exponents a random and dynamic
analogue of real part of an eigenvalue of a deterministic matrix
The aim of this section is to propose a random and dynamic analogue
of imaginary part of an eigenvalue called rotation number
The basic geometric idea which is due to San Martin
is as follows
While the MET assigns to each tangent vector v TxM at a good base
point x
M a number v which measures the exponential growth
rateofT t xvast
we will assign to each two dimensional
1

ROTATION NUMBERS
tangent plane p TxM a number p which measures the rotation of
T t x insidethemovingplanept x T t xpwithrespect
to a parallel transported reference direction These numbers were shown
to exist in the deterministic case for linear conformal and Killing vector
elds San Martin
and in the random case provided the Lyapunov
spectrum of under is simple see Arnold and San Martin
Here
the analogy goes even further As v is a random variable which takes
on only nitely many di erent basic values realized in canonical directions
similarly p takes on only nitely many basic values the rotation numbers
realized in canonical planes
For d xt A t xt written in polar coordinates r
xx
arctan x
x reads
rt hAtututirt r r
t hAtutvti
where
ut
cos t
sintSvt
sin t
cost Shutvti
h i the standard scalar product of R
The rotation number of the solution starting in r is de ned to be
the linear growth rate of the unreduced i e lifted angular part i e by
lim
T
T
T lim
T TZThA t ut vtidt
if the limit exists and its Lyapunov exponent is the exponential growth
rate of the radial part i e
lim
T logrT
T
lim
T TZThA t ut utidt
if the limit exists
In case of a constant matrix A with eigenvalues A these limits
exist for any initial values and yield
Im A for any initial r
if rotation is counterclockwise and if we use the canonical orientation
of R by the standard frame e e and
ReA
1

CHAPTER RDS ON HOMOGENEOUS SPACES
the smaller one of these numbers being realized if and only if we start in
the corresponding eigenspace
The appearance of time averages in formulas
and
calls
for using the ergodic theorem If xt A t xt and A t is periodic almost
periodic or stationary stochastic with nite mean then the limits in
and
exist See e g Johnson and Moser for the almost periodic
case and Subsect
for the stationary stochastic case
The rotation number plays a basic role in the theory of one
dimensional random Schrodinger operators as the density of states see
e g Bougerol and Lacroix Carmona and Lacroix
and Pastur and
Figotin
We will rst develop the deterministic concept of rotation number for
a vector eld on a Riemannian manifold M of dimension d
Whether
the proposed concept is sensible or not will be tested against the following
for M R and xt A t xt it should reduce to the above elemen
tary concept
for M Rd and xt Axt the rotation number p of an eigenplane
p corresponding to a pair of complex conjugate eigenvalues of A should yield
the imaginary part of the eigenvalues modulo sign
We stress that our concept is in nitesimal in the sense that we accu
mulate in nitesimal rotations in the time interval T as already clearly
visible in formula
thus
it is a continuous time concept i e does not work for discrete time
but see Remark iii
it needs a generator vector eld in the deterministic case RDE
or SDE in the random case as the rotation numbers will be de ned by
Furstenberg Khasminskii type formulas
Linear Case
LetM Rdwithd
EachA Rd dgeneratesa ow t etA
Gl d R which by the left action of Gl d R naturally induces a ow on the
Grassmann manifold G d of two dimensional subspaces planes of Rd
Moreover Gl d R also acts on the Grassmann manifold G d of oriented
planes see Subsect
WehavedimG d dimG d d
tinducesaowonG ddenotedbyp pt G tp
We now want to measure the in nitesimal rotation by t inside the
plane p t For this purpose we pass to the principal bundle St d
G d with structure group SO R where St d is the Stiefel manifold
oforthonormal framesn u v ofRd and n p span u v isthe
canonical projection which maps n to the plane p oriented by the frame n
We have dim St d d By xing the standard bases in R and Rd
1

ROTATION NUMBERS
we obtain a one to one correspondence between St d and the set of d
matrices n with n n I
GldR alsoactsontheleftonStdbytheactionn
uv
gn
gu gv where
Stdnuv
uv
u
kukv hvui
hu uiu
kv hvui
hu uiukA St d
denotes orthonormalization of the linearly independent pair u v in such
a way that u v and u v have the same orientation This action
covers the above action of Gl d R on G d Hence t induces a ow
n nt St tnonStdwhichcoversthe owG onG d
In order to measure the rotation of the frame n t St t n inside
theplanept G tp nt weneedareferenceframent inthe
ber over p t This reference frame is obtained by parallel transporting n
by means of a connection in St d G d Suppose such a connection is
chosen and since SO R acts transitively on the bers of St d G d
from the right we have
ntntgtgtSOR
see Fig
Figure Rotation of the frame n t inside the plane p t relative to
nt
Now g t SO R uniquely corresponds to an angle t for which
we need a di erential equation analogous to t hAut vti in the case of
R We then would de ne the rotation number of the plane p under the
ow t asin
There is the following canonical choice of a connection on St d
G d see Narasimhan and Ramanan
Ifnt Stdisadieren
tiable path then di erentiating n t n t I yields
ntntntnt ntnt ntnt
Thus the tangent space TnSt d at n can be identi ed with the set of d
matrices w such that n w is skew symmetric Therefore the expression
n
ndn TnStd so Rdenesanso Rvalued formin
St d This form is the connection form of the canonical connection on
St d G d and its kernel is the horizontal subspace Hn of TnSt d
Hn Vn The connection is invariant under the left action of SO d R on
Std iegHn Hgnforallg SOdR
1

CHAPTER RDS ON HOMOGENEOUS SPACES
The interpretation of n w
n w is as follows It measures the in
nitesimal deformation of the frame n u v
p without mov
ing the plane p in the direction of w TnSt d A horizontal curve
nt ut vt inStdisthusoneforwhichbothut andvt are
orthogonal to the plane p t span u t v t Hence from the point of
view of an observer inside p t n t does not rotate
Eachd dmatrixAgivesrisetoavector eldAonSt d by
And
dtSt tnjt
so that
nt utvt Ant
According to the above n A n measures the in nitesimal rotation of
n
p in the direction of w A n so that the following expression
o ers itself as the de nition of the rotation number of the plane p G d
under the action of the ow t generated by the matrix A
p lim
T TZT
ntAntdt
provided the limit exists and is independent of the element n
p Strictly speaking p so R which is identi ed with R via
Lemma i The vector eld on St d induced by A is given for
n uv StMbythed matrix
An Au hAuuiuAv hAvviv hAuvi hAvuiu
ii The connection form n A n is given by
nAn
hu vi
hu vi
hAu vi
hAu vi
iiiIfgtinnt ntgt see
is represented as g t
cost sint
sintcost
then
t hutvti hAutvti
Proof iBydenition An
d
dt
etAn jt To calculate this
replace u v in
by etAu etAv and di erentiate with respect to t
1

ROTATION NUMBERS
ii This immediately follows from the formula n w n w and
iii Di erentiating
with respect to t and applying n t to
both sides gives the following di erential equation in SO R
gtgtntAntgI
But using the representation of g t we obtain
gt gt
t
t
so that
follows from
Since only obvious alterations of this procedure are necessary to treat
nonautonomous linear equations instead of autonomous ones we nally
arrive at the following de nition
De nition Rotation Number Consider the linear di erential
equation xt A t xt in Rd for d with t A t locally integrable
The rotation number p of the oriented plane p G d under the ow
generated by the above equation is de ned by
p lim
T TZThut vtidt lim
T TZThA t ut vtidt
provided the limit exists where h i is the standard scalar product in Rd
and nt ut vt is the ow induced in St d with arbitrary initial frame
nn
p
Notethatthe owntinSt d solvestheequationnt ut vt Atnt
more precisely
ut Atut hAtut utiut
vt Atvt hAtvt vtivt hAtut vti hAtvt utiut
It turns out that this de nition passes our test
Proposition Rotation Number is Independent of Frame
i The existence or nonexistence of the limit in
and its value in
case of existence are independent of the particular choice of n
p
ii p changes sign with the orientation of p
Proof iLetnn
p withn n Then
ut cos tut sin tvt vt
sin tut cos tvt
1

CHAPTER RDS ON HOMOGENEOUS SPACES
where t t is continuous with t
since n nt is injective
Using hut vti hut vti which follows from di erentiating hut vti
we obtain
hut vti t hut vti
whence
TZThut vtidt T T
TZThut vtidt
Since always T T as T
the existence or non existence of
the limit and its value in case of existence is independent of the chosen
frameii If n
uvandn
u v have di erent orientation repeat the
calculation in i with u v
For d
obviously reduces to
Further in the autonomous
case xt Axt every plane p Rd has a rotation number in particular
the imaginary parts of eigenvalues can be recovered as rotation numbers of
their eigenplanes for details see San Martin Sect
Example Consider in R
ABB CCA
In this case the integrand in
is constant and equal to p n An
If ij span ei ej then
ij fori
and
j but
spancose sinecose sine
cos cos
sin sin
always modulo the sign depending on the orientation of the corresponding
plane
Nonlinear Case
Let f be a complete C vector eld on a Riemannian manifold M of di
mension dim M d
Soxt fxtgeneratesaow ttRof
di eomorphisms of M The linearized ow T on T M is generated by
vt T f vt hence its vertical component by vt rf xt vt with vt de
noting the absolute derivative
1

ROTATION NUMBERS
We now want to de ne the rotation number p of an oriented plane
p TxM under T T induces a ow denoted by G on the Grassmann
bundle
GM xMGTxM
where G TxM is the Grassmann manifold of oriented planes in TxM
We want to measure the in nitesimal rotation of T t inside the plane
p t For this purpose we pass to the principal bundle St M G M
with structure group SO R where St M is the Stiefel bundle
StM xMStTxM
and St TxM is the Stiefel manifold of orthonormal frames n u v in
TxMh ix
T alsoinducesa owSt onStMwhichcoversG onGM
denedforn uv StMby
nt utvt Sttn
TtuTtv
where is de ned as in the linear case by
Parallel transport of n is now accomplished by the following canonical
choice of a connection on St M G M We take the composition of the
Levi Civita connection on T M for moving along the manifold with the
canonical connection from the linear case for moving inside St TxM
G TxM for xed x It will be of fundamental importance that this
connection is compatible with the Riemannian metric h i i e we can
di erentiate the scalar product by the usual product rule
The comparison of n t St t n with the parallel transported refer
enceframentyieldsnt ntgtwithgt SO RseeFigure
withSt d St andG d replacedwithStMSt andGM and
the angle t corresponding to g t satis es a di erential equation which
we will now determine
The ownt ut vt St tnisgeneratedbynt Stfnt
whereStfistheliftofftoStM denedby
Stfn d
dtSt tnjt
whose horizontal component is the horizontal lift of f to St M which by
de nition does not contribute to rotation and whose vertical component
is
Stfnv rf n
where rf isdenedasin
with rf in place of A Hence
ntrfnt
1

CHAPTER RDS ON HOMOGENEOUS SPACES
with
rf n rfu hrfu uiurfv hrfv viv hrfu vi hrfv uiu
From this as above in the linear case
t hut vti hrfut vti
This suggests to de ne the rotation number of p G M by
p lim
T TZThut vtidt lim
T TZThrfut vtidt
provided the limit exists Clearly we obtain the autonomous version of
De nition
as a particular case For M Rd with the standard scalar
productandfx Ax wehave rfu Au
Now Proposition
remains true since our connection is compatible
with the Riemannian metric meaning that d
dt hut vti hut vti hut vti
so that the proof for the linear case remains valid Hence
the existence or nonexistence of the limit in
and its value in
case of existence are independent of the particular choice of n
p
p changes sign with orientation of p
We stress that even if M is compact rotation numbers depend in general
on the Riemannian metric for examples see
They are therefore
not as robust as the Lyapunov exponents
Remark Other Approaches i There are other general
izations of the two dimensional linear rotation number to some higher
dimensional systems assigning roughly speaking a rotation number to
a ow in the symplectic group see Ruelle
Johnson
Delyon
and Foulon
However only one number is obtained even for higher
dimensions
ii To measure rotation inside a k dimensional oriented subspace of
the tangent space we could consider the principal bundle StkM Gk M
k
d with for k non Abelian structure group SO k R which
would lead to a skew symmetric rotation matrix
sokR FortheSDE
case this is treated by Ru no Chap
iii Ru no Chap also built a bridge between Poincare s orig
inal de nition of rotation number for orientation preserving homeomor
phisms of S and the continuous time linear concept for xt A t xt in
R by proving a sampling theorem If h is Poincare s rotation number
of the discretized system with time increment h on S and if is the ro
tation number of the linear system then modulo conditions h h
ash

ROTATION NUMBERS
Rotation Numbers for RDE
Let M be a Riemannian manifold of dimension d and assume that the
random vector eld f x satis es the conditions of Theorem
which
ensures the existence and uniqueness of a local C RDS on M generated
by
xtftxt
The linearized RDS T solves
vtTftxvt
equivalently with vt denoting the absolute derivative
vtrft xvt
and is a linear bundle RDS over
restricted to the invariant
set graph E
M where E is the set of never exploding initial
values x M for equation
see Subsect
We will de ne the rotation number of an oriented plane p TxM
in complete analogy to the case of an autonomous vector eld of Subsect
The linear RDS T induces an RDS G on the Grassmann bundle
G M and an RDS St on the Stiefel bundle St M which covers G
TheRDSSt isgeneratedbyanRDEnt Stf t x nt whose
horizontal component is the horizontal lift of f and whose vertical compo
nent rf is explicitly given in
with
ntrftxnt
In view of
it is evident to de ne the rotation number p of p by
p
lim
T TZThut vtidt
lim
T TZThrf t utvtidt
provided the limit exists As made clear above Proposition
remains
valid
Lifts of Invariant Measures to the Stiefel Bundle
Let be an ergodic invariant measure for necessarily supported by
graph E which satis es krfk L
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Then the IC of the MET are satis ed see Theorem
providing us
with the Lyapunov spectrum S f i di i
pg a invariant set
graph E of full measure and a splitting
TxMEx
Epx
x
As we want to use the ergodic theorem to prove the existence of the limit in
the rst task is to determine all invariant measures St of St
which are lifts of as every number
St ZStMhrfuvidSt
quali es as a possible rotation number
As usual we freely identify a measure St on St M with marginal
on M with its factorization St x obtained from
StddxdnSt xdnddx
Recall that St is invariant under St if and only if for each t R
SttxStxSttx
as
We also need to consider the RDS F induced by T on the ag bundle
FM xMFTxM
whereF TxM isthemanifoldof agsf V p inwhichVandp
are subspaces of TxM of dimension and respectively
We now need for the remainder of this subsection the following
Assumption Let the RDS under have simple Lyapunov spectrum
See Remark
for a discussion
Under this assumption all Oseledets spaces Ei are one dimensional
For each pair i j
ij di jthespacesEi xand
Ej x uniquely de ne a measurable random plane eld pij x
spanEi xEj x andarandomageldfij x Ei x
pij x The invariance of the Ei s implies the invariance of pij and fij
or equivalently
GijxdppijxdpFijxdffijxdf
areliftsof MoreoverF ijliftsG ijform GMto F M
WenowliftF ijfurtherfrom F Mto StM
1

ROTATION NUMBERS
Givenaagf V p F MWeselectn uv StMfrom
fasfollows Chooseu Vkuk andv pkvk hu vi These
two vectors give rise to four possible orthonormal frames of p selected from
f namely n
n
uvn
uvn
uvn
uv
Here n and n have the same orientation while n and n also have the
same but opposite orientation
Lemma iLetG fg idg g ggbethegroupofdieo
morphisms ofStMdenedbygin ni i
ThenG Z isan
Abelian group which is isomorphic to the Klein group of four elements G
acts freely on the left on St M
ii ThegroupGcommuteswiththeRDSSt andF M StMG
HenceStMisa foldcoveringofF M
Proof i can be easily checked
ii follows from the de nition of St and the fact that G commutes
with
Lemma Lift of Measure from Flag to Stiefel Bundle Let
fxV
p x F Mbeaninvariantrandomagielet
Ftxfxftx
Then the invariant measure F
x
f x can be lifted from
F M to St M to yield exactly N f ergodic invariant measures where
Nf
or More speci cally
i N f There is exactly one ergodic lift St given by
StxX
gGgnxX
gG
gnx
where n x
u x v x is an arbitrary measurable selection from
fxwithuxVx
ii N f There are exactly two di erent ergodic lifts St and
St given by
StxX
gH
gnxStxX
g GnH
gnx
whereH Hi fg gigfori
is a two element subgroup of G and
n
x
u
xv
x is some measurable selection from f x
with u
xVx
Moreover
1

CHAPTER RDS ON HOMOGENEOUS SPACES
aforHfgggorHfgggStkxk
charges points
with di erent orientation
bforHfgggStkxk
charges points with the same
orientation
iii N f There are exactly four di erent ergodic lifts described
by the Dirac measures
Stkx
gknxk
where n
x
u
xv
x is an invariant measurable selec
tionfromf x withu
xVxieSttxn
x
ntx
Proof The same reasoning as in Remark gives the existence of an
ergodic lift of f x to St M By Lemma ii all elements of the
setfggg Garealsolifts Thesetofg Gforwhichg
is obviously
a subgroup H of G
Now the following possibilities arise
i N f H G Take some measurable selection n x from
f x and write
xX
i
ixginx
ixX
i
ix
Equating weights in g
yields
and
a s while
equating weights in g
gives
hence i x
iiNf H Hifggigi
To x ideas consider
H H the other cases are similar Put
andgg
These two measures are ergodic hence singular so that there are subsets
k xinthepointbreoverf xforwhichi x k x
ik
Now take a measurable selection n
x
x such that
u
xVxSinceg
g
g
has the
form
x
xnx
xgn x
Applying g to both sides of
gives
As a result
ngn
gngn
iii Nf
H fid g The four ergodic measures i gi
i
are all di erent hence singular Their support i fnig
is thus one point and i
gin In particular St t x ni x
nit xfori

ROTATION NUMBERS
With this lemma we have determined all possible lifts of the Dirac measure
supported by the canonical ag f fij Ei pij pij span Ei Ej
from the ag bundle F M to the Stiefel bundle St M
On the other hand fij projects to the canonical invariant plane pij
G M Applying the usual argument the possible lifts of the corresponding
Dirac measure to G M are either
one measure pij
pij or
two Dirac measures on pij and pij
Finally on the principal SO R bundle St M G M the projec
tion identi es all frames of the same orientation
cases i and ii a of Lemma
on St M project to case
on
G M while
cases ii b and iii project to case
Proposition Rotation Number of Invariant Measure Let
be the local C RDS on M generated by the RDE
Let be an
ergodic invariant measure satisfying the integrability condition
Assume that under has simple Lyapunov spectrum
Thenhrf i L St foranyliftSt of from Mto
St M i e the rotation number of St
St ZStMhrfuvidSt
is nite
More speci cally consider the Nij N fij measures St k
ij con
structed in Lemma from the canonical frame fij Ei pij Then
we have
i for Nij
St ij
iiaforNij andcasea St k
ij fork
iibforNij andcaseb Stij
St ij
iii for Nij
Stij Stij
St ij
St ij
Proof The integrability assertion follows from jhrf u vij krfk
L iWehave
St ijZMhrfuvi hrf u vi hrf u vi hrfu vid
which vanishes since u v hrf u vi is bilinear
ii and iii are also simple consequences of bilinearity

CHAPTER RDS ON HOMOGENEOUS SPACES
Rotation Numbers for Canonical Planes
As a rst step towards an MET for rotation numbers we now prove that
rotation numbers exist for canonical planes
pij x
spanEi xEjx ijijd
However the MET does not supply these planes with a canonical frame or
orientation We hence start by measurably selecting an arbitrary frame
nij uv
pij from pij To determine to which invariant measure
on St M the orbit St t nij is attracted we construct a canonical frame
nij u v inpijfromnij u v asfollows
If proj ij pij Ei denotes projection onto Ei along Ej take
nij uv projiju v
We can assume without loss of generality that proj ij u
otherwise we
use v u to start with Note that since u Ei nij is a selection from the
ag fij i e it is one of the four canonical frames selected from fij in the
way used in Lemma
and nij and nij give pij the same orientation
Theorem Rotation Numbers for Canonical Planes Let
the RDE xt f t xt on a Riemannian manifold M with d dim M
satisfy the assumptions of Theorem
and denote by the local C RDS
generated by it Let be an ergodic invariant measure which satis es
krfk L implying the validity of the IC of the MET Assume that
the RDS under has simple Lyapunov spectrum
Then each canonical plane pij span Ei Ej
i j d oriented
with an arbitrary frame nij has a rotation number pij ij the absolute
value of which is a xed non random number which is uniquely determined
by the unoriented pij and the sign of which is uniquely determined by the
orientation
More speci cally ij is determined as follows Let fij Ei pij be
the corresponding canonical ag and let Nij
be the number of ways
it can be lifted to an invariant measure on St M
iIfNij
or if Nij
but the measures charge frames with
di erent orientation then ij irrespective of any orientation
ii If Nij but the measures charge frames with the same orienta
tion or if Nij
then
ijx
St ij
x Aij
St ij
x Aij
where
St ijZStMhrfuvidSt ij
1

ROTATION NUMBERS
Aijfxnijx suppStijg
Nij
f x nij x suppSt ij suppSt ijg Nij
and nij is the canonical frame associated with nij
Proof Ifnij x nij x thennij x suppSt k
ij x for some
measure hence by the ergodic theorem
ij lim
T TZThrf t x ut vtidtZStMhrfuvidSt k
ij
exists
Ifnij x
nij x we prove that with nij t
ut vt and
nijt utvt
lim
T TZThrf t x utvtidt
lim
T TZThrf t x utvtidt
We have
jhrfuvi hrfuvij krfkduu dvv
krfkd u u
the latter being true because for two orthonormal frames u v and u v
inthesameplanedvv duu
By Lemma i following this proof
lim
t tlogdut ut
ji
uu
uu
so that part ii of Lemma
gives the equality
The remaining assertions immediately follow Lemma
and Propo
sition
To prepare the lemma needed in the above proof we come back to the RDE
for and
respectively
for T
By the real noise version of the statements preceding Theorem
us
ing polar coordinates in T M x M Tx M
turns into the equiv
alent coupled RDE
stSftxst
1

CHAPTER RDS ON HOMOGENEOUS SPACES
for the angular part where Sf is the vector eld on SM induced by T f on
T M equivalently
strft xsthrft xststist
and
rthrft xststirt
for the radial part The RDE
generates an RDS S on SM over
Inserting S into
and solving it gives the Lyapunov exponent
of v Tx M as a time average writing s
v
kvkandst S ts
xv lim
T TZThrf t x st stidt
Lemma Let be generated by
and let be an ergodic
invariant measure for which
hence the MET holds with cor
responding invariant set of full measure Let for
x
vx
p
ivixTxMp
i Ei x anddene
iminfi vigjminfiivig
j p if the latter set is empty
iIfdss ks sk distanceinTxMandifwewrites
v
kvk
si
vi
kvikst S tsandsi t S tsi thenfor
x
lim
t tlogdst si t
ji
p
ii Iff SM Rissuchthat
jfxsfxsjLxdss withLL
and if for some lift S of
lim
T TZTft xsitdtZSMfdSSas
then also
lim
T TZTft xstdtZSMfdS Sas
1

ROTATION NUMBERS
Proof i Since
dssi ks sik sin ssi
where s si is the angle between s and si and since
jsin ssijkv vik
kvk kvi k
the statement follows immediately from the obvious manifold version of
Corollary
ii By assumption
TZTft xstft xsitdt
TZTLt xdstsitdt
By i there is for any
aTx
suchthatdst si t
c
forallt T wherecRML xd
ThusforT T
TZT
TLt xdstsitdt cTZT
TLtxdt
Bytheergodictheorem TRTLdt c as sothereisaT x
for which
TZT
TLdt cforallT T
Finally choose T x
such that for all T T
TZTLt xdstsitdt
This makes the right hand side of
less than for all T
maxT T
Rotation Numbers for Arbitrary Planes
Our nal steps towards an MET for rotation numbers are now i to prove
the existence of the rotation number for an arbitrary plane and ii to show
that this rotation number is a random variable whose possible values are
just the canonical rotation numbers ij
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Let now p p x G TxM be an arbitrary plane We will determine
the strongest of the canonical planes pij x in TxM such that p has
a component in direction of pij
Let as usual v
d
iviTxMd
i Ei x be the decomposition
of a vector according to the Oseledets splitting Introduce the subspaces
Fix
dX
jiEjxFdfg
We de ne the following indices for p G TxM
iixp
minfi there exists v p with vi g
iixp
minfi i there exists v p with vi g
iixp
minfi i vi forv pwithvi g
d
if above set is empty
jjxp
minfi there exists v p Fi with vi g
jjxp
minfijvpFi withvig
d
if above set is empty
We have
iijjd
andiid
The next three lemmas are analogues of Lemma
onGMFM
and St M respectively
Lemma Theorbitspt G tpandpij t G tpij
satisfy
lim
t tlogdptpijt
max i
ijj
ij
jj
ij
where d p p is the distance of the two planes p p TxM considered
as elements of the projective space P TxM
Proof We represent a plane p span u v as the element in P TxM
corresponding to u v TxM The Oseledets splitting in T M is given
by i j Ei Ej see the manifold version of Theorem
The relative
position of p to this splitting is described by our indices as
puiuiu
vjvjv
whereui Ei ui Ei u Fi
vjEjvjEjandvFj
Hence
puivjuivjuivj
1

ROTATION NUMBERS
where
represents an element whose exponential growth rate is strictly
smaller than min i j i j The strongest component of p is
ui vj For the second strongest component we have to distinguish the
following cases
ii jTheneitherai j
i jorbi j
i j In case a Lemma applied to T M yields
lim
t tlogdpt pijt
ij
ij
ij
i
imaxi
ijj
In case b again by Lemma
lim
t tlogdpt pij t
jjmaxi
ijj
iiijNowui vj
so that the second strongest component
is ui vj thus
lim
t tlogdpt pij t
ij
ij
jj
We now look at frames inside the planes and study their exponential con
vergence Let n u v be a frame in p Assume without loss of generality
that ui proji u
Let v be the only unit vector in p Fj such
that u v and ui v have the same orientation put vj proj j v and
nij ui vj
ui vj
Note that ni j is one of our standard frames in pi j since ui Ei and n
and ni j have the same orientation
As an intermediate step we consider the orbits of
fVpspanupfijEipij
underF inF M
Lemma Theorbitsft F tfandfij t F tfij
satisfy
lim
t tlogdftfij t max i
ijj
where
dffdVpVpdVVdpp
with d the distance in P TxM and d the distance in P TxM
1

CHAPTER RDS ON HOMOGENEOUS SPACES
Proof By de nition i i We have to distinguish two cases
ii j Theni i IndeedinthiscaseV Ei Fi Since
pFi
Fj and p is spanned by V and p Fi it follows that
pEiFiThusiisoii
By Lemma
lim
t tlogdVtEit
i
i
i
i
This and Lemma
give the result
iiijThenijInfactV Ei Fi Ei FjSince
pFi
Fj it follows that p Ei Fj hence i j and nally
i j The result follows again by combining Lemma
for d and
Lemma
for d
The nal step now consists of lifting this result from F M to St M
Lemma Theorbitsnt St tnandnij t St tnij
satisfy
lim
t tlogdnt nij t max i
ijj
wheredn n dff
Proof By Lemma
dftfij t issmallforlarget Forevery
such t there exists an open set Ut of F M that trivializes the bundle
StM F Mandsuchthatft fij t Ut Letuscheckthat
nt andnij t areinthesamecomponentof Ut
For large t d V t Ei t is small and moreover by our choice of
ui dut ui t issmallinT t xM Thisimpliesthatnt iseither
in the component which contains ni j t ui t vj t or in the com
ponent which contains ui t vj t However by our choice of v and
since orientation is unaltered by projection and orthonormalization the
rst possibility is true Thus for large t
dntnijt dftfijt
and the assertion follows from Lemma
We are nally in a position to derive the main result of this subsection
Theorem MET for Rotation Numbers Real Noise
Case Let the RDE xt f t xt on a Riemannian manifold M
with d dim M satisfy the assumptions of Theorem
and denote

ROTATION NUMBERS
by the local C RDS generated by it Let be an ergodic invariant
measure which satis es krfk L implying the validity of the IC of
the MET Assume that the RDS under has simple Lyapunov spectrum
Thenhrf i L St foranyliftSt of from Mto
St M and there exists a invariant set
M of full measure such
that on for any plane p G M the rotation number of p
p lim
T TZThrf t ut vtidt
existsHerent ut vt St tnwithanarbitraryn
p
The quantity is a random variable on G M which takes on
only nitely many values namely the d d
rotation numbers ij of the
properly oriented canonical planes pij span Ei Ej for i j d
determined in Theorem
More precisely
pij
where i j is the rotation number for pi j the strongest canonical plane
contained in p and oriented by ni j
Proof Let be the subset of the validity of the MET for which the
ergodic theorem on St M for any lift St holds The only statement
which needs to be proved is that
lim
T TZThrf t x utvtidt
lim
T TZThrft xuitvjtidt
But this follows from Lemma
since
jhrfuvi hrfui vjij krfkduui dvvj
and for large t and a certain constant c by Lemma
dutuit dvtvjt cdntnijt
exponentially fast
Theorem
tells us that the rotation numbers ij of the canonical
planes pij span Ei Ej are the basic ones and that any other plane p
has a rotation number p that picks the value ij whenever p has pij as
its strongest component We thus have exactly the same situation as for

CHAPTER RDS ON HOMOGENEOUS SPACES
the Lyapunov exponents of tangent vectors in the classical MET Also note
the striking similarity between
and
Also the rotation number p for an oriented plane p G M is
uniquely determined while for an unoriented plane it is only determined
up to its sign
Particular cases
i Our results apply also to the deterministic situation xt f xt
where is a measure invariant under the local ow t generated by f on
M There is however in general no simpli cation
iiIf dxd
x dxPd ie x isanonrandom xedpointof
f we can restrict our considerations to the tangent space Tx M Rd
andthustoG Rd F Rd andSt Rd Inthiscasevt T t x v
satis es
vtAtvtA
fx
xxx
iii For d there exists a unique rotation number for and
The particular case dx d x dx P d leads back to the concept of
rotation numbers for linear systems in R by which we started our discussion
in Subsect
Choosing the standard frame n e e in R and an
invariant measure on S realizing
R
EhAuvi
Remark Lyapunov Spectrum not Simple At this mo
ment we are not able to prove the existence of rotation numbers for a
situation where the Lyapunov spectrum is not simple This is due to the
lack of knowledge about the dynamics of the cocycle T inside a higher
dimensional Oseledets space Ei As Example
shows there can be
continuously many di erent rotation numbers in one Oseledets space
The assumption of simple spectrum is however generically true for
truly random situations as noise tends to split multiplicities Evidence for
this claim has been found in many situations see e g Arnold and Nguyen
Dinh Cong for products of random matrices and Remark
Rotation Numbers for SDE
With the preparations of Subsect
and with the real noise case of Sub
sect
in mind the concept of rotation number in the white noise case
should now be rather evident For more details examples and discussions
see Ru no
and Arnold and Imkeller
We restrict ourselves to the linear case since it already reveals the nec
essary modi cations of the real noise concept
1

ROTATION NUMBERS
Let
dxt A xtdt mX
j Ajxt dWjt
mX
j Ajxt dWjt
be a linear SDE in Rd with d generating the linear RDS Then the
RDS St induced on the Stiefel manifold St d of orthonormal frames
n u v is generated by the SDE
dnt
mX
j Ajnt dWjt
where the lift Aj n to St d of the linear vector eld Aj x in Rd is explicitly
given by the d matrix
Aju v Aju hAju uiuAjv hAjv viv hAju vi hAjv uiu
The SDE for ut and vt are hence respectively
dut
mX
j Ajut hAjut utiut dWjt
dvt
mX
j Ajvt hAjvt vtivt hAjut vti hAjvt uti ut dWjt
If nt
ntgt where nt is the parallel transported initial frame n and
gt
cos t
sin t
sintcost SORthen
dthdutvtivtdut
mX
j hAjut vti dWjt
and we naturally arrive at the following de nition of the rotation number
of the oriented plane p G d under
p lim
T TZTh dut vti lim
T
mX
j TZThAjut vti dWjt
provided the stochastic integrals in
make sense and the limit exists
We again have the statements of Proposition
since dhut vti
h dut vti hut dvti
1

CHAPTER RDS ON HOMOGENEOUS SPACES
It is more di cult to assure the existence of the rotation number in
the SDE case because of the following technical problem which we already
encountered in Subsect
If we mimic the procedure in Subsect
leading to the MET for rotation numbers we have to solve the SDE
for random initial values e g nij chosen in pij span Ei Ej which are in
general not F measurable This calls for the use of anticipative calculus
We quote the main result of work see
which is still in progress
Theorem MET for Rotation Numbers Linear White
Noise Case Consider the linear SDE
and assume that the
smallest closed subgroup of Gl d R generated by the matrices A Am
isequaltoSldR orGl dR Then
i The Lyapunov spectrum of is simple The distributions of the
forward ag the backward ag and of the Oseledets spaces E Ed
have C densities with respect to the canonical Riemannian volume of the
respective state spaces
ii The rotation numbers pij
ij of the canonical planes pij
span Ei Ej
i j d oriented by some nij exist and are given by
ijZStdRuv DuvdStij
where St ij is a certain lift of the invariant Dirac measure in fij Ei
pijfromF d toSt d
Ruv hAuvi mX
jRjuv
Rj u v hAju Ajvi hAju vihAjv vi hAju uihAjv ui hAju vi
hAju uihAju vi
and D u v is an additional Stratonovich Skorokhod correction term
iii The rotation number p for every plane p G d exists and
is a random variable on G d which takes on the values ij More
speci cally
pij
where i and j are determined as in Theorem
ivInparticular ZSt dRuvdSt
1

ROTATION NUMBERS
where St is the marginal of the F measurable invariant measure St
from ii
Further for each deterministic plane p G d
p
Pas
v In dimension d the unique rotation number of R with canonical
orientation is given by
ZShAuvi mX
j hAju vi hAjv vi hAju uiAd u
where is a solution of the Fokker Planck equation on S and v
uu
Proof We will only sketch the very technical proof and refer to for
details
i For the simplicity of the spectrum see Remark
The existence
of the densities was proved by Imkeller
ii Hard work using Malliavin calculus is needed to prove the existence
of pij and an explicit form of the correction term D u v
If however n is F measurable e g constant then hAj ut vti is a
semimartingale and
ZThAjut vti dWjt ZThAjut vtidWjt ZT Rj ut vt dt
where the Stratonovich Ito correction term can be calculated by standard
methods to be equal to
Taking time averages the Ito integral cancels due to the following
lemma for a proof see e g Liptser
Theorem
applied to Mt
R thAj us vsidW js
Lemma Let M be a local square integrable martingale Then
Z dhMit
t
lim
tMt
t
Pas
For an F measurable n u v
p we arrive at
p lim
T TZTRutvtdt
existence provided
1

CHAPTER RDS ON HOMOGENEOUS SPACES
iii The di cult part of the proof is to carry over Lemma
to time
averages of anticipative Stratonovich integrals
iv Since p is F measurable the correction term D u v vanishes
and we can use
and the ergodic theorem
Further due to the existence of densities see i for each xed p
ip andjp Pashence p
Pas
v For d further terms in R u v vanish as can be easily checked
Since by our assumptions the hypoellipticity condition
is valid the
Fokker Planck equation on S either has a unique solution or has exactly
two symmetric solutions which give the same average

Part III
Smooth Random
Dynamical Systems
1

Chapter
Invariant Manifolds
Summary
This longest and technically most complicated chapter of the book is de
voted to the theory of invariant manifolds of RDS Our aim is to give an
up to date picture which is as complete and as reliable as possible
For the whole chapter we use a dynamical method which today is
widely used in the deterministic case and was carried over to RDS by Wan
ner
The idea is to work with a scale of Banach spaces of orbits
with certain exponential growth rates by which invariant manifolds can be
characterized The advantage of this method is besides yielding global
invariant manifolds and their regularity that it also gives the Hartman
Grobman theorem for RDS thus technically unifying the whole chapter
After explaining the problem Sect
and doing various preparations
to simplify the main work Sect
the long Sect is devoted to
the construction of global invariant manifolds Theorem for unstable
manifolds Theorem
for stable manifolds and Theorems
and
for center manifolds The moral of the two key technical lemmas
and is that all one has to do is solving a ne di erence equations
by the variation of constants formula
The outcome of these constructions are invariant manifolds which are
just Lipschitz continuous no matter how smooth the RDS under consider
ation is We will improve the regularity by playing with the parameters
in the scale of Banach spaces To accomplish e g Ck stable manifolds for
a Ck RDS one has to place two parameters a b into the gap between
the two Lyapunov exponents at which the spectrum is split such that also
ka b which for k is not always possible and is the origin of the gap
conditions Theorem
1

CHAPTER INVARIANT MANIFOLDS
In Sect we complete the picture by showing that the splitting of the
Lyapunov spectrum between any two exponents yields foliations of invariant
manifolds Theorem
These can be used as nonlinear coordinates
to topologically decouple the nonlinear RDS into blocks according to the
linear splitting of the MET Theorem
There is just one more step
to topologically linearize all blocks having non zero Lyapunov exponents
Hartman Grobman Theorem
Due to very restrictive conditions global invariant manifolds rarely ex
ist In contrast local invariant manifolds usually do exist Theorem
and some can be dynamically characterized and extended to global objects
Theorem
We hope that after all this the reader will enjoy studying the examples
in Sect
The Problem of Invariant Manifolds
Suppose is a Ck RDS k with two sided time on a d dimensional
Riemannian manifold M Let be an ergodic invariant measure of such
that T satis es the IC of the MET Consider the Lyapunov spectrum
S fidii
pg of under and the invariant splitting TxM
p
i Ei x where x is from the invariant set
M of good
points which has full measure
Now choose
f pg and consider
Ex
iiEixTxM
This is a subbundle of T M of dimension d Pi i di which is invariant
under T
TtxExEtx
where t x
t t x is the corresponding skew product ow
The problem of invariant manifolds consists of bending E down
from T M to the manifold M that is of nding nonlinear analogues
of E More precisely We seek local Ck submanifolds M x with
x M x with the following properties
iLocalclosenessofM andE TxM x E x
ii Invariance under
tMxMtxlocally
iii Dynamical characterization We aim at characterizing the points
y M xbytheexponentialgrowthrateofd t y t x as
t
where this rate should be related to the Lyapunov exponents of
of tangent vectors v under T collected in See Fig
1

THE PROBLEM OF INVARIANT MANIFOLDS
Figure Invariant manifold M x tangent to E x
The invariant manifolds are of utmost importance for the study of the
dynamical behavior of For example they allow us to reduce an RDS to
a lower dimensional manifold to study stability Since we try to obtain the
nonlinear object M from the linear object E invariant manifold theory
is part of what is called local theory
Particular cases
i If contains all negative vanishing positive non positive or non
negative Lyapunov exponents we obtain the random versions of the fol
lowing ve classical invariant manifolds the stable center unstable
center stable and center unstable manifold denoted respectively by Ms
Mc Mu Mcs Mcu We also call the random versions of these manifolds
classical stable manifold classical center manifold etc
ii If is any interval from the spectrum i e if for some i j p
ij fi
jgthenEij k i k jEkandMijtangenttoEij
is called the center manifold For i we obtain the unstable manifold
M j For j p we obtain the stable manifold Mip and for i j we obtain
the Oseledets manifold Mi tangent to the Oseledets space Ei The invariant
manifold M p x has dimension d and will be a random neighborhood
of x possibly equal to M We sometimes add the attribute generalized to
distinguish these manifolds from their classical versions
iii We could select more general subsets from the spectrum of
under eg f pgifp
However it will in general not be
possible to obtain smooth or even Lipschitz continuous invariant mani
folds tangent to such an E as deterministic counterexamples show see
Aulbach and Wanner Sect But see Remark
Our program for this chapter is to construct the global and local center
manifolds Mij of a Ck RDS under an invariant measure We stress
that this construction is based on the information provided by the MET
for T and hence depends in particular crucially on
Invariant manifold theory for RDS based on the MET is an important
part of smooth ergodic theory It was started in
with the pioneering
work of Pesin
He constructed the classical stable and unstable
manifolds of a deterministic di eomorphism on a compact Riemannian
manifold preserving a measure which is absolutely continuous with respect
to the Riemannian volume His technique to cope with the non uniformity
of the MET random norms slowly varying functions can be adapted
to RDS and will also be used by us Ruelle
removed the smooth
ness condition on the measure and generalized the construction to Hilbert
manifolds
1

CHAPTER INVARIANT MANIFOLDS
Ruelle s work paved the way for the rst classical stable invariant man
ifold theorem for RDS generated by an SDE on a compact manifold due to
Carverhill the invariant measure is still restricted to a solution of the
Fokker Planck equation Boxler developed a classical center manifold
theory for general RDS in Rd
Dahlke
was the rst to construct the center manifolds Mij
for a discrete time RDS on a compact manifold in particular the invari
ant manifolds tangent to a single Oseledets space A similar result for a
deterministic di eomorphism was obtained by Pugh and Shub
Wanner
constructed global and local center manifolds
by a method developed earlier by Aulbach and Wanner see Wanner
Aulbach and Wanner
for constructing invariant subbundles of
nonautonomous dynamical systems We will adopt Wanner s technique
here The drawback however of this method is that it only yields Lipschitz
continuous manifolds even for Ck RDS In order to prove that the invariant
manifolds are indeed Ck provided a gap condition holds if k
we
carry over a method developed by Vanderbauwhede and Van Gils
and
Hilger based on a contraction mapping theorem for a scale of Banach
spaces to the random case The same method was also used by Siegmund
to prove the Ck property of integral manifolds for nonautonomous
di erential equations
Classical stable and or unstable manifolds for RDS under various con
ditions and for various purposes were also constructed by Brin and Kifer
Gundlach and Arnold and Kloeden A new method using a
random xed point theorem is used by Schmalfu
For an account of
Pesin theory see Katok and Strelcyn for the deterministic case and
Liu and Qian for the random case
Let us mention some related work for nonautonomous dynamical sys
tems For skew product ows over a continuous DS on a compact space a
program analogous to the above can be formulated but now based on expo
nential dichotomy and the Sacker Sell spectral theory see Remark
in
place of the MET Essential parts of such a program have been carried out
seeegSell
for generalized center manifolds and Yi
and
Chow and Yi for the classical manifolds However due to the built in
continuity and uniformity constructions for skew product ows are though
similar technically simpler than those for RDS See also Bronstein
and
Colonius and Kliemann for more nonautonomous constructions
1

REDUCTIONS AND PREPARATIONS
Reductions and Preparations
Our aim is to construct the invariant manifold M x as a graph
ExymyEcxintheExEcxcoor
dinate system where c is the complement of in the spectrum S
ieEc x
i i Ei x on a manifold M the graph describes
expMx
To facilitate the actual construction we go through a series of prepara
tory steps to simplify the state space the invariant measure and the
linearized cocycle T and nally make certain standing assumptions on
constants
Reductions
Reduction of the State Space from M to Rd
By Nash s theorem every smooth Riemannian manifold M of dimension d
can be isometrically embedded into R d into R d if M is compact en
dowed with its standard Riemannian structure We hence have a mapping
gMhiNgMRdhiS
which is an isometric di eomorphism between M and N In particular N
is an embedded closed submanifold of R d with
huvi hTxguTxgviS forallx Muv TxM
Now if is a Ck RDS on M with invariant measure satisfying the IC
of the MET then
g isaCkRDSonN gM withinvariant
measure g also satisfying the IC Moreover the Lyapunov spectrum
is preserved under g since kT gk and the Oseledets splitting of is
Tg Ei
It is well known that every closed smooth submanifold N of R d has
a tubular neighborhood
UrN
xNfyRnkyxkrxgrN
If N is compact then r can be taken to be constant We want to con
struct an extension of from N to R d which is trivial outside a
tubular neighborhood Ur N t x x such that its Oseledets split
ting at Tg x R d is T g Ei x with the exception that the possibly
trivial Oseledets space corresponding to a Lyapunov exponent has to
be augmented by the d dimensional normal space Tg x N As a re
sult S
S f g where the possibly zero multiplicity of the
Lyapunov exponent has to be increased by d
1

CHAPTER INVARIANT MANIFOLDS
At present we only know how to do this extension for the case where N
is compact time is continuous and has a generator which we consider
nowFor any x R d let p x be the nearest point in N such that the line
betweenxandpx isorthogonaltoTp xN Let R R beaC bump
function with
and
outside rr IfanRDS onNis
generatedbyanRDExt f t xt oranSDEdxt Pm
jfjxt dWjt
we extend the generator from N to R d as follows In the RDE case
denefxkpx xkfpx
xRd
and in the SDE case de ne
fjx kpx xkfjpx
xRdj
m
The corresponding RDS has the same regularity as it leaves N and
the measure invariant and has the Lyapunov spectrum described above
For details in the SDE case see Carverhill pp and
Reduction of General Invariant Measure to Dirac Measure at
For RDS in Rd there is the following improvement of Lemma
Lemma Every Invariant Measure is Dirac at Let be
a Ck RDS in Rd over the DS with invariant measure Consider the
following extension of
RdFFBt
tx
P
Then
tv
txv
tx
is a Ck RDS over which has invariant measure
Further T satis es the IC of the MET with respect to if and only
if T does for
and S
S
while for the Oseledets splitting
EixxEi
The simple proof is left as an exercise
This lemma enables us to do the construction of the invariant manifolds
at a xed point of our cocycle This is without loss of generality For
if we have constructed the invariant manifold M
for atx
we can recover the invariant manifold M x for at any good point
x by putting
Mxx
M
xMx
One easily checks that M is invariant if and only if M is invariant
1

REDUCTIONS AND PREPARATIONS
The result is that we can assume that is a Ck RDS in Rd with
t
hence D t
t is a linear cocycle over and
we can split at into a linear cocycle and a nonlinear part
tx
txtx
where t x
tx
t x contains the terms of higher than
linear order satis es t
Dt
and
x
Discretization of Time
As the main proofs will be done for discrete time only we consider in Sects
to in case of continuous time the embedded discrete time cocycle
nx
nx
nxnZ
where n
andD n
and show that the discrete
time objects invariant manifolds and foliations decoupling and linearizing
homeomorphisms are indeed also the continuous time ones
The generators of the cocycles and are given by their time one
mappings With
A
Fx
x
is equivalent to the random di erence equation
xn
nxnAnxnFnxn
xxnZ
where A
GldR F
and DF
Preparations
Random Norms Block Diagonalization of Linear Part
We assume throughout that the linear cocycle in
satis es the IC
of the MET and that without loss of generality the MET holds on all of
without exceptional set
Our next aim is the improvement of contraction and expansion proper
ties of through the use of random norms see Sect
and its simpli
cation through the choice of an adapted basis
LetRd E
Ep be the Oseledets splitting of Recall
that for a given constant
we can construct a new norm kxk which
crucially improves the uniformity in the behavior of because of
eitjtjktjEikt
eit jtj forallt T
1

CHAPTER INVARIANT MANIFOLDS
By choosing an adapted basis see Corollary
we de ne a coordinate
transformation P such that
t
PttP
diag t
pt
is Lyapunov cohomologous to and block diagonal where the blocks are
cocycles on Rdi with one point spectrum f i dig Note that the cocycle
is for continuous time in general only measurable with respect to t This
is however irrelevant at this stage as the construction of random norms
which requires continuity in t is done on the original
The main advantage of over is that its Oseledets spaces are obvi
ously deterministic namely
Ei fg fgRdifg fgRd
We will tacitly use the identi cation Ei Rdi and jEi
ijRdi
The nonlinear cocycle
in the new coordinates is
tx
txtx
Again is in general only measurable with respect to t
The random coordinate transformation P induces a random norm
on Ei Rdi via
kxik kxik kP xik
where
Ei xi
xi
xi Rdi
and P xi Ei We extend this de nition to Rd by taking the
norm which allows simpler estimates than the norm but is related to it
by deterministic constants
kxk pX
ikxik x
p
ixixiRdi
By Lemma
there is for each
an slowly varying random
variable R
such that
RkPkR
Hencethe normk k de nedinRdby
and
is in the same
relation to the standard norm k k as the original random norm k k For

REDUCTIONS AND PREPARATIONS
each there exists an slowly varying random variable C
such that
CkxkkxkCkxk
By our de nitions the cocycle also satis es the uniform estimate
under the norm
As a result for an interval
ij fi
jg ijp
from the spectrum of S S and with
Eij Eij
jk iEi Rdi
Rdj
we have the estimates
ej tktjEijkt
eitt
and ei tktjEijkt
ejtt
From now on we drop the bar over the transformed objects and the su
perscript of the norm We hence have arrived at a nonlinear cocy
cletx
t x t x onthenormedbundleRdkk
which is measurable in t and Ck in x It satis es t
and
Dt
t where the linear cocycle
tdiagt
pt
is block diagonal and satis es the estimates
and
for any ij
A Scale of Banach Spaces
We follow Wanner and many others and de ne Banach spaces of
functions with a geometrically weighted sup norm which allows for expo
nential growth of their elements
De nition Weighted Banach Spaces Consider Rd k k
letTTRand
De ne for each
X fg T Rd measurable kgk sup
t
tkgtk t g
X fg T Rd measurable kgk sup
t
tkgtk t g
1

CHAPTER INVARIANT MANIFOLDS
X fg T Rd measurable kgk
sup kgk kgk
g
and
X
X
WewriteX E etcifRdisreplacedbyE etc WewriteX Z
etc if we want to emphasize that time is Z etc We denote by
X fg T Rd measurable kgk sup
t
tkgtk g
etc the corresponding Banach spaces based on the nonrandom Euclidean
norm
An element g X grows at most like t forward in time i e there is
a random variable C
such that
kgtktCtt
and is said to be
bounded An element g X grows at most
like t backward in time i e there is a random variable C
such
that
kgtktCtt
and is called
bounded Both properties characterize an element
gX
as being
bounded see Fig
Note that the
spaces also depend on
which is suppressed in our notation
Figure The space X
for
Lemma Scale of Banach Spaces i For each
and
each
XXandX
are Banach spaces
ii For each
they form a scale of Banach spaces in the following
sense If and
we have the continuous embeddings
X
Xkkkk
X
Xkkkk
X
X
kk
kk
iii Ifgis
bounded then for each
it is
bounded in
the Euclidean norm of Rd Conversely if g is bounded in the Euclidean
norm then for each
it is
bounded
Similarly if g is
bounded or
bounded
Aulbach and Wanner Sect use the name quasibounded
1

REDUCTIONS AND PREPARATIONS
Proof i is clear
ii The continuous embeddings follow from the inequalities between the
respective norms which immediately follow from their de nitions
iii We have g X if and only if there is a random variable C
suchthatkgtk t C tforallt
Further for each
the random and Euclidean norm are related by B k k k k
B k k where B is a slowly varying function Hence for any t
kgtkBtkgtktBCetK
t
ifwechoose log andK BC
Similarly for all other cases
Remark Weighted Banach Spaces and Lyapunov Index
Let
g lim sup
n
nlogkgn k g limsup
n
nlogkgn kn
be the Lyapunov index of g n in the Euclidean and the random norm
respectively Then we have the following elementary relations
iIfg X or X thentheLyapunovindexofgsatises
glogandg log
iiglog
g log
gX
for all
gX
for all
Selection of Constants
In this last preparatory step we choose and then x two parameters related
to the Lyapunov exponents
pof
Choice of We rst choose the constant
which enters into the
construction of the random norm k k small enough so that the intervals
i
ii
p do not overlap and in the hyperbolic case all
i
none of them contains Henceforth we will suppress writing
justk k
It turns out that it is more convenient to consider the exponentiated
intervals
ii
ei ei
i
p
By our preparations the block i of the linear cocycle satis es
kitkt
eit
tit
1

CHAPTER INVARIANT MANIFOLDS
and
kitkt
eit
tit
More generally for any i j with i j p
ij fi
jg
and Eij the corresponding deterministic invariant subspace
ktjEijkt
tit
and
ktjEijkt
tjt
By the cocycle property for t t
implies
ktjEijkt
tit
and
implies
ktjEijkt
tjt
Choice of In order to cope with the growth due to the nonlinearity F
of the di erence equation
we introduce a constant such that
min
p
pp
i e the closed intervals
i
i
i
p
do not overlap In the hyperbolic case we assume in addition that none of
these intervals contains
The interval
i
i
i
i
p
is called the spectral gap between i and i See Fig This choice of
and remains unaltered during the proofs
Figure Selection of constants for the construction of invariant mani
folds
To describe this preliminary work with a single word we introduce the
following notation
1

GLOBAL INVARIANT MANIFOLDS
De nition Prepared RDS Let be a measurable RDS on the
normed bundle Rd k k with two sided time and the property t
Let denoteameasurablelinearRDSon Rdkk if isC inxwe
always choose t
Dt
and de ne the nonlinear part of
by
tx
tx
tx
so that
tx
txtx
t
Assume that
diag
p is block diagonal and has a spectral
theory such that
and
hold Let the constants and be
chosen as above We call such an RDS more precisely the linear part
with respect to the random norm k k prepared
We summarize our ndings from above
Proposition Linear Equivalence to Prepared RDS Every
C RDS with two sided time and xed point such that the linear RDS
t
Dt
satis es the IC of the MET is linearly equivalent to
a prepared RDS i e there is a measurable map P
GldR such
that
Pt
t
tP
P de nes a Lyapunov cohomology between the linear parts and
More speci cally the random norm k k and the random coordinate
transformation P can be chosen in such a way that for each
there
is an slowly varying random variable B
such that
BkxkkxkBkxk
and
BkPkB
In particular if x X or X then for each
or
PxP
xX
orX
respectively Similarly in the reverse direction
Global Invariant Manifolds
In this inevitably very technical section we construct unstable stable and
center mainfolds for a Ck RDS with xed point
1

CHAPTER INVARIANT MANIFOLDS
In a rst step Subsects
to
we construct manifolds which
are just Lipschitz continuous even if the RDS is Ck Later Subsect
we will show that these manifolds have in fact the same regularity as the
RDSThe constructions are rst done for discrete time and then carried over
to continuous time in Subsect
Throughout Subsects
to
we do not need to assume that is
C Rather we only need to assume now that is prepared in the sense of
De nition
The corresponding generating random di erence equation
is
xn
nxnAnxnFnxn
nZ
where F
For given ij S
i j p we seek an invariant manifold
Mij of which is a graph in the deterministic Eij Eij coordinate
system where
Eij fg fg Rdi
Rdj fg fg Rdi
Rdj
with random norm
kxijk X
ikjkxkkxij xi
xj xk Rdk
and
Eij Rd
Rdi fg fg Rdj
Rdp
with random norm kxijk X
k ijkxkk
and
Rd Eij Eij kxk kxijk kxijk x xij xij
The basic idea of the construction of Mij is to collect the initial values
x of orbits
x of similar asymptotic behavior as the orbits
xij on Eij
Construction of Unstable Manifolds
Main Result Structure of Proof
We will rst construct the unstable manifolds M j for j p as Lipschitz
continuous invariant graphs
Here is our main result
1

GLOBAL INVARIANT MANIFOLDS
Theorem Global Unstable Manifold Theorem Let the
RDS with discrete time T Z be prepared and generated by the di erence
equation
Assume
kFxFykLkxykforallxyRd
where
L
Choose any j with j p let
jf
jg be
the corresponding spectral interval of put E
Ejx
xj
x
xjEEjEjpx
xj xj
xp choose a
constant a in the spectral gap to the left of
aajleftj
j
and de ne the set
M
MjfxRd
xXag
Theni M is a topological manifold of dimension dim E d
dj called the unstable manifold corresponding to the interval
It is
independent of the particular choice of a
ii M isagraphinE E
Mfx
m
x
xEg
with the following properties
m
E E is measurable
m
m is globally Lipschitz continuous in the random norms of E and
Ekm
x
mykDLkxyk
where the constant
DLLLL
is independent of j
iii M is invariant under i e
nM
Mn
nZ
1

CHAPTER INVARIANT MANIFOLDS
See Fig
Figure The unstable manifold M and its graph m
Remark i It is important to note that F is part of a cocycle
hence the norm k k on the left hand side and k k on the right hand
side of
ii Since F
the Lipschitz condition
implies sublinear
growth of F
kFxkLkxkxRd
iii IfF x
then M
E with trivial graph m
x
In this case the result is true even for
iv We will call a de nition of the form
and
a dynamical characterization of the corresponding set Hence the unstable
stable center manifolds are dynamically characterized by exponential
growth conditions on their orbits
The proof of Theorem
is quite involved so we will break it up into a
number of simpler steps
By our preparations the linear part A is block diagonal
AAA
where A diagA Aj A diagAj
Ap
and the nonlinear part can be written as
FF
F
FBF
FjCA F BFj
Fp CA
Our random di erence equation
takes the form
xnAnxnFnxnxn
x
x
xnAnxnFnxnxn
x
x
The two equations are coupled through their nonlinear parts and from now
on will be called respectively the unstable and stable equation
Again by our preparations putting
j
j
knkn
nnknkn
nn
1

GLOBAL INVARIANT MANIFOLDS
where we have
for our choice of By our assumption
kFxFykLkxyk
We will now construct the unstable manifold M of
and
asagraphx
x
m
x in the E E coordinate system the
result of this construction is formulated in Proposition
The main
idea is to assure for each and x E the existence of a unique element
x
m
x
E which is characterized by the growth property
xxXa
where a
is arbitrary
Here is a step program for constructing the unstable manifolds M
Solve the unstable equation
for some initial value x but
for the generally wrong stable part
n
nZ
Xa E The result is
xXaE
Insert this solution
x into the stable equation
It turns
out that there is exactly one solution
xXaEwhich
picks its initial value x
x
Prove that the operator T x Xa E
Xa E dened
by
x is contracting hence has a unique xed point
xXaE
For each and x
E the point x
m
x
x
E is the uniquely determined point for which
x x Xa dynamical characterization
The function x
m
x is measurable and independent of
the particular value of a
M has the other properties Lipschitz continuity invariance
claimed in Theorem
We will give an operator formulation of this program in Subsect
Two Key Technical Lemmas
We will rst deal with the crucial step of the above program
Lemma First Key Technical Lemma Consider the random
di erence equation
xnAnxnfnxnfn
nZ
1

CHAPTER INVARIANT MANIFOLDS
in Rd k k with measurable Gl d R valued A and measurable f and f
Assume that there are constants
and L such that for each xed
kAk
and
fn
kfnxfnykn
Lkx ykn
Let
L
Then the following assertions hold
iIff
X then there exists exactly one solution of
which has the property
X and
kk
Lkf k
Moreover
n is measurable for each n Z
ii If moreover f
X then the solution of part i satis es
X and
kk
Lkf k
iii IfM
supn
nkf n k n equivalently f
X
then all solutions of
on Z satisfy
xXand
kxk
kxk
LM
kxk
Lkf k
iv Two solutions and of
de ned on Z satisfy
kn
nkn
nk
kn
Proof i We prove the existence and uniqueness of a bounded
solution of
in three steps
Step Let f and L hence f Then
reduces to
the linear random di erence equation
xnAnxnx
Note that this di erence equation is in general only solvable forwards in time and
does not necessarily generate an RDS
1

GLOBAL INVARIANT MANIFOLDS
whose solution is
n
n
nZ
where is the linear cocycle generated by A The trivial solution
is
certainly bounded We show that this is the only one
Indeed since by assumption k n k n
nforalln
any
other bounded solution satis es for each n by the cocycle prop
erty
kk knknknkn
nknkn
n
nknkn
nkk
Letting n
yields
Hence the trivial solution is the only
solutioninX ItisinfactinX
Step Letnowf X butstillL hencef
Now
is the a ne equation
xn Anxnfn
x
A bounded solution of this equation is certainly unique For if
are solutions in X then
X satises n
An
n hence
by step
As to the existence the variation of constants formula gives the solution
n
nX
i
niifi
nZ
In particular
X
i
ifi
These expressions make sense For example for n
nknkn
nX
i
ni ikfiki
X
n
nkf k
kfk
1

CHAPTER INVARIANT MANIFOLDS
nally
kk
kfk
Step Consider now the full equation
We de ne the nonlin
ear operator
TX
X
T
where is the unique solution in X of the a ne equation
xn Anxnfn
where
fn
fn
nfn
i e with the possibly wrong sequence n in the nonlinearity of the
right hand side
We have to make sure that f
X provided f
X Indeed for each n by
nkfnkn
nkfnkn
nkf n
nkn
nkfnkn
nLkn kn
nkfnknLnknknkk
Thus
kfkkfkLkk
Hence by step the wrong a ne equation
has a unique
bounded solution hence the operator T in
is well de ned Step
also yields the estimate
kTk
kfk
hence together with
kTk
Lkkkfk
The operator T is for each a contraction in the Banach space X
This can be seen as follows
T
T
X
1

GLOBAL INVARIANT MANIFOLDS
solves the a ne equation
n
An
nfn
where f n
fn
nfn
n
An estimate completely analogous to the one leading to
yields
kfkLk
k
hence f
X
Applying step to the bounded solution of
gives
kk
kfk
Recalling the de nition of and combining
and
nally
gives
kT
Tk
Lk
k
If
L T is a contraction hence has a unique xed point
X
By de nition of T
solves equation
on Z hence
onallofZ
It remains to estimate the norm of the xed point
By
kk
Lkkkfk
whence k k
Lkf k
which is
To complete the proof of part i we still have to show that
n
is measurable for each n Z We do this by appealing to the following
useful lemma
Lemma Measurability of Fixed Point Let for all
B k k be a Banach space of sequences
nnZnRd
such that the evaluation mapping
n is continuous for all n Let for
each
TBB
1

CHAPTER INVARIANT MANIFOLDS
be a contraction such that if n is a sequence of measurable functions
for which
Bthenn
T
n is measurable for all n
Denote by
B the unique xed point of T
Then
n is measurable for each n
Proof We start with n
which is obviously measurable and
recursively de ne
kn
Tk
n
whose right hand side hence left hand side is measurable by assumption
By the Banach xed point theorem for all
lim
k
k
The continuity of the evaluation mapping
n implies that also
n
lim
k
kn
and so as a pointwise limit of measurable functions
n is measurable
We apply this lemma as follows Choose B X
The evaluation
mapping is clearly continuous since
k nkn
nkk
Further
n
T
n
nX
i
niifi
where
fn
fn
nfn
Since f and f are assumed to be measurable the measurability of n
implies that of n
This completes the proof of part i of the lemma
ii Letnowf
X In step the bounded solution is
de ned on all of Z and the estimate leading to
also holds for all
n Zimplying k k
kfk
This can be used in step to derive
with instead of from
which
follows
1

GLOBAL INVARIANT MANIFOLDS
iii We now solve equation
with initial value x
x
forwards in time and assume that M
By the variation of constants formula for all n
n
n
nX
i
niifi
i
fi
where the summation is not present if n
By the assumptions
andk n kn
nkknX
i
niLki ki kfiki
hence
nknkn
kkLnX
i
ikiki
nX
i
iM
With the abbreviations
n
nknkn
nkknX
i
iM
kk
we arrive at
n
n LnX
iin
We now require a discrete time version of the Gronwall Bellman lemma
Lemma Discrete Time Gronwall Bellman Lemma Let
ban
andcn forn
and suppose that
an cn bnX
i
ain
Then for all n
an
bnc
nX
i
bnici ci
1

CHAPTER INVARIANT MANIFOLDS
A proof can be found in Wanner
pp
Applying this to
yields for n
n
Ln
nX
i
Lniii
After some elementary manipulations we obtain for n
nknknkk
LM
This estimate holds trivially also for n
Hence altogether putting
x
kkkxkkxk
LM
which is
iv If and are solutions of
onZ henceonZthen
solves
xn Anxnfn xn
nZ
wherefnxfnx
nfn
n
Equation
is of the type of equation
withf andf
clearly satis es conditions
We thus have for all n Z in particular
for n
n
An
nfn
n
whence k n kn
Lknkn
and after n stepsk k
Lnknkn
equivalently k n k n
L
nkk
Now the assumption
L implies
n
L
nforalln
consequently
kn
nkn
nk
k alln

GLOBAL INVARIANT MANIFOLDS
For step of the program for the construction of the unstable manifold we
need the following lemma which is dual to Lemma
Lemma Second Key Technical Lemma Consider the ran
dom di erence equation
xnAnxnfnxnfn
nZ
in Rd k k with measurable Gl d R valued A and measurable f and f
Assume that there are constants
and L
such that for each
xed
kAk
and
fn
kfnxfnykn
Lkx ykn
Then we have
o For all
andnZxAnxfnxfn
de nes a bimeasurable bijection of Rd hence every initial value problem for
equation
can be solved on all of Z
Let
L
Then the following assertions hold
iIfM
supn
nkf n k n equivalently f
X
then there exists exactly one solution of
which has the
property
X and
kk
LM
Lkfk
Moreover
n is measurable for each n Z
ii If moreover f
X then the solution of part i satis es
X and
kk
Lkfk
iii Iff
X then all solutions of
on Z satisfy
xXand
kxkkxkLkfk
iv Two solutions and of
on Z satisfy
kn
nkn
nk
kn
1

CHAPTER INVARIANT MANIFOLDS
Proof o We refer to Exercise
for the characterization of those
time one mappings which de ne an RDS with two sided discrete time
Let andn Zbe xedandforeachz Rdde nethe operator
Tn z Rdkkn
Rdkkn
by
TnzxAn
zAnfnxAnfn
We have
kTn zx Tn zykn
kAnkn
nkfn x fn ykn
Lkx ykn
Since we have assumed that L
the operator Tn z is contracting
hence has a unique xed point n z As a pointwise limit of measurable
functions this xed point is measurable in z for each n
Bydenition n z fullls
An
nzzfn
nzfn
hence for all n z there exists a unique measurable solution n z of
equation
zAn
nzfn
nzfn
i The proof is analogous to the one for Lemma i Assumption
implies that the linear cocycle generated by A satis es
knkn
n foralln
Further in the analogue of step L the unique solution
X
of the a ne equation
xn Anxnfn
is
nX
in
niifi
nZ
in particular
X
iifi
1

GLOBAL INVARIANT MANIFOLDS
That this makes sense if f X can be seen as follows For n
nknknX
in
niniikfiki
X
in
niM
M
We thus obtain k k
M
which is the analogue of
The proofs of ii iii and iv are analogous to their counterparts in
Lemma
hence are omitted
Construction of the Unstable Manifold M
By means of the two Key Technical Lemmas we are now able to construct
a random graph x
x
m
x in the E E coordinate system
the elements of which are dynamically characterized in a way that quali es
them as members of the unstable manifold M
We collect the main part of the work in the following proposition
Proposition Graph of Unstable Manifold Let the condi
tions of Theorem
besatisedFixa left j
j
Then for each and x
E there exists a unique element
x
m
x
E which depends measurably on
x and
is independent of the particular value of a such that the cocycle
generated by the coupled pair of equations
and
has
the property that
xm
x
Xa E equivalently
xm
xXa
Proof We follow the program steps to formulated at the beginning
of this subsection
Step We choose
Xa E and consider the unstable
equation
with the wrong stable component in the right hand side
xnAnxnFnxn
n
x
xE
Equation
satis es the assumptions of the Second Key Technical
Lemma
with
fnxFnx
n
Fn
n
1

CHAPTER INVARIANT MANIFOLDS
and
fn
Fn
n
Indeed
is satis ed for A furthermore f n
and
kfnxfnykn
Lkx ykn
so that also
holds Lemma o tells us that each initial value
problem for
can be solved on all of Z
We now make sure that we can apply part iii of the Second Key Tech
nical Lemma by proving that our assumption
XaE
implies that f
Xa E Wehaveforalln Z
ankfn kn
L
aanknkn
thus
kf ka
L
ak ka
Hence Lemma iii applies If
a
L which holds true
for all a
under the condition L
then any solution
xof
isinXa E and
kxkakxk aLakfka
The last two estimates yield
kxkakxkL
Lakka
Step The arbitrary
Xa E and the corresponding
solution
x Xa E of step are now inserted into the non
linear part of the stable equation
We obtain
xnAnxnFn
nx
n
nZ
The assumptions of the First Key Technical Lemma
are satis ed for
with f hence L and
fn
Fn
nx
n
provided a which is automatically the case
Thus Lemma i applies if f
Xa E which we are
going to verify now
1

GLOBAL INVARIANT MANIFOLDS
Wehaveforanyn Z
ankfn kn
L
aankn xkn
anknkn
implying
kf ka
L
akxkakka
We have estimated the rst term on the right hand side of the last inequality
in
resulting in
kf ka
L
akxkLa
aLakka
Hence by Lemma i there exists a unique solution
x
XaEof
and its norm satis es
k xka
a
akfka
Together with the estimate
we obtain
k xka
L
a kxk
La
a
Lakka
Step Steps and allow us to de ne for each xed and x E
the operator
TxXaEXaE
by
Tx
x
We will now prove that T x is contracting
We choose initial values x x E and
XaEand
determine the solutions
x and
x of equation
according to step The di erence
satis es
ynAnynFnyn
Fn
1

CHAPTER INVARIANT MANIFOLDS
with initial value
yx
x Equation
satis es the
assumptions of the Second Key Technical Lemma
with
fnyFny
Fn
and fn
Fn
Fn
provided L
a which is implied by L
Hence Lemma iii applies to
and yields
kkakx
xkL
Lak ka
Now let
Tx
and
Tx
The di erence
Xa E solves
xnAnxnFn
Fn
Equation
satis es the assumption of the First Key Technical
Lemma
with f and
fn
Fn
nx
Fn
nx
The same reasoning which lead to
yields
kf ka
L
akx
xkLa
aLakka
We hence can apply Lemma i which results in
kTx
Txka
a
akfka
The latter combined with
nally gives
kTx
Txka
L
akx
xk
La
a
Lak ka
We now look for an upper bound in
which holds for any value of
a
An elementary calculation shows that for such an a
L
a
L
La
a
La
LL
1

GLOBAL INVARIANT MANIFOLDS
The nal estimate is thus For all a
and L
kTx Txka
Lkx xk LLk ka
For the particular choice x
x
x
reduces to
kTx
Txka
L
Lk ka
Thus T x is contracting if L
which we have assumed
Hence there is a unique xed point
xXaEofTx
Step Put m
x
xE
Starting our procedure with determining the corresponding with
initial value
x andusingthedenitionofT x weobtaina
solution
xm
x
of the pair of equations
and
such that the point x
m
x E is the uniquely determined point see step for which
xxXaE
As
xm
x Xa E is automatically true see step
and as
kx xkkxkkxk
hence
kxkakxkakxkakxka
it follows that even
xm
xXa
Conversely
xx
Xa implies that
xx
Xa E Thelatteristhecaseifandonlyifx
m
x Conse
quently for prescribed and x E x x m
x is the uniquely
determined initial value for which the orbit of the cocycle has the property
x Xa We have thus proved the dynamical characterization of
the graph
Mfx
m
x
x EgfxRd
xXag
Step The measurability of x
m
x follows again as in
the proof of Lemma i by applying Lemma
That m x does not depend on a
can be seen as
follows Assume a a
with a
a By Lemma
a
a implies Xa
Xa Hence if is a bounded then
it is a bounded By the uniqueness of the a bounded solution
ma
x
ma
x

CHAPTER INVARIANT MANIFOLDS
Step of our program remains to be carried out
Completion of the proof of Theorem
i That M is a topological manifold with the corresponding di
mension is a consequence of the fact that it is a Lipschitz continuous graph
inE E
ii The measurablity of m was proved in Proposition
The trivial solution
is a bounded hence by unique
ness m
For the Lipschitz continuity of m
we utilize the estimate
which for the xed points takes the form
k
ka
Lkx
xk LLk
ka
Our assumption L
implies
LL
L
L
hence
k
ka
L LLkx
xk
This implies in particular that
km
x
m
xk L LLkx
xk
iii Invariance of M
We have to prove that
nM
M n foralln Z
This is equivalent to the two relations
aM
M
b
M
M
Indeed since n is a cocycle of homeomorphisms of Rd a and b
imply
M
M
M
M
from which
follows for n
and n
andthusforalln Z
Proofofa Letx M
We want to show that
x
M
By the dynamical characterization x M if and only if
x Xa By the cocycle property
n
x
n
x
1

GLOBAL INVARIANT MANIFOLDS
Since the left hand side hence the right hand side is in Xa
x
M Proof of b By the cocycle property
n
x
n
x
If x M then the right hand side hence the left hand side is in
Xa thus
xM
This completes the proof of Theorem
As a by product we have the following dynamical properties of orbits of
starting on M
Corollary Let the conditions of Theorem
be satis ed
Choose x M
Then for any
j
kn xkn
n LLkxk n
andhence k n xkn
nL
Lkxk n
Proof We again utilize the estimate
which for n
and x
xx
xx
m
x and instead of a yields
knxkn
nk xk nDLkxk
On the other hand by
knxkn
nkxk LLk xk
n DLLLkxk
Adding up we obtain
kn xkn
n LLkxk n LLkxk
since
DLDLLL
LL
Now choose n By the cocycle property and the rst estimate
kxkkxkknn
nxk
n LLkn xkn
1

CHAPTER INVARIANT MANIFOLDS
which is equivalent to
kn xkn
nL
L kxk
Corollary Flag of Unstable Manifolds Let the conditions
of Theorem be satis ed Then we obtain a nested sequence of unstable
manifolds M M
Mj
Mp Rd
corresponding to the backward ltration of the MET for
EVp
EjVpj
EpVRd
The above inclusions are embeddings of submanifolds
Proof The inclusions follow from the dynamical characterization of the
unstable manifolds and the embeddings of the corresponding Banach spaces
Lemma
That the inclusions are embeddings follows from the representation of
the manifolds as graphs
Construction of Stable Manifolds
The reader will certainly be able to imagine that the stable manifolds can
be constructed in a completely analogous dual procedure with the help
of the two Key Technical Lemmas now we need the ner estimates using
M in Lemma iii and i We will hence omit the proof and
immediately state the nal result
Theorem Global Stable Manifold Theorem Let the
RDS with discrete time T Z be prepared and generated by the di erence
equation
Assume
kFxFykLkxykforallxyRd
where
L
Choose any i with i p let
ip fi
pg be the
corresponding spectral interval of put E Eip x
xi
xp
E
EipEix
x
xi
choose a constant b in the
spectral gap to the right of
b bi right
i
i
1

GLOBAL INVARIANT MANIFOLDS
and de ne the set
M
Mip fx Rd
xXbg
Theni M is a topological manifold of dimension dim E di
dp
called the stable manifold corresponding to the interval It is indepen
dent of the particular choice of b
iiM isagraphinE E
Mfx
m
x
xEg
with the following properties
m
E E is measurable
m
m is globally Lipschitz continuous in the random norms of E and
Ekm
x
mykDLkxyk
where the constant
DLLLL
is independent of i
iii M is invariant under i e
nM
Mn
nZ
As another by product we have the following dynamical properties of orbits
of starting on M
Corollary Let the conditions of Theorem
be satis ed
Choose x M
Then for any
i
kn xkn
n LLkxk n
andhence k n xkn
nL
Lkxk n
1

CHAPTER INVARIANT MANIFOLDS
Corollary Flag of Stable Manifolds Let the conditions of
Theorem
be satis ed Then we obtain a nested sequence of stable
manifolds
MppMpp
Mip
Mp Rd
corresponding to the forward ltration of the MET for
Epp Vp
Eip Vi
EpVRd
The above inclusions are embeddings of submanifolds
We nally combine the Corollaries
and
in the following corollary
the result of which is shown in Fig
Corollary Disjoint Decomposition of Spectrum Let the
conditions of Theorems
and
be satis ed assume that the Lya
punov spectrum of is not one point and choose j p Denote
byM
the unstable manifold corresponding to the spectral interval
f
jg and by M the stable manifold correspond
ing to the complementary spectral interval f j
pg
Then
M
Mfg
Figure Stable and unstable manifolds do not intersect except at
Proof Let x M
M
Since x M Corollary
applies For
j
kn xkn Cnkxkn
where C
L
L
Since x M Corollary
applies For
j
kn xkn Cnkxkn
Combining these estimates and using kxk
n C foralln
which is a contradiction

GLOBAL INVARIANT MANIFOLDS
Construction of Center Manifolds
We now complete our program of constructing global invariant manifolds
by proving the following theorem
Theorem Global Center Manifold Theorem Discrete
Time Let the RDS with discrete time T Z be prepared and generated
by the di erence equation
Assume
kFxFykLkxykforallxyRd
where
Lp
Considerforanyijwith i j ptheinterval ij fi
jg
from the spectrum of choose constants
aj left j
j
bi right
i
i
and de ne the set
Mij fx Rd
x Xajbig
see Fig
so that
Mij
Mj Mip
Theni Mij is a topological manifold of dimension dim Eij di
dj
called the center manifold corresponding to the interval ij It is indepen
dent of the particular choice of aj and bi
iiForij pMp
RdForij pMij isa
graph in Eij Eij
Mij fxij mij xij xij Eijg
with the following properties
mij
Eij Eij is measurable
mij
Note that p
1

CHAPTER INVARIANT MANIFOLDS
mij is globally Lipschitz continuous in the random norms of Eij and
Eij kmij xij mij yijk CLkxij yijk
where the constant
CL
DL
DLDLLLL
is independent of i j
iii Mij is invariant under i e
n Mij
Mij n
nZ
Figure Dynamical characterization of the center manifold Mij
Proof Assume the situation of Theorems
and
To single out nontrivial cases we assume that i j p and intro
ducethenotationsE Ei E EijandE Ej p sothat
Rd E E E These are nontrivial subspaces
Under the conditions Lip F
LandL
we have con
structed the following invariant manifolds
MMjm
EEELipm
DL
and
M Mip
m
EEELipm
DL
We now de ne for each and x E the operator
TxEEkkEEkk
by
Txxx
m
xxm
xx
We check under which condition this is a contraction Using the Lipschitz
continuity of m and m we obtain
kTxxxTxyyk
km
xx
m
xyk
km
xx
myxk
DLkx ykkx yk
DLkx x
yyk
1

GLOBAL INVARIANT MANIFOLDS
Hence T x will be a contraction if D L
equivalently L L
Since we also need L
the latter is the case if and only if
Lp
Under this condition there exists a unique xed point
xx
m
x
m
x
m
xEE
which depends measurably on
x BydenitionofTx this xed
point has the properties
m
x
m
xm
x
and
m
x
m
m
xx
means that the point x
m
x RdliesonM and
means that it lies on M
Thus
M
Mijfxm
xxEg
M
MfxRd
x Xaj Xbig
Clearly m
by uniqueness
We will now calculate the Lipschitz constant of m
Letx y
E Since the Lipschitz constants of m
and m
are known to
beDL weobtain
km
x
m
yk
km
xm
x
mym
yk
DLkx ykkm
x
m
yk
and
km
x
m
yk
km
m
xx
m
m
yyk
DLkx ykkm
x
m
yk
Addition of both estimates yields
km
x
myk
DLkxykkm
x
myk
1

CHAPTER INVARIANT MANIFOLDS
Since we made sure that D L
we nally obtain
km
x
mykDL
DLkx yk
We leave the proof of the invariance of M as an exercise to the reader
Remark iIfF x
then Mij
Eij with trivial graph
mij xij
In this case the result is true even for
ii For i j we obtain an invariant manifold Mi tangent to the
Oseledets space Ei which we call Oseledets manifold
iii Note that the restriction on the Lipschitz constant of F is less severe
for the construction of the unstable and stable manifolds than it is for the
center manifolds
We now combine Corollaries
and
and obtain a more precise
dynamical description of the orbits on the center manifold see Fig
Corollary Let the conditions of Theorem
be satis ed
Choose x Mij
Then for any choice of aj and bi such that aj j
i
bi
Can
jkxkkn xkn Cbn
ikxk n
and
Cbn
ikxkkn xkn Can
jkxk n
where C LL
Figure Dynamical description of an orbit in Mij
The Continuous Time Case
Our program is completed if the RDS has discrete time However the
continuous time case can easily be reduced to the discrete time case
Theorem Global Center Manifold Theorem Continuous
Time Let
tx
txtxtR
beanRDSwith t
which is prepared according to De nition
Assume
sup
tkt x
tyktLkxyk
1

GLOBAL INVARIANT MANIFOLDS
where
Lp
Then the invariant manifolds Mij constructed for the embedded discrete
time RDS
nx
nx
nx
are also the invariant manifolds for the continuous time RDS
In
particular
t Mij
Mij t forallt R
and
Mij fx Rd
xXbiRXajRg
Proof i We rst prove the invariance of Mij First observe that
sup
tktkt
e
c
Consider x Mij First take n Z and
t
We write
n
n By the cocycle property
ntx
nttx
tn
nx
tn
nx
tn
nx
Hence
knttxknt
cLkn xkn
c Lbn
ik xkbi Z
Analogously for n Z
knttxknt
c Lan
jk xkaj Z
By the discrete time dynamical characterization of Mij
t x Mij t forallt
Representageneralt Rast t s
sIfsxMs
then
tx
ts
sxMijts
Mij t
by the discrete time invariance
1

CHAPTER INVARIANT MANIFOLDS
So far we have proved that
t Mij
Mij t forallt R
Since t is a homeomorphism of Rd the invariance of Mij follows
as in the proof of Theorem iii
ii We nally prove the continuous time dynamical characterization It
su ces to show that
sup
nZbn
ikn xkn
sup
tRbt
ikt xkt
and the analogous statement on Z R
Nowforn Z and t asabove
bnt
ikntxknt
c LBibn
ikn xkn
where
Bi
sup
tbt
i
max bi
This means that
holds which nishes the proof
Higher Regularity
Main Result Structure of Proof
Consider a prepared RDS with discrete time given by
nx
nx
nxnZ
where n
equivalently generated by the random di erence
equation
xnAnxnFnxn
nZ
where
A
Fx
xF
In the previous subsections we have constructed the global center manifolds
Mij of tangent to Eij provided the nonlinear part F has a Lipschitz
constant Lip F
L with L small enough Tangent to Eij
meant so far that Mij fxij mij xij xij Eijg is a Lipschitz
continuous graph in the Eij Eij coordinate system of Rd with a small
Lipschitz constant
1

GLOBAL INVARIANT MANIFOLDS
Suppose our prepared RDS is such that
DikRd for
some k
in particular F is Ck and A
D
hence
DF
Our method does not allow for using this additional in
formation to prove that Mij is more regular than Lipschitz and is in
fact tangent to Eij in the classical sense i e T Mij
Eij equivalently
Dmij
We will now show that the invariant manifolds Mij are indeed Ck if the
RDS is for the case k under the provision that the spectral gaps to the
left and to the right of the block ij f i
jg are wide enough
gap condition Siegmund gives a detailed and clear exposition for
the case of nonautonomous deterministic di erential equations
De nition Let F
Rd Rd be a measurable function
i If F is continuous we say that F
Cb if there is a nite
random variable L
such that for all
kFksup
xRdkF xk L
ii LetF beCkforsomek Wesaythat
F
Ck
bCk
bRdkk Rdkk
if for each q with q k there is a nite random variable Lq
such that for all
kF kq
sup
x RdkDqF xk Lq
HereDqF x istheqthderivativeofF atx
DqFxLqRd
Rdkk Rdkk Lq
where L q is the space of continuous q linear operators endowed with the
operator norm and Rd
Rd has norm kxk sup i qkxik
WehaveL LRdkk Rdkk L
LRd RdRd
LRdL andLq LRdLq ingeneral
iii The space
Ck
bEijkk Eijkk
is de ned analogously as the space of random functions between the corre
sponding spaces for which the derivatives of order q k are bounded
by nite random variables

CHAPTER INVARIANT MANIFOLDS
Suppose F
Cb and there is a deterministic constant L such that
kFksup
xRdkDF xk L
By the mean value theorem this implies
kFxFykLkxyk
Hence the Unstable and Stable Manifold Theorems
and
hold
provided L
while the Center Manifold Theorem
holds pro
vided L
p
Theorem Global Smooth Invariant Manifold Theorem
A Discrete time case Let k and let be a prepared RDS with
discrete time T Z such that
Di k Rd Let the generating
di erence equation be
xnAnxnFnxn
nZ
where A
D
F
and DF
Assume that
F
Ck
b and that there is a deterministic constant L such that
kFksup
xRdkDF xk L
where the bound L satis es
L
p
Consider for any i j with i j p the center manifold
Mij fxij mij xij xij Eijg
corresponding to the interval ij f i
jg from the Lyapunov
spectrum of
Then the following assertions hold
i Mij is a C manifold more precisely
mij
CbEijkk Eijkk
with kmij k sup
xij Eij kDmij xij k
LL
ii The manifold Mij is tangent to Eij i e T Mij
Eij equi
valently Dmij
iii Higher regularity Suppose k and the following gap conditions
are satis ed
1

GLOBAL INVARIANT MANIFOLDS
The spectral gap right
i
i
to the right of ij is wide
enough such that we can choose two numbers b b right with b b
for which moreover also bq b for every q
k no condition
for i i e for the unstable manifolds
The spectral gap left
j
j totheleftof ijiswide
enough such that we can choose two numbers a a
left with a a
for which moreover also a aq for every q
k no condition
for j p i e for the stable manifolds
Then there exists a constant k
suchthatifL k Mij isa
Ck manifold more precisely
mij
Ck
bEijkk Eijkk
In particular if for q k Dq F
then also Dq mij
B Continuous time The statements in part A are also valid for a
prepared continuous time RDS
tx
txtxtR
where t
DikRd t
Dt
t
and
Dt
provided
Ck
b wherenowfor q k
kkq
sup
t sup
xRdkDqt xkt
and there is a deterministic constant L such that
kksup
t sup
xRdkDtxkt L
and L satis es the condition L k from A
Remark When are the Gap Conditions Satis ed i If
int right
then the rst gap condition is automatically satis ed for
all k N In particular the stable manifolds Mip with i in particular
the classical stable manifold Ms are Ck if the remaining conditions are
metIf int left
then the second gap condition is automatically
satis ed for all k N In particular the unstable manifolds M j with
j in particular the classical unstable manifold Mu are Ck if the
remaining conditions are met
ii Let some Lyapunov exponent vanish i
and consider the clas
sical center manifold Mc Mi i tangent to Ei Then
ie
e
i
and right
i
left
i
It
will hence not always be possible to satisfy the gap conditions for a given
k

CHAPTER INVARIANT MANIFOLDS
Part B of Theorem
is a by product of Theorem
We hence
can restrict ourselves to the proof of the discrete time case part A
We will prove Theorem A only for the unstable manifolds M
M j for j p There is a similar dual proof for the stable manifolds
M Mip hence the smoothness result for Mij follows from Mij M j
MipThe reader should recall the notations of Subsect
in particular
ExEx
j
j and
a aj left
Our random di erence equation
takes the form of the following
coupled pair of the so called unstable and stable equation see
and
xnAnxnFnxnxn
x
x
xnAnxnFnxnxn
x
x
The corresponding linear cocycles satisfy
knkn
nnknkn
nn
Our assumption
assures that Theorem
holds We now give
an operator formulation of its proof
Operator formulation of the construction of the unstable man
ifold In order to nd the invariant manifold M we have solved the
following equation which for convenience we write down for all n Z in
Xa given
andx E
n
nx
Pn
i
niiFi
n
n
Pin
niiFi
n
n
nX
i
niiFi
i
where for reasons of display we have used the abbreviation
FiFi
i
in the
equation
This equation is derived from steps and of our step program of Sub
sect
where we have inserted the explicit solution of the initial value

GLOBAL INVARIANT MANIFOLDS
problem of the unstable equation
and the explicit form
of the unique solution in Xa of the stable equation
see First
Key Technical Lemma
The graph of the invariant manifold is then
obtained as m
x
We write the above equation in the following succinct operator form
SxKNF
Xa
The operators S K and NF are de ned as follows
SEXa
x
x
KXa
Xa
where again for n Z but used only for n
K
nBB
B
Pn
i
nii i
n
n
Pin
nii i
n
Pn
i
niiiCC
CA
Because of its importance for the study of regularity the last operator
NF deserves a name of its own
De nition Nemytskii Operator i Let f n Rd Rd
be a sequence of functions The Nemytskii operator corresponding to this
sequence is de ned as
NfRdZRdZNfnfnn
ii For the case f n x F n x the Nemytskii operator NF
corresponding to the random function F is de ned by NF
n
Fn
n Clearly NF
ifF
Note that the Nemytskii operator NF is solely determined by the non
linear part F of the RDS
In the language of these operators Proposition
states that for
each and x equation
has a unique solution
xXa
which depends measurably on
x In other words the operator
I K NF is invertible in Xa and the unique solution can be rep
resented as
x
IKNF
Sx
We rst prove that the de nitions of S and K make sense
1

CHAPTER INVARIANT MANIFOLDS
Lemma i The operator S E X is a bounded linear
operator for any in particular for
a left and
kSk
ii The operator K X
X is a bounded linear operator
for any
in particular for
a left and
kKk max
in particular kK ka
for all a left
Proof i Since by Lemma ii X
X for
and the
inclusion mapping is a bounded linear operator with norm it su ces
to prove the claim for
Sxn
n x is de nitely linear For n
kSxnknknknkxknkxk
whence kS k
ii First observe that
kKkkKkkKk
Let na We rst consider the
component We have
kX
in
nii iknX
in
nikiki
Hence for
nkkn X
in
niikiki
It follows that
kK
k
kk
b We now consider the
component We have for
nk nX
i
nii ikn
nX
i
niikiki
1

GLOBAL INVARIANT MANIFOLDS
whence k K
k
kk
Altogether for
kKkmax
kk
For any a left
is a uniform bound for the
norm
The Nemytskii Operator
Since bounded linear operators are C equation
indicates that the
smoothness of the Nemytskii operator NF determines the smoothness
of the unstable manifold M
Indeed if NF Xa
Xa were Ck the implicit function the
orem would imply that the right hand side hence the left hand side of
hence x
x
m
x isCk
However even if F is Ck NF is in general not di erentiable for a
deterministic counterexample see Siegmund Sect
The way out
is to play with the scale parameter a
left and consider NF
as an
operator from Xa into a bigger space Xb
whereb left butb a
For k this is the origin of the gap condition
Proposition Regularity of Nemytskii Operator i Con
tinuity Let F
Cb with deterministic bound L Then for all a
and c
the operator NF Xa
Xc iswelldened andis
continuous for c
ii Lipschitz continuity Let Lip F
L for some determin
istic constant L Then for all b a NF
Xa
Xb is
Lipschitz continuous with
kNF
NF kb
Lak ka
iii Di erentiability Let F
Cb with deterministic bound L
Thenforall b a NF Xa
Xb is C more precisely
NF CbXa Xb
with
DNF
n DFn
nn
n
nXa
and kDNF kL Xa Xb
akDNF k b
a
akF kb La
1

CHAPTER INVARIANT MANIFOLDS
where on the right hand side of the rst inequality DNF
is inter
preted as an L Rd valued sequence in X b
a
iv Ck smoothness Let F
Ck
b for some k with determin
istic bound L and random bounds Lq for q k Choose a b such
that b aqforallq
k ThenNF Xa
Xb isCk
more precisely NF Ck
bXa Xb
with
DqNF
Xa
Lq Xa
Xa Xb
Lq
given by
DqNF
n DqFn
n
and we have
kDqNF kL q
aqkDqNF k b
aq
aqkF kq Lq
aq
for q k where in the center term of the last inequality Dq NF is
interpreted as an L q valued sequence in X b
aq
Proof i a NF well de ned First note that since kF k L
for any sequence Rd Z
kNF k
sup
n kNF
nkn
sup
nkFn
nkn
L
Thus
NF
RdZX
Xc
c
and nally NF Xa
Xc iswelldenedforanya
i b Continuity We want to prove continuity of NF
at the
arbitrary but xed reference point Xa
For Xa andN Z
kNF
NF kc
sup
ncnkFn
nFn
nkn
max max
NnkFn
nFn
nkn cNL
where we have used the fact that c
Let
be given To control the in nitely long tail of the sequence
Fn
n we now need to make the crucial assumption that c
1

GLOBAL INVARIANT MANIFOLDS
under which we can choose an N Z big enough such that cN L
The nitely many remaining terms are estimated by making use of the
continuity of F
Since F is continuous there exists a
such that
max
NnkFn
nFn
nkn
provided C
max
Nnkn
nkn
Now
kka
sup
nankn
nkn
sup
Nnankn
nkn
CaC
whereC a minaN a
Summing up for
choose
Ca
Then
kka
kNF
NF kc
proving the continuity of NF Xa
Xc foranya
and c
as claimed
ii Lipschitz continuity For b a
kNF
NF kb
kNF
NF ka
sup
nankFn
nFn
nkn
L sup
nankn
nkn
Lak ka
Since NF
it also follows that
kNF kb
Lakka
iii Di erentiability
1

CHAPTER INVARIANT MANIFOLDS
Step Letagain b aandletF
Cb with constant L
Hence DF
Cb RdLRd Wecanthusdeneacandidatefor
DNF
namely
NF
XaRdXb
a LRdkk Rdkk
where
NF
n DFn
n
By part i of this theorem
NF
isdenedforall b a
and continuous for all b a Further again for all b a
kNF kb
akDFkbL
Step There is however a second interpretation of NF
for
xed Xa
namely as a linear mapping between Banach spaces
NF
Xa
Xb
de ned by
NF
n DFn
nn
Weclaimthatforb aNF
LXa Xb
and moreover
kNF kLXa Xb
akNF kb
a
Indeed
kNF kb
a sup
n
ba
nkDF n
nkn
nanknkn
kka
akNF kb
a
Step Let now b a We now prove that NF
is di erentiable
at any Xa
and that DNF
NF
Let
Xa
andb a Bydenition
kNF
NF
NF kb
sup
nbnkFn
n
nFn
n
DFn
n
nkn
sup
nbnkkn
1

GLOBAL INVARIANT MANIFOLDS
Since F
Cb we can apply Taylor s formula in Rd
FxyFxZDFxtydty
to obtain
kFn
n
nFn
n DFn
n
nkn
kZDF n
ntnDFn
n dtnkn
It follows that
sup
nbnkkn
kka
aZsup
n
ba
n
kDF n
ntnDFn
nkn
ndt
kka
a
sup
t kNF
tNFkb
a
By step Xa
NF
Xb
a is continuous provided
b a Thatisfor
there exists a
such that
kNF
tNFkb
a
wheneverk t ka kka
As a result NF
Xa
Xb is di erentiable for all
b aandDNF
NF
Further since by step Xa
DNF
Xb
a is continuous by
Xa
DNF
L Xa Xb is continuous Finally by
and
NF CbXa Xb
iv Ck smoothness By induction on q
k
Proof of the Smoothness of Invariant Manifolds
With the knowledge on the operators S K and NF summarized
in Lemma
and Proposition
we return to equation
SxKNF
1

CHAPTER INVARIANT MANIFOLDS
Proposition Let Lip F
L and
iKNFX
X is Lipschitz continuous with Lipschitz
constant
LipK NF
l
L
min
In particular for
a left
lL
iiForlIKNFX
X is a homeomorphism
and its inverse
IKNF
X
X
has Lipschitz constant l
iii The solution set of equation
inEXis
fx
Sx
xEg
and its projection onto Rd E E
Mfx
m
x
xEg
is given by
m
x
Sx
We also have
m
xX
n
nnFn
Sxn
Proof i This immediately follows from Lemma ii and Proposi
tion ii putting b a
ii IfLipK NF
l then I K NF is invertible and
IKNFX
nKNF
nX
X
A global Lipschitz constant of
is obtained from
kK NF
n
KNF
nklnkk
asPn ln
l Hence
is in particular continuous thus a homeo
morphism
iii This is just the formal expression of our procedure leading to The
orem
to determine M Since
is Lipschitz and all the other
operations in
are linear and bounded hence C the Lipschitz
continuity of m
can be read o

GLOBAL INVARIANT MANIFOLDS
For the Ck smoothness of m
an inspection of
reveals that
we need
to be Ck Our hope is that this follows from the fact that
NF
Xa
Xb isCkwhenever b aqforq
k
Proposition iv provided we can meet the gap condition i e we
can choose b a left with the properties of Theorem iii
The smoothness of
can be deduced from a contraction principle
on a scale of Banach spaces which was developed for the deterministic case
which su ces since is xed here e g by Vanderbauwhede and Van
Gils Theorem Hilger Theorem
and Chow and Yi see
also Siegmund
It allows to show smooth dependence of a xed point
on parameters
We follow Vanderbauwhede and Van Gils
and quote their Theorem
without proof
Theorem Contraction Theorem on Scale of B Spaces
Let Y Y Y and be Banach spaces with norms denoted respectively by
kk kkkkjjsuchthatY iscontinuouslyembeddedinYand
Y is continuously embedded in Y We denote the embedding operators by
J Y Y and J Y Y Consider a xed point equation
yfy
where f Y
Y satis es the following hypotheses
HJfY
Y has a continuous partial derivative Dy Jf
Y
LYY with
DyJfy Jfy fyJforally Y
for somef Y
LY andf Y
LY
HfY
Y y fy fJy hasacontinuous
partial derivative D f Y
LY
H There exists some
such that
kfy fyk ky ykforallyy Y
and
kfyk kfyk forally Y
H It follows from H that
has for each
a unique
solution y y Y Suppose that y J y for some continuous
y
Y
1

CHAPTER INVARIANT MANIFOLDS
The hypotheses allow us to consider the following equation in L Y
Afy ADfy
because of H this equation has for each
a unique solution A
LY
Then under H to H the solution map y
Yof
is
Lipschitz continuous and y J y
Y isC with
Dy
JA for all
As an application of this theorem we can nally prove Theorem
Proof of Theorem
We restrict ourselves to the C case the
Ck case for k is in principle proved by induction on q
k using
Theorem
at its full strength see Vanderbauwhede and Van Gils
Sect for the autonomous deterministic case and Siegmund for the
nonautonomous deterministic case
Wenowx
and suppress it for ease of notation Put E
with generic element x Y Y Xa with generic element y
Y Xb where a b left with b a The embedding operators are
J id Xa Xa andJ Jab Xa Xb Ourequation
is
fx SxKNF
We now verify the conditions H to H in our case
H ThefunctionJabf Xa E Xb denedbyJabf x
KNF Sx has a continuous partial derivative with respect to which
is even independent of x and can be represented by
DJabfx JabfxfxJab
where for each Xa and x E
fx KDNF LXaXa
and
fx KDNF LXbXb
This holds by Lemma ii and since by
DNF
LX X forany
HffXaE Xadenedbyx SxKNFhas
a continuous partial derivative with respect to x which is even independent
of andx andisgivenby
DxfXa E LEXa
xS
1

GLOBAL INVARIANT MANIFOLDS
This follows from Lemma i
H a We rst prove that f is uniformly contracting Indeed by
Proposition i for any a left
kfxfxka kKNF KNFka
Lk ka
and L
ifandonlyifL
which follows from
b The other two estimates follow by Lemma ii and from the
estimates
and
H Since in our case Y Y hypothesis H follows from H
Hence Theorem
applies The result is as follows Denote as
above by x
x
Sx
Xa the unique solution map of
equation Sx KNF i e
I KNF Then
S E Xa is Lipschitz continuous and
JabSE
Xb isC
Thederivativeofx Jab Sx isgivenby
DJabSx JabAx LE Xb
where A x L E Xa is the unique solution of
IKDNFSxAS
ieAx
IKDNFSx S
Denoting by X Rd the projection
andby Rd E
the projection x x x
x
mx
JabSx
the derivative at x is
Dmx
JabAx
whence
kDm xkkAxk C
where C LL yielding part of the bound of Theorem i More
over
Dm
Ja bA
Since S
hence S
and thus DNF S
DFn
by assumption we have A S and Dm
ieM istangent
toE
1

CHAPTER INVARIANT MANIFOLDS
Exercise Higher Order Derivatives We can derive higher
order derivatives of m at x
by di erentiating the de ning operator
equation Sx Sx KNF Sx
atx
cf Siegmund
Korollar
Remark Holder Continuity Inspecting the proof of Hilger
for the deterministic case one can convince oneself that the following
result holds If the RDS is Ck
b and the gap conditions are augmented
by the requirements that also bk b and a ak then the invariant
manifolds are also Ck The moral is that there is no loss of regularity in
the construction of invariant manifolds
Final Global Invariant Manifold Theorem
In this subsection we undo our preparations We carry over the main
invariant manifold results from Subsects
to
to our original un
prepared Ck RDS with xed point considered in Rd with Euclidean
norm
By Proposition
is linearly equivalent to a prepared RDS denoted
by
Pt
t
tP
Theorem Global Invariant Manifold Theorem Let be
aCkRDS k withtwosidedtimeand t
represented by
tx
txtx
where t
Dt
hence t
andD t
Supposethat satisestheICoftheMET LetS f i di i
pg
be the Lyapunov spectrum Rd E
Ep be the splitting of
and ij be any interval from S
Consider the corresponding prepared RDS presented by
tx
txtx
and assume that satis es the conditions of the Smooth Invariant Manifold
Theorem
Let for any
i j pMij betheCkcenter
manifold of and put
Mij
P Mij
Theni Mij is invariant under
t Mij
Mij t
1

HARTMAN GROBMAN THEOREM
ii Mij is an embedded Ck submanifold of Rd tangent to Eij
Ei
Ej of dimension di
dj given by a Ck graph in the
Eij
Fij coordinate system where Fij
k ijEk
iii Mij is dynamically characterized as follows For any aj
int left j
j
bi int right
i
i
Mij fx Rd
x Xaj big
Proof Everything carries over from the prepared RDS via the linear
isomorphism x P x The dynamical characterization follows from
Proposition
and Lemma
Invariant Foliations and the Hartman
Grobman Theorem
Our aims in this section are the following i We rst exhibit a whole fo
liation of invariant manifolds corresponding to a section of the Lyapunov
spectrum of into a stable and unstable part ii We use this foliation
as a new coordinate system to also decouple block diagonalize the non
linear part of the RDS iii We nally linearize all blocks with a non zero
Lyapunov exponent Hartman Grobman theorem
We basically follow Wanner Sects
Invariant Foliations
We assume the situation of Sect
in particular the conditions of the
Global Center Manifold Theorems
T Zand
TR
Suppose the Lyapunov spectrum of is not one point and is split into
the disjoint nonvoid intervals f
jgand fj
pg with corresponding splitting Rd E E The unstable
manifold M and stable manifold M are given by
MfxRd
xXg
MfxRd
xXg
where
j
j
and the manifolds do not depend on the
particular value of By Corollary
M
Mfg
We now want to study the asymptotic behavior of orbits which start o
those manifolds
1

CHAPTER INVARIANT MANIFOLDS
For example consider the case where is hyperbolic and the spec
trum split at equivalently in the gap with int Then the orbits
starting on M approach exponentially fast forwards in time choose
while the orbits starting on M approach exponentially fast
backwards in time choose
However if x M
Mwe
expect that
x approaches M
in fact a certain reference orbit
P xonM
exponentially fast forwards in time and a refer
ence orbit
P x on M exponentially fast backwards in time
This is indeed the case and these reference points P x M are
uniquely determined see Fig
We will prove that all initial values with the same point y P x
form an invariant manifold M y and all initial values with the same
point z P x form an invariant manifold M z These two man
ifolds have exactly the point x in common so that P x P x are
nonlinear coordinates of x All these manifolds hence form a foliation i e
have local product structure We will use the corresponding nonlinear co
ordinate system in Subsect
to decouple the nonlinear RDS into a
and a part
Theorem Foliation by Invariant Manifolds
A Discrete time case Let be a prepared RDS with xed point
generated by the random di erence equation
xnAnxnFnxn
nZ
which satis es the conditions of the Global Center Manifold Theorem
Assume that the Lyapunov spectrum of is not one point and
split into nonvoid and with corresponding invariant manifolds M
and M
Then there are uniquely de ned measurable mappings P
Rd Rd
such that for each
iP
P
Rd Rd is continuous and satis es P
PRdMandP
P i e P are nonlinear
projections Further
kPxkDLkxkDLLLL
ii The mappings P are invariant i e
Pt
t
t P forallt Z
iii The preimages of any point x M
iethesets
MxPfxgxM
1

HARTMAN GROBMAN THEOREM
MxPfxgxM
are graphs E z
m
xz EandEz
m
xz
E respectively which are measurable in x z continuous in x z
and Lipschitz continuous in z with constant D L hence are embedded
submanifolds of Rd
iv ThemanifoldsM x x M
are dynamically character
ized as follows For arbitrary
MxfzRd
z
xXgxM
MxfzRd
z
xXgxM
In particular M
M
v The manifolds M are invariant i e
tMxMttxforalltZxM
vi If the RDS is Ck k and satis es the hypotheses of Theorem
then the invariant manifolds M x are embedded Ck submani
folds of Rd and the graphs x z
m
x z are k times continuously
di erentiable with respect to z
B Continuous time case Let
tx
txtxtR
be a prepared RDS satisfying the conditions of the Global Center Manifold
Theorem
Assume the situation described in part A
Then all assertions of part A hold for Z replaced with R
Figure Foliations by invariant manifolds
Proof We restrict ourselves to the proof of part A Part B can be
reduced to part A by arguments similar to those used for the same purpose
in Subsect
The fact that we want to study the RDS with respect to an arbitrary
reference solution
x in general not the trivial solution forces us to
make an excursion to general nonautonomous random di erence equations
Step Let x Rd be xed Our ultimate aim is to prove that there is
a unique point P x M with the property that the di erence of
the orbits with initial values x and P x satis es
x
PxX
1

CHAPTER INVARIANT MANIFOLDS
To see this we have to study for the xed parameter k x
ZRd
the di erence equation of perturbed motion i e
znAnznfnkxznzlznZ
where denoting by
k x the solution of the original random di er
ence equation
with initial value xk x
fnkxzFnz
nkxFn
nkx
which is measurable in
x z continuous in x z and satis es
fn kx
kfnkxzfnkxzkn
Lkz zkn
Denote the solution of
with initial data l z and parameters k x
by
kxlz
Note that is a solution of
with parameter values k x if and
only if
kxisasolutionof
In particular the solution
of
corresponds to our reference solution
xof
With this well known trick the study of our original RDS in a neigh
borhood of the arbitrary reference solution
x is transferred to the
study of the neighborhood of the trivial reference solution z
of the
nonautonomous equation
While this trick in general fails to simplify the situation it is successful
in our case It can be easily checked that the two Key Technical Lemmas
and
and all constructions based on them remain valid if the
correct di erence equation
generating an RDS is replaced with
which has the wrong nonlinear part hence does in general not
generate an RDS Wanner
has formulated all results for this more
general case of a random di erence equation in his Chap
The price
we have to pay is that the invariant manifolds for the xed point z
become time and parameter dependent hence bundles For instance the
unstable and stable manifolds are now the bundles
Mkxfnz
mnkxz
nZzEg
of graphs and are dynamically characterized as
Mkxfnz
kxnzXg
Wewillmakeuseofthefactthat n kx
n kk xlaterinstep of
the proof
1

HARTMAN GROBMAN THEOREM
The graphs have the property m n k x
and have the same
Lipschitz constant namely D L for any k x with respect to the norm
kkk
In the remainder of the proof we need slight variations of arguments
already presented in great depth in Sect
so we take the liberty of
omitting some of the details
The measurability of x z
m n kx z followsasbefore
from Lemma
but now with replaced with x
We now verify the continuity of x z
mnkxzThe
continuity with respect to x is a consequence of Lemma
of Wanner
on the continuous dependence of a xed point on a parameter This
fact and Lipschitz continuity with respect to z uniformly in x yield joint
continuity with respect to x z as claimed
The general ow property for the solution of
and the dy
namical characterization of M k x obviously imply that M k x
is invariant in the following sense If n z
M kx then
ppkxnzMkxforanypZ
The dynamical characterization of M k x also has the consequence
thatforany kx
MkxMkxfn
nZg
Step We now transfer the constructions from
back to the original
equation
thereby dropping the tilde from M and m
We de ne manifolds M k x via the graphs
mnkxy
mnkxy
nkx
nkx
These graphs m enjoy the same measurability continuity and Lipschitz
conditions as those proved above for m
The manifolds de ned by these graphs are dynamically characterized as
Mkxfnz
nz
kxXg
They are invariant in the sense that if n y M k x then
ppnyMkxforanypZ
In particular since
k
forallk Z
Mk
M
fnz
nzXg
fnz
mn
z
nZzEg
Step We now construct the base points P k x and for reasons
of symmetry restrict ourselves to P k x
1

CHAPTER INVARIANT MANIFOLDS
Theideaistojustintersectthek berofM kx withthek ber
ofM
To detect points in the intersection de ne an operator
TkxEEz
mkkxmk
z
It has contraction rate D L
hence has a unique xed point
k x E which is measurable and depends continuously on x
By the de nition of the operator T k x the xed point ful lls
kx mkkxmk
kx
thus kmk
MkxM
De ne now
Pkx
kxmk
kx Rd
Henceforevery kx P kx istheuniquelydenedpointinRd
such that
kPkxM
and
kP kx M kx ieforwhich
kx
kPkxX
The base point P k x is similarly constructed
For the proof of
note that
k kxkk
DLkxkk
so
kP kxkk
DLk kxkk
DLkxkk
Analogously for kP k x k k
Step We nally use the fact that
generates an RDS i e
nkx
nkk
x
nkkx
This implies for the graphs of the n th ber of M k x that
mnkx
mnkk
x
1

HARTMAN GROBMAN THEOREM
We hence can put without loss of generality k
thus consider just
bundles M depending on x only Their ber is denoted by
Mxfz
m
xz
zEg
and is dynamically characterized by
MxfzRd
z
xXg
In particular we recover
M
M
fz Rd
xXg
Using
the construction of P k x in step also yields
PkxPk
x
We hence put
PxP
xMxM
Since obviously P x x for all x M
x M isitsownbase
point We collect all z Rd with this prescribed base point x M
MxfzRdPzxgPfxg
By
MxfzRd
z
xXg
and by
M x isagraph
z
m
xz
m
xz
With
and
we nally have obtained the objects guring in
Theorem
The manifolds M x have the regularity claimed in iii The dy
namical characterization in iv also gives v Parts i and ii immediately
follow from our construction
Part vi basically follows as the Ck property of M in Subsect
from a contraction theorem in a scale of Banach spaces which gives
smooth dependence of the xed point on the parameter z However we
refrain from giving more details Recall that for k the spectral interval
has to be wide enough to satisfy the two gap conditions of Theorem
Remark As deterministic counterexamples show even for smooth
RDS the continuous dependence of M x on x can in general not be
improved to C dependence

CHAPTER INVARIANT MANIFOLDS
Topological Decoupling
We continue to consider the situation of the last subsection For each
x Rd we have constructed the base points
Px
xm
xM
Px
xm
xM
where
x are the xed points of certain operators T x E E
see step of the last proof
Since P x lies on a graph in E E it is uniquely determined by
its coordinate
x E Similarly P x is uniquely determined
by
x
We thus have a mapping
g
g
Rd Rd
xgxgx
x
xEERd
such that by Theorem
x g xismeasurableandx g x
is continuous
Our original prepared RDS is for discrete time generated by the pair
of random di erence equations
xn
AnxnFnxnxn
xn
AnxnFnxnxn
coupled through their nonlinear parts F and F
We claim that if is a solution of
and
with initial
value x x then
ngn
n
is a solution of
zn
AnznFnznmnzn
zn
AnznFnmnznzn
with initial value z g x
Indeed let xn
n x solve
Then by the invari
anceofP Theorem ii P n xn zn m n zn arealso
solutions of
on M n Hence zn solves
and zn
solves
Now note that the pair
is simpler than the original pair
since their nonlinear parts hence their right hand sides are
decoupled from each other
1

HARTMAN GROBMAN THEOREM
Summing up we assign to each orbit
x a pair of two orbits
the orbit
P x moving on M and being identi ed with its E
coordinate given by
and the orbit
P x movingonM
and being identi ed with its E coordinate given by
We will now prove that this assignment is in fact one to one and onto
more precisely g is continuously invertible hence a random homeomor
phism so that it de nes a topological equivalence between and
Proposition Topological Decoupling into Two Blocks
A Discrete time Assume that the conditions of Theorem A are
satis ed Then
i The mapping g Rd Rd de ned above is a random homeomor
phismofRdie g
g
Homeo Rd with g
For
z z z the inverse g z is the unique point in the intersection of
the manifolds M z
m
z
andM z
m
z from
the invariant foliation
ii g de nes a topological equivalence between the RDS generated by
and
and the RDS generated by
and
ie wehave
gt
t
tg
tZ
iii The random homeomorphism g satis es the estimates
Bkxk kg xk Bkxk
Bkxk kg xk Bkxk
whereB BL
As a consequence if is arbitrary a solu
tion
xof
and
is
bounded if and only if the
corresponding solution
gxof
and
is
bounded
iv The invariant manifolds M of are mapped by g to the
linear subspaces E
gM
E
hence E are the corresponding linear and deterministic invariant man
ifolds of
B Continuous time case Assume that the conditions of Theorem
B are satis ed
Then all assertions of part A hold for represented by the pair
txx
tx
txx
txx
tx
txx
1

CHAPTER INVARIANT MANIFOLDS
and represented by the pair
tz
tz
tzm
z
tz
tz
tm
zz
Proof Since we again apply methods which have been elaborated in de
tail before we permit ourselves to be quite brief In particular we restrict
ourselves to the discrete time case leaving the reduction of the continuous
time case to the discrete time case as an exercise to the reader
i By de nition g is continuous We now prove that for any given
zz
z
Mz
m
zMz
m
z
is a one point set A point y y is in this intersection if and only if it
is a xed point of the operator
TzEEEE
yym
z
m
zym
z
m
zy
This operator is contracting with constant D L
and by the usual
argument its unique xed point g z is measurable with respect to z
and continuous with respect to z From the way we constructed P x
itisclearthatg g
gg
id Rd From this it follows that
g andg areonetooneandonto
ii That
de nes a topological equivalence follows from the fact
that g is a random homeomorphism
iii is derived in the usual manner from the fact that both g x and
g
z are xed points of contracting mappings
iv is an immediate consequence of iii and the dynamical characteri
zation of the unstable and stable manifold
By using Theorem
in an induction on i
p we arrive at the
best possible topological decoupling result based on the MET
Theorem Topological Decoupling
A Discrete time case Assume that the conditions of the Global Center
Manifold Theorem
are satis ed Let the prepared RDS be generated
by the p coupled random di erence equations
xinAinxinFinxn
xpnnZi
p
1

HARTMAN GROBMAN THEOREM
Then there exists a random homeomorphism g
Rd Rd with
g
such that is topologically equivalent to the RDS
gt
t
tg
tT
where is generated by the following set of p decoupled random di erence
equations
zinAinzinFinzinminzinnZi
p
Here
MifzimiziziEig
is the Oseledets manifold tangent to Ei
ii The random homeomorphism g satis es the estimates
Bkxk kg xk Bkxk
Bkxk kg xk Bkxk
whereB BL
iii All invariant manifolds Mij
i j p of aremappedby
g to the linear subspaces Eij
g Mij
Eij
hence the Eij are the corresponding linear and deterministic invariant
manifolds of
B Continuous time case Assume that the conditions of the Global
Center Manifold Theorem
are satis ed Let the prepared RDS be
represented by the p coupled equations
itx
xp
itxiitx
xpi
p
Then the statements of part A hold for
where is now rep
resented by the following set of p decoupled equations
it zi
itziitzimizii
p
Remark More Invariant Manifolds but not Lipschitz In
Theorem
we have constructed the p
p
invariant man
ifolds Mij with Lipschitz continuous graphs corresponding to spectral
intervals ij
1

CHAPTER INVARIANT MANIFOLDS
But the RDS generated by
has p
nontrivial invariant
subspaces E
i Ei where f pg is nontrivial The homeo
morphic images
gEM
are topological manifolds which are invariant for However these man
ifolds are in general not Lipschitz continuous as deterministic counterex
amples show cf Aulbach and Wanner Sect
We can now derive the random version of the reduction principle which is a
key statement of center manifold theory Suppose the Lyapunov spectrum
of is cut at a negative point
Then the stability behavior of
equations
and
for t
can be reduced to the one of the
unstable equation
De nition Stability Let be an RDS of continuous mappings
in Rd A reference solution
x is called stable with respect to the
random norm k k if for any
there exists a
such that
sup
tktx
t xkt
whenever kx x k
The reference solution is called asymptotically
stable if it is stable and in addition if for any x with kx x k
lim
tktx
t xkt
If the reference solution is stable and if there exists a c
such that for
anyxwithkx xk
ktx
t xkt
ectfortTx
it is called exponentially stable
Corollary Reduction Principle Assume that the top Lya
punov exponent of satis es
andtake f gand f
pg Then the trivial solution
of
is stable unstable asymptotically stable exponentially
stable if and only if the trivial solution of the reduced equation
on E is stable unstable asymptotically stable exponentially stable
Proof Consider discrete time Proposition iii implies that is
stable etc for if and only if it is stable etc for
1

HARTMAN GROBMAN THEOREM
Equation
describes an RDS on the stable manifold M
Since
Corollary
implies that every solution
of
tends to exponentially fast more precisely
knxknCnkxkn
xM
Becausekn xknkn xknkn xkn
the stability behavior of is exactly re ected by that of
Hartman Grobman Theorem
We can now improve the result of the last subsection Not only is the
RDS topologically equivalent to an RDS which is block diagonal but
even to an RDS whose blocks corresponding to non vanishing Lyapunov
exponents are linear This is the key statement of a random version of the
Hartman Grobman theorem due to Wanner
We prepare its proof by a lemma which is an application of the First
Key Technical Lemma
Except for the nal theorem we will restrict
ourselves to discrete time
Lemma Consider the random di erence equations
xn Anxnfn xn
and
xn Anxnfn xn
in Rd k k with measurable Gl d R valued A and measurable f and f
Assume that there are constants
M andL
such
that for each xed
kAk
fin
sup
x Rdkfi n xkn
Mi
and kfin xfin ykn
Lkx ykn i
Finally assume that all solutions of
and
exist on all of Z
and that the solution with initial data k x denoted by i k x is
continuous with respect to x
Then for every k x there exists a uniquely determined point
H kx Rdsuchthat
kH kx
kxX
1

CHAPTER INVARIANT MANIFOLDS
where XfZRdkk sup
n Zknkn g
The function k x H k x is measurable continuous with respect
to x and satis es the estimate
kH kx xkk
ML
Further if is a solution of
then
H
is
a solution of
Proof Consider the random di erence equation
xn Anxnfnkxnfn
kx
where
fnkxfnx
nkxfn
nkx
and
fn
kx fn
nkxfn
nkx
Equation
depends on the parameter k x it is solvable on all of
Z and satis es all the assumptions of the First Key Technical Lemma
with
and
kfkxkM
Hence by the Key Technical Lemma ii there is a uniquely deter
mined initial data k k x depending measurably on x and con
tinuously on x for which the solution
kxkxof
isinX and
k
kxkxk
M
L
hence
k kxkk
ML
Note that
is a solution of
for the parameter values k x if
and only if
kxisasolutionof
Hence
Hkx
x
kx
1

HARTMAN GROBMAN THEOREM
is the uniquely determined quantity for which
kH kx
kxX
Moreover the mapping x H k x is measurable and continuous
with respect to x since
x
kxisso Byitsdenitionand
H satis es the required estimate
As to the last assertion let
be an arbitrary solution of
set x
kand n
nkHkxThen
isa
solution of
which is bounded
Nowletn Zbe arbitraryandputx H n
n We proved
above that x is the unique point in Rd for which
nx
n
nX
The identity
n
n implies
n
nX
hence by uniqueness
nxandxHn
n
n Since n Z was arbitrary we are done H
is indeed a
solution of
namely
As an intermediate step we linearize an RDS with a stable xed point by
means of the last lemma
Proposition Topological Linearization of Stable RDS Let
be a prepared RDS with discrete time and xed point generated by the
random di erence equation
xnAnxnFnxn
nZF
Suppose that the linear RDS generated by A has top Lyapunov exponent
so that by our preparations kA k
e
Assume further that the nonlinear part F of satis es the following
conditions There are constants L
and M for which
sup
xRdkF xk M
and
kFxFykLkxykxyRd
Then the RDS and are topologically equivalent i e there exists a
random homeomorphism h Rd Rd with h
such that
ht
t
th
tZ
1

CHAPTER INVARIANT MANIFOLDS
Moreover h has the property of being near identity in the following sense
khxxkMLkhxxkML
and h x Rd is the uniquely determined point for which
hx
xX
Proof We apply Lemma
twiceFirstwithfn x F n x
andf yieldingH kx H k
x and a second time with
f andfn xFnxyieldingHkxHk
x Put
hxHxhxHx
Both h and h are measurable with respect to
x and continuous with
respect to x
Wearedoneifwecanprovethath h
This is accomplished by applying Lemma
a third time now with
fn x fn x Fn x Thereishenceauniquemapping
h x such that
hx
xX
namely h x x On the other hand the above de nitions of h and h
say that also
hx
xX
as well as
hhx
hxX
But then
hhx
xX
By the uniqueness statement of Lemma
hhxhxxallxRd
Similarly
hhx
x allx Rd
Hence h h
That h and h map respective orbits to each other follows from the
last statement of Lemma

HARTMAN GROBMAN THEOREM
Remark There is an obvious dual version of Lemma
and
Proposition
applying to an RDS generated by
and its
linear part but now for the case that the smallest Lyapunov exponent
of satis es p Then
pepkAk
and
the constant L has to satisfy L
Now the Second Key Technical
Lemma has to be used We leave details to the reader
Proposition Topological Linearization of Hyperbolic
RDS Let be a prepared RDS with discrete time and xed point gen
erated by the random di erence equation
xnAnxnFnxn
nZF
Suppose that the linear RDS generated by A is hyperbolic
Assume further that there is a constant M such that
sup
xRdkF xk M
andaconstantL suchthatL CL
p whereCL isgiven
in Theorem
and
kFxFykLkxykxyRd
Then the RDS and are topologically equivalent i e there exists a
random homeomorphism h Rd Rd with h
such that
ht
t
th
tZ
Moreover for any i j p the center manifolds Mij of are
homeomorphically mapped onto the linear subspaces Eij the center mani
folds of
h Mij
Eij
Further the unique invariant measure of is the Dirac measure at
Proof We rst apply Theorem
according to which is topologically
equivalent via a random homeomorphism g to the RDS generated by
the system
of p decoupled equations
Now each block satis es the conditions of Proposition
or its dual
see Remark
Hence the RDS i generated by the i th equation is
topologically equivalent to the linear RDS i generated by Ai Denote the
corresponding random homeomorphism by hi The random homeomor
phism
hxhgx
hpgpx
1

CHAPTER INVARIANT MANIFOLDS
serves our purpose
A measure is invariant if and only if the measure
h
is
invariant But the only invariant measure of a hyperbolic RDS is cf
Corollary
We now treat the classical C case and also undo the preparations
Theorem Hartman Grobman Theorem Let be a C
RDS with discrete or continuous time and xed point so that t x
Dtxtxwitht
andD t
Suppose
that the linear RDS D satis es the IC of the MET and is hyperbolic
Assume that the nonlinear part t for discrete time F
of the corresponding prepared version of ful lls the following
conditions
a There exists a constant M such that
forT Z
sup
xRdkF xk M
forT R
sup
t sup
xRdkt xkt M
b There exists a constant L such that L C L
p and
forT Z
sup
xRdkDF xk L
forT R
sup
t sup
xRdkDtxkt L
Then is topologically equivalent to its linear part D at x
ie
there exists a random homeomorphism h with h
such that
ht
t
Dt
h
tT
Moreover all invariant manifolds Mij are homeomorphically mapped
by h onto their linear counterparts Eij
Further the unique invariant measure of is the Dirac measure at
Proof i We rst assume that the RDS is prepared The mean value the
orem implies that the assumptions of Theorem
are satis ed Hence
there is a random homeomorphism h such that
ht
t
Dt
h
which also maps the invariant manifolds Mij homeomorphically onto
Eij
1

LOCAL INVARIANT MANIFOLDS
ii If is not prepared we use Theorem
and switch to the corre
sponding prepared RDS If P denotes the random linear isomorphism
of Theorem
then
h
P
hP
certainly de nes a topological equivalence of and D as a simple
calculation shows Since P Mij
Mij the proof is completed
Remark Hartman Grobman Theorem Non Hyperbolic
Case In the non hyperbolic C case the equation of the decoupled
system
and
which corresponds to the vanishing Lya
punov exponent cannot be simpli ed further Under the assumptions of
Theorem
with the only exception that now i
for some i
the RDS is topologically equivalent in general not to D
s c ubutto
s c u where scu jEscuEscu
the stable center unstable space of and e g for discrete time czc
czc Fczc mczc Ec
zc
mczc Es
Eu theC
graph of the classical center manifold Mc of
A measure is invariant if and only if the image measure is
invariant which is the case if and only if supp
Ec
andisc
invariant Hence nontrivial invariant measures of are supported by the
classical center manifold Mc of In particular if has an invariant
measure which is di erent from then the point cannot be hyperbolic
Remark Topological Versus Smooth Linearization We
stress that even for smooth RDS the linearizing homeomorphism is in gen
eral not a di eomorphism The problem of linearizing a C RDS by means
of a C di eomorphism is part of the normal form problem treated in Chap
It turns out that even in the hyperbolic case obstructions to lineariza
tion in the form of resonances appear
Local Invariant Manifolds
Although the outcome of our invariant manifold theory developed in Sect
is very satisfactory this is not at all the case for the assumptions we
have to impose as some of these drastically restrict its use in applications
Recall that for a Ck RDS with two sided time and xed point we put
t
Dt
and write t x
txtxwhere
is Ck in x and has the properties t
andD t
We
rst need the not very restrictive and inevitable assumption that satis es

CHAPTER INVARIANT MANIFOLDS
the IC of the MET without this assumption there would be no invariant
manifold theory for RDS at all
Our further assumptions and their drawbacks are
First we prepare the RDS We block diagonalize by means
of a linear random coordinate transformation P whose construction
i e measurable selection needs the knowledge of the Oseledets spaces of
which is hardly ever available Then we introduce the technically very
convenient random norm k k the construction of which also needs the
knowledge of Oseledets spaces as well as Lyapunov exponents altogether
an extremely unrealistic situation
Next we impose a global Lipschitz condition with a deterministic con
stant L on the nonlinearity of the prepared RDS but formulated in
terms of the random norm This constant L has to be su ciently small
where smallness is measured in terms of the gaps in the Lyapunov spectrum
of For the Hartman Grobman theorem we need in addition a uniform
deterministic bound M on also measured by means of the random norm
However there is a satisfactory way out the construction of local in
variant manifolds under conditions which are quite general and are met in
most applications to which this section is devoted Our method will be
to replace the nonlinearity of by a new one
which is of the same
regularity class and agrees with in a neighborhood U of but satis es
the assumptions for the existence of global invariant manifolds We call
the intersection of the global manifolds for with the neighborhood U the
local manifolds of The problem then is what those manifolds mean
dynamically for the RDS
Local Manifolds for Discrete Time
The existence of local invariant manifolds can be proved in great generality
for discrete time We hence start again with a Ck RDS with xed point
generated by
xnAnxnFnxn
nZ
where A
D
F
and DF
and the linear
cocycle generated by A satis es the IC of the MET In view of Proposition
we can without loss of generality assume that is prepared in the sense
of De nition
Truncation of Nonlinear Part
Lemma Let F
Rd Rd be a Ck function with F
and DF
Then for any xed M
and L
there

LOCAL INVARIANT MANIFOLDS
exists a random variable
and a function F
Rd Rd
such that
aF x F xforallx B where
BfxRdkxk g
bF
Cb more precisely
kFksup
xRdkF xk M
void if M
and F
Ck
b more precisely
kFksup
xRdkDF xk L
Proof i We x a C cut o function Rd Rd with the following
properties
C sup
xRdkxk
x
x kxk
x
kxk kxk
Note that all derivatives of are bounded so
Cq
sup
xRdkDq xk q
ii Nowdenefor
xRdRd
x
x
x kxk
kxk x kxk
Clearly supx Rd k x k and the chain rule yields
sup
xRdkDxkC
iii We now construct a Euclidean ball KR fx Rd kxk R g
in which F and DF have certain prescribed bounds
Denote by B
the slowly varying function which connects
the Euclidean and the random norm
BkxkkxkBkxkeB
B
eB
See De nition
for the spaces Ck
b
1

CHAPTER INVARIANT MANIFOLDS
First choose R
such that in case M
R
maxfr kF xk M
eB for all kxk rg
This is possible since F
and F is continuous Clearly R is
measurable In case M
take R
Now choose R
such that
R
maxfr kDF xk L
CeB for allkxk rg
Again this choice is possible since DF
and DF is continuous
In addition R is measurable
iv WenowdeneR
minR R
B and
FxFRx
and verify the assertions about F
Since R is C F is of the same regularity class as F hence Ck
Clearly
FxFxforallxKR
BydenitionkF xk B kF xk eB kF xk
and
sup
x RdkF xk sup
kxkRkFxkM
eB
hence
kFksup
xRdkF xk M
Similarly
kFksup
x RdkDF xk eB sup
x RdkDF xk
CeB
sup
kxkRkDF xkL
Putting
R
B
F x F x alsoholdsinthe ball
B with respect to random norm since
B fxRdkxk gKR
Alsofor q kkF kq supxRdkDqF xk Lq
hence F
Ck
b

LOCAL INVARIANT MANIFOLDS
Construction of Local Manifolds
Proposition Local Invariant Manifolds Discrete Time
Let be the prepared Ck RDS generated by
andletL p
Choose
L according to Lemma case M
and put
U
BfxRdkxk g
Theni x A x F x is the time one mapping of a prepared
Ck RDS which satis es the assumptions of the Global Smooth Invariant
Manifold Theorem
Denote the corresponding global manifolds by
Mij ii We have
nx
nxforallxUnTU xTU x
where
TUx
supfnZ kxUkforallkng
TU xinffnZkxUkforallnkg
denote the moment before the rst exit forwards backwards in time from
the neighborhood U
iii ForeachN Nthereisaradius N
such that setting
UN
BNfxRdkxkNg
nx
n xforallxUN jnjN
Equivalently
lim
xTUx
lim
xTUx
iv We call
M loc
ij
Mij U
the local invariant manifold of corresponding to the spectral interval ij
fi
jg tangent to Eij It is locally invariant under in the
following sense If x M loc
ijthennxMloc
ij n foralln
TUxTUx
See Fig
Figure Local invariant manifold
1

CHAPTER INVARIANT MANIFOLDS
Proof i We show that
AF
DikRd
As in the proof of part o of the Second Key Technical Lemma
we
prove invertibility of by studying for each z
Rd the operator
TzRdRdxAzAFx
Since kA k
p and was chosen in Subsect
to satisfy
p the operator is contracting with uniform rate L
By the well known fact that the xed point x z of a uniform contrac
tionf X Z XofclassCkdependsinaCk wayontheparameterz
see Chow and Hale Theorem
the result follows
iiIfxnUn thenFn xn Fn xnimplyingxn
xn similarly backwards in time Hence the orbits of and coincide at
timesn iftheydosoattimen andifxn U n
iii Choose an arbitrary N N and consider for jnj N the mappings
x
n x Since they are continuous even Ck and n
there exists a N
for which
kn xkn
n foralljnjN
provided x UN
BN fx Rdkxk NgForthosex
clearlyTU x NandTU x N
iv The local invariance immediately follows from the invariance
ofMij andthefactthat n x and n x coincideforn
TUxTUx
Remark i Although the ball U
B only depends on
the function F depends on the particular cut o function in the proof of
Lemma
and so do its invariant manifolds Mij This is the case
even for the part of these manifolds in the intersection with U where
F F since an orbit starting in U can go in and out in the course of
time Thus the local invariant manifolds M loc
ij generally depend on the
cut o function
ii There is no general dynamical characterization of local invariant
manifolds Hence so far they are purely geometrical objects tangent to the
subspaces Eij characterized not uniquely by their local invariance
iii The local invariance can also be expressed as follows For N
N take UN
according to Proposition iii Since n UN
U n forjnj N wehave
n Mloc
ij
M loc
ijnforallNnN
on the intersection of both sides with the set n UN

LOCAL INVARIANT MANIFOLDS
There is however the following observation which is very important in
bifurcation theory namely that small solutions always lie on certain local
invariant manifolds irrespective of the cut o function
Proposition Let the conditions of Proposition
be satis ed
Letx U
iIfTU x
then x Mloc
j for any local unstable mani
fold for which j and for any cut o function
ii IfTU x
then x Mloc
ip for any local stable manifold
for which i
and for any cut o function
iii IfTU x
andTU x
then x Mloc
ij for any
local center manifold for which j
i and any cut o function In
particular if i for some i and TU x
then x Mloc
c
M loc
ii
a classical local center manifold and for any cut o function
Proof iWehaveTU x
if and only if
kn xkn
n foralln
hence n x
n x forthisx U andalln Z Now
MjfxRd
xXag
wherea j j
j
is the dynamical characterization
of this manifold We try our x U with TU x
Then
k xka
k xka
sup
nank xkn
sup
nan
n
where we have used the fact that a
and
As a result x
M loc
j no matter which cut o function was used for
The proofs of ii and iii are completely analogous
The moral of Proposition iii is that those initial values whose
orbits never escape the ball U forwards and backwards in time this is the
meaning of small here are the common core of all local center manifolds
We collect our ndings in the following theorems
Theorem Local Invariant Manifold Theorem Discrete
Time Let be the Ck RDS with discrete time and xed point gen
erated by
Assume that D satis es the IC of the MET
Then for each interval ij from the Lyapunov spectrum of D has
local invariant Ck manifolds M loc
ij tangent to Eij
This like the other conditions in parts ii and iii of this proposition is a condition
on only
1

CHAPTER INVARIANT MANIFOLDS
Proof We switch to the prepared RDS n
Pn
n
P
and construct its local invariant manifolds M loc
ij according to
Proposition
Then M loc
ij
P Mloc
ij are the corresponding
local invariant manifolds of
Theorem Local Hartman Grobman Theorem Discrete
Time Let be a C RDS with discrete time and xed point gener
ated by
Assume that D satis es the IC of the MET
i Suppose is hyperbolic Then is locally topologically equivalent to
i e there exists a random homeomorphism h with h
and a
neighborhood U of such that
hn
nx
nhxforallxU
andalln TU x TU x
ii If is not hyperbolic then is locally topologically equivalent to
the RDS
s c u where scu jEscuEscudenotesthe
stable center unstable space of and c
c Fczc mczc Here
Ec
zc mc zc Es Eu is the classical C center mani
fold of a corresponding cut o RDS
Proof A brief outline of the proof for the case i will su ce We rst
move to a prepared RDS by a linear isomorphism P then use Lemma
and replace the nonlinearity F
with a cut o F satisfying the
assumptions of the global Hartman Grobman Theorem
We next lin
earize the RDS
F according to Theorem
by a random home
omorphism which maps all Mij homeomorphically to Eij Let B be
the closed ball inside of which
Then U
P int B serves
our purpose
Remark i The local topological equivalence in Theorem
can be brought into the following form For each N N there exists a
neighborhood UN of such that in case i
hn
nx
nhxforallxUNjnjN
ii Small invariant measures We have the following extension of the
global statements in Theorem
and Remark
If a invariant
measure is small in the sense that for some N N supp UN
then necessarily
in the hyperbolic case while h
isc
invariant in the non hyperbolic case In particular if has a small
nontrivial invariant measure then cannot be hyperbolic This is relevant
for bifurcation theory of RDS see Theorem

LOCAL INVARIANT MANIFOLDS
Dynamical Characterization and Globalization
In extension of Proposition
we would expect that small enough initial
values x of orbits which decay exponentially fast to have TU x
and are natural candidates for elements of any local classical stable manifold
M loc
s
tangent to Es
i i Ei This is indeed the case provided the
size of U n can be controlled in the sense that it cannot shrink to zero
at a geometric rate This gives rise to an additional assumption on the
nonlinear part F of which is not present in the deterministic case
De nition Tempered Ball The ball U
Bfx
Rd kxk g is called tempered with respect to if is a tempered
random variable in the sense of De nition
ieif
lim
n
nlog n
Since
our condition excludes the case lim inf log n n
Also the property of U being tempered or not is a property of the nonlinear
part F of
Proposition Dynamical Characterization of Local Mani
folds Consider the prepared Ck RDS generated by
Choose a
constant L p construct the ball U
B according to Lemma
and local invariant manifolds M loc
ij according to Proposition
Assume in addition that U is tempered
i Consider the local stable manifold M loc
ipforiinUand
select bi right
Then there exists a closed ball V
Br U such that
M loc dyn
ip
M loc
ip
V
has the following properties
For any x Mloc dyn
ip
TUx
Further M loc dyn
ip
is dy
namically characterized by as follows For x V
x Mloc dyn
ip
x Xbi
Consequently M loc dyn
ip
is a piece of the local stable manifold which is
dynamically characterized solely by hence is independent of the choice of
the cut o function
1

CHAPTER INVARIANT MANIFOLDS
Finally M loc dyn
ip
has the following forward invariance property
n Mloc dyn
ip
M loc
ip n foralln Z
ii Consider the local unstable manifold M loc
jforj inU
and select aj left
Then there exists a closed ball V
Br U such that
M loc dyn
j
M loc
j
V
has the following properties
For any x Mloc dyn
j
TUx
Further M loc dyn
j
is
dynamically characterized by as follows For x V
x Mloc dyn
j
x Xaj
Consequently M loc dyn
j
is a piece of the local unstable manifold which is
dynamically characterized solely by hence is independent of the choice of
the cut o function
Finally M loc dyn
j
has the following backward invariance property
n Mloc dyn
j
M loc
j n foralln Z
Proof iFixiwith i
choose bi
and a cut o function and
construct Mip Choose x Mip We have x Mip if and only if
x Xbi By Corollary
kn xkn Cbn
ikxk n
whereC LL Henceforx Mip B C
kn xkn bn
in
Clearly
bn
i
n
logbi nlog n
and
log bi
Now we use our assumption that U is tempered We have
nlog n
n
hence there exists an ni
such that
bn
i
n for all n ni implying
kn xkn
nforxMip BCnni
1

LOCAL INVARIANT MANIFOLDS
Now choose a ball V
Br B Csuchthatforallx V the
latter is true also for the remaining nitely many indices
nni
This is possible because n
and x
n x is continuous
We now prove that M loc dyn
ip
V Mip serves our purpose
Take x Mloc dyn
ip
Then by construction n x
nx
U n foralln thusTU x
As to the dynamical characterization note that x M loc dyn
ip
Mip is equivalent to
x Xbi which because of TU x
is equivalent to
x Xbi
The forward invariance property immediately follows from the for
ward invariance of M loc
ip
and the fact that the orbit n x of
x Mloc dyn
ip
never leaves U n
The proof of ii is completely similar
Remark i The proof shows that uniformly on M loc dyn
ip
kn xkn bn
i foralln
ii We have ags of dynamically characterized local stable for i
and local unstable manifolds for j
similar to Corollaries
and
iii We always have M loc
p
U For
M loc dyn
p
V
U is a ball around such that
holds for some b and
uniformly for all x V This uniform exponential decay to also holds
in the Euclidean norm
It remains to be investigated when the nonlinearity F of admits a tem
pered ball U
B such that
sup
kxkkDFxkL
Put
Mr
sup
kxk rkDF xk
The function r M r is continuous and increasing with M
and the biggest possible is
supfrMrLg
It is hence the steep increase of M
thus of DF
near which is
dangerous as it forces to be small
1

CHAPTER INVARIANT MANIFOLDS
Lemma Su cient Condition for Temperedness Let
Mr
sup
kxk rkDF xk
rq
forsomeq and r r wherelog L P Thenlog L P
hence U B is tempered We can in particular choose q and
sup
kxk rkDF xk
Proof Wehavelog L P ifandonlyifforany
X
nPflog ngX
nPfMenLg
But for each
there is some
such that
PfMen Lg PfLenqg
Pflog logL n qg
Pflog
ng
the latter from a certain index n on
The second statement follows from the rst one and the mean value
theorem
Dahlke
introduced an integrability condition on the second deriva
tive of a random di eomorphism on a compact manifold for exactly the
same reason as we do to obtain a dynamical characterization of local stable
and unstable invariant manifolds see his Lemma
In Proposition
we have obtained a dynamical characterization of
certain local invariant manifolds for RDS which do not satisfy the global
assumptions This dynamical characterization is exactly the same as the
one by which we de ned invariant manifolds at the outset in Sect We
hence could try the de nition
Mip fx Rd
x Xbigi bi right
and
MjfxRd
xXajgj
aj left
1

LOCAL INVARIANT MANIFOLDS
for stable or unstable invariant sets for By the cocycle property
these sets are indeed invariant i e
n Mip
Mip n and
nMj
Mj n foralln Z
In particular if i is the biggest negative Lyapunov exponent or if
j is the smallest positive exponent we can de ne the classical stable
and classical unstable set Ms and Mu as particular cases of
andThe above de nition also makes formally sense for i
and j
but then the corresponding orbits are permitted to grow exponentially fast
hence no connection with our local constructions in a neighborhood of
can be expected
There is however a second concept to arrive at stable and unstable
invariant sets namely to globalize a local manifold i e to start from a local
manifold as a nucleus and collect all initial values whose orbits eventually
enter this local manifold
Proposition Globalization of Local Manifolds Assume
that the conditions of Proposition
are satis ed
i The set
Mip
n
n Mloc dyn
ip
n
of initial values whose orbits eventually enter M loc dyn
ip forwards in time is
an immersed submanifold of Rd of the same regularity and dimension as
M loc dyn
ip ii The set
Mj
n
n Mloc dyn
j
n
of initial values whose orbits eventually enter M loc dyn
j backwards in time
is an immersed submanifold of Rd of the same regularity and dimension as
M loc dyn
j
Proof We only consider i We need to prove that Wn
n contains
an increasing subsequence where the closed sets
Wn
n Mloc dyn
ip
n
n n Mlocdyn
ip
nn
We do not know yet whether these sets are manifolds
1

CHAPTER INVARIANT MANIFOLDS
consist of those initial values whose orbits are in M loc dyn
ip
n atthe xed
time n and then never leave M loc
ip afterwards forwards in time
Itsu cestoshowthatforW
M loc dyn
ip
there exists an n
n
such that
M loc dyn
ip
n
M loc dyn
ip
n
and then to repeat the argument
By the construction of V
Br in Proposition
kn xkn bn
i forallx Mlocdyn
ip
We now choose n
such that bn
i
rn
which is in nitely
often possible since lim supn log r n n we possibly have
liminfn logr n n
as V need not be tempered For this
n the above is true
Consequently for each Mip is the monotone union of the discs
n
M loc dyn
ip
hence is an injective immersion of the Euclidean
space Eip such that T Mip
Eip see Pugh and Shub p
The next theorem is the climax of invariant manifold theory for RDS It
says that the above two concepts of de ning stable and unstable invariant
sets are the same
Theorem Stable and Unstable Invariant Manifold The
orem Let beaCkRDSk
with discrete time and xed
point generated by
Suppose that D satis es the IC of the
METLetSD f
pg be the Lyapunov spectrum and let
RdE
Ep be the splitting of D
Consider the corresponding prepared RDS n
Pn
n
P
choose a constant L p and construct the ball U
B
according to Lemma
assume that U is tempered construct the dy
namically de ned local stable and unstable invariant manifolds M loc dyn
ip
and M loc dyn
j
according to Proposition
and consider M loc dyn
ip
P Mloc dyn
ip
and M loc dyn
j
M loc dyn
j
iForeachisuchthat i
Mip
Mip
Mip
and Mip is called the stable manifold of the RDS corresponding to the
spectral interval ip f i
pg It is of the same regularity class
as Mloc
ip
and tangent to Eip at It is dynamically characterized by
Mip fx Rd
x Xbig
1

LOCAL INVARIANT MANIFOLDS
where bi is any number less than in the spectral gap right
i
i
to the right of i Mip is invariant under
ii For each j such that j
Mj
Mj
Mj
and M j is called the unstable manifold of the RDS corresponding to
the spectral interval j f
jg It is of the same regularity
class as M loc
j and tangent to E j at It is dynamically characterized
by
MjfxRd
x Xajg
where aj is any number bigger than in the spectral gap left j
j to the left of j M j is invariant under
iii We have ags of stable for i
and unstable for j
manifolds similar to Corollaries
and
but now immersed in
each other
Proof We only deal with part i and can assume without loss of gener
ality that is prepared
We rst prove that Mip
Mip Let x Mip Then by
de nition of Xbi there exists a number C x such that
kn xkn bn
iC x foralln
It again follows from the fact that lim supn log r n n that there
exists an n n
x for which bn
iCxrnhence
kn
xkn
rn
iey
n
xVn
Now by the cocycle property
n y Xbi n Butthenby
Proposition i y M loc dyn
ip
n implying x Mip
Conversely let x Mip Then there exists an index n
n
x
for which y
n
x Mloc dyn
ip
n Hence again by Proposition
i n y Xbi n and by the cocycle property
x
Xbi which proves x Mip
Remark Alternative Dynamical Characterizations
i We also have
Mip fx Rd
x Xbig bi int right
1

CHAPTER INVARIANT MANIFOLDS
and
MjfxRd
x Xajg aj int left
ii The classical stable and unstable manifolds Ms and Mu are
particular cases of the above If D n
is hyperbolic the following
dynamical characterization is equivalent to the ones given in Theorem
and part i of this remark
MsfxRd
nx
asn
exponentially fastg
and
MufxRd
nx
asn
exponentially fastg
either in the Euclidean or the random norm
Corollary Asymptotic Stability in Probability Assume
the conditions of Theorem
and
Then the classical stable
manifold Ms is a neighborhood of its domain of attraction Moreover
is asymptotically stable in probability
Proof We only need to prove that is asymptotically stable in prob
ability for stochastic stability theory for SDE see Khasminskii
and
Arnold
Using Remark ii for the Euclidean norm we have for some b
uniformly for x V
kn xkCbnforalln
For each given
choose rst n such that C bn
with high
probability then choose x small enough to make the nitely many remaining
terms
with high probability using continuity in x and n
This implies
lim
xPfsup
nkn xkg forall
i e stability in probability
Further since V
Br with r
Pas
lim
xPflim
n
nxglim
xPfxVg
i e we have asymptotic stability in probability

LOCAL INVARIANT MANIFOLDS
Local Manifolds for Continuous Time
As local invariant manifolds can be constructed for local RDS we start for
T RwithalocalCkRDS
DRdtx
txDR
Rd
see De nition
hence
tDt
Rt
Dtt
is a Ck di eomorphism De ne the strictly invariant set
E
tRDt
Rd
of never exploding initial values see Sect
To go on we assume that possesses an invariant measure for a
su cient condition see Corollary
Then necessarily E
P a s in particular E
We can reduce to the Dirac measure by Proposition
but
now the extension is not to Rd but to graph E We have v
Dt
Dt
xforallt R hence E Wenowwriteagain
instead of and x instead of v
We hence can assume that t D t
Rt isalocalCkRDS
with t
hence both D t and R t are open neighborhoods
of Weput t
Dt
which is a global linear RDS for which
we assume that the IC of the MET are satis ed and write
tx
t
tx
t
Dt
In view of Proposition
we can always pass to a prepared if necessary
Theorem Local Invariant Manifold Theorem Local
Hartman Grobman Theorem Continuous Time Let the con
tinuous time RDS given by
satisfy the following conditions
i There exists a global Ck RDS
tx
txtx
t
Dtx
which satis es the su cient conditions of the Global Invariant Manifold
Theorem
ii On a closed ball U
B fx Rdkxk gwehave
tx
txforallxUtTU xTU x
1

CHAPTER INVARIANT MANIFOLDS
and
lim
xTUx
lim
xTUx
Here
TUx
supftR r xUrforallrtg
and
TU xinfftR rxUrforalltrg
Then the continuous time analogue of the Local Invariant Manifold The
orem holds for and its local invariant manifolds M loc
ij enjoy all
the properties stated in Subsects
and for the discrete time case
Further if satis es the conditions of the Global Hartman Grobman
Theorem
then the continuous time analogue of the Local Hartman
Grobman Theorem
holds for
The discrete time proofs hold verbatim for the continuous time case with
only one exception The fact that is now a local RDS causes the following
modi cation of Invariant Manifold Theorem
a The stable manifold Mip satis es
Mip
E
tDt
and ful lls instead of the full invariance property
t Mip
MiptRt
t
b The unstable manifold M j satis es
Mj
E
tDt
and
tMj
MjtRt
t
Note that it follows from assumption ii of the last theorem that for
each T there exists a radius T such that putting UT
BT
tx
t xforallx UT
and jtj T
The question remains as to which RDS meet the requirements of Theo
rem
In the discrete time case every RDS has local invariant mani
folds of the same regularity and satis es the local Hartman Grobman theo
rem This is due to the availability of truncation techniques for the nonlin
ear part in the neighborhood of a xed point Those truncation techniques
do not seem to be available for a general continuous time RDS
It is not known to the author whether such techniques exist for deterministic ows
1

LOCAL INVARIANT MANIFOLDS
However what we really truncate in the discrete time case is the non
linear part of the generator time one mapping of the RDS This proce
dure has of course its continuous time analogue We will hence investigate
whether RDE and SDE can be cut o on the level of their vector elds so
that the RDS they generate meet the requirements of Theorem
For the RDE case a quite satisfactory answer is given by Wanner
Sect for the Lipschitz case which can be carried over to the Ck case
we use the less clumsy t instead of t
Theorem Local Invariant Manifolds for RDE Let
xtAtxtftxtf
Df
where f LP Ck and A L P entailing that
generates a local
Ck RDS of type
and the RDE xt A t xt generates a linear
RDS which satis es the IC of the MET
Assume further that diag
p is prepared with
it Ait it
i
p
Then all assertions of Theorem
hold
We refrain from reproducing the very technical and long proof Its key idea
is as follows Replace f with a specially manufactured f in such a way that
xt A t xt f t xt generates a global Ck RDS By the variation
of constants formula can be represented as
txtxZt
sfs
s xds
The Gronwall Bellman lemma applied to the last expression and to the
corresponding expression for D shows that the assumptions i and ii of
Theorem
are satis ed
We have discussed in Subsect see Corollary
when assump
tion
can be satis ed
For the case of an SDE Carverhill following Ruelle
estab
lished the existence and regularity of classical local stable manifolds for SDE
on compact Riemannian manifolds and for invariant Markov measures
Carverhill s result was used by Baxendale Theorem to prove
the existence of classical local stable manifolds for
dxt
mX
j fxt dWjt fj
j
m
1

CHAPTER INVARIANT MANIFOLDS
in Rd for the Dirac measure by replacing the fj with compactly supported
fj which agree with the fj inside a ball
However general criteria are still missing but work is in progress e g
by Mohammed and Scheutzow
Examples
In this section we collect several mainly pedagogical examples in which
invariant manifolds can be constructed explicitly without reference to gen
eral existence theorems In cases where we write generators we restrict
ourselves to the white noise case
Example Explicitly Solvable Scalar SDE The explicitly
solvable scalar SDE
dxt
xt xN
tdt xtdWt
RN
is treated in great detail in Subsect
Example
and Subsect
to which we refer All invariant measures their Lyapunov exponents
and their invariant manifolds can be explicitly determined
Example A ne RDS with Hyperbolic Linear Part We
refer to Subsect
for a detailed presentation of the following Let
tx
t x t beana neRDSinRdwhoselinearpart
satis es the IC of the MET and is hyperbolic and whose additive part
satis es the assumptions of Theorem i Recall that all these assump
tions are automatically satis ed for the a ne RDS generated by
dxt
mX
j Ajxt bj dWjt
with dxt Pm
j Aj xt dW jt hyperbolic Then there exists a unique
invariant measure
whose spectrum splitting stable and unsta
ble invariant manifolds and invariant foliation were explicitly determined
in Corollary
Example Induced RDS on Sd P d and Gk d Let
be a linear RDS with two sided time and let be the nonlinear C RDS
induced on Sd by via
ts
ts
ktsksSd
1

EXAMPLES
This RDS is extensively treated in Sect We determine all invariant
measures in Theorem
and for each such measure we explicitly deter
mine the spectrum and the splitting in Theorem
On this basis one
can easily construct the corresponding invariant manifolds
Suppose for simplicity that the ergodic invariant measure real
izes the simple Lyapunov exponent i and
s
s
Sd
Ei the latter is always possible over an extension of see Theo
rem
Hence is hyperbolic under with spectrum S
f j i dj j pj igandOseledetsspaces
Ejs
spanvhvsis
vEj
Ts Sd
s
ji
The Oseledets manifold tangent to Ej s
clearly is
Mjs
Sd
spanEjs ji
where span E s
span v
svER
The stable
manifold is
Mss
Sd
span jiEj s
and the unstable manifold is
Mus
Sd
span jiEj s
We leave the analogous case of the RDS induced by a linear one on pro
jective space P d to the reader The RDS induced on the Grassmannian
manifold Gk d can be treated similarly on the basis of Subsect
Example Fractional Linear Transformations The mapping
which assigns to each one dimensional subspace spanned by x
y Rnfg
the value x
y R R f gis ahomeomorphismbetweenP andR The
action A induced by a matrix A a b
cd GlRonPandthe
corresponding fractional linear transformation z fA z
azb
czdonR
commute A fA Assume that the linear cocycle t on R
with two sided time satis es the IC of the MET and is hyperbolic with
spectrum
and splitting R E
E Then has exactly
two invariant measures namely the random Dirac measures at E with
exponent
and stable manifold P n fE g and at E
with exponent
and unstable manifold P n fE g Hence
the cocycle t f t on R with values in the three dimensional

CHAPTER INVARIANT MANIFOLDS
Lie group of fractional linear transformations has the two random xed
points zi
Ei
which are the only invariant measures with
corresponding stable and unstable manifolds
Example Three RDS on the Unit Circle A Additive
noise LetS R Z
boundary points identi ed and con
sider the SDE
dxt fxtdt fxt dWt bxtdt dWt
onS wherefx bx d
dxfx
d
dx and where b is smooth with
b
b and
Equation
generates a global C RDS
i The generator of the corresponding di usion process for time R
L bxd
dx
d
dx
is elliptic hence there is a unique stationary measure with a smooth positive
density x solving L
given by
x CexpZxbydy Zx
exp Zybududy
where
expZbxdx Zexp Zxbydy dx
The Lyapunov exponent of T which solves dvt b xt vtdt under is
Zbx xdxZx
xdx
where the second equality follows by integration by parts or from formula
We have
if and only if b is not constant in which case the
F measurable invariant measure dx corresponding to consists of
nitely many atoms of equal mass Le Jan Proposition and Crauel
Proposition
The stable manifold is an open neighborhood of
these points which cannot contain the whole segment between two adjacent
points if is a Dirac measure the stable manifold is a proper subset of
S We have
ifandonlyifbx b Rifandonlyif x
in which case the RDS consists of the random rotations
txxbtWt
mod
1

EXAMPLES
The invariant Markov measure corresponding to
x dx is hence
dx
x dx It is the unique invariant measure and all of S
is the center manifold for any x For the particular case b and
the generator of the di usion process is L
d
dx hence dxt dWt
de nes Brownian motion on S
ii Similarly the di usion process de ned by
for time R has
generator
L bxd
dx
d
dx
and its unique stationary measure has density x given by
with
b replaced by b The formula for the corresponding Lyapunov exponent
for
is given by
with b instead of b Finally the Lyapunov
exponent of is see Sect
Zbx xdxZx
xdx
We have
if and only if b is not constant in which case the
F measurable
corresponding to is concentrated on nitely many
atoms of equal weight and
ifandonlyifbx bifandonlyif
x
x
leading back to
B Gradient Brownian ow on S The following example is a particular
case of Example but studying it together with A is quite instructive
Consider the SDE
dxt sin xt dWt cos xt dWt
on S which generates the gradient Brownian ow on S The forward as
well as backward generator is with f x sin x d
dxfx cosxd
dx
Lfxfx
d
dx
so that the one point motions are Brownian motions on S and thus are
statistically identical with the one point motions generated by
for
b and
Both Fokker Planck equations have the unique solution x
but
the Lyapunov exponents of the corresponding invariant measures
x
and
x
are directly by formula
or by Theorem
below Z div sinxd
dx div cosxd
dx dx
1

CHAPTER INVARIANT MANIFOLDS
andThe stable manifold of is Ms x
Snfx gandthe
unstable manifold of is Mu x
Snfx g
Comparing cases A b and
and B we learn that Brownian
motion on S can be the one point motion of completely di erent RDS and
that invariant measures and their Lyapunov exponents cannot be seen by
the generator of the corresponding di usion process Case B also shows
that the same RDS can be stable under one invariant measure but unstable
under another one
C Carverhill s noisy north south ow An example related to A and
B was treated by Carverhill Consider on S R Z x the SDE
dxt
sin xt dt dWt
The two stationary measures and are given by
with
andbx
sin x for
respectively b x sin x for By
and
and
But for this example one can prove as in B that the corresponding
invariant measures
and are Dirac measures at x
F and
x F and the stable and unstable manifolds are as in B
Example Gradient Brownian Flow on Compact Manifold
Embedded in Rd Let M be a k dimensional compact Riemannian man
ifold isometrically embedded in Rd d
A standard way of obtaining
Brownian motion on M is to consider the vector elds fj x which are or
thogonal projections to TxM of the unit vector elds ej x restricted to M
equivalently fj x grad ej x The SDE
dxt
mX
jfjxt dWjt
on M has generator L
see Elworthy
hence the one point
motions are Brownian motions on M and the unique stationary measure
is normalized Riemannian volume dx
volMdx
The global ow of C di eomeorphisms of M global C RDS
generated by
is called gradient Brownian ow and its ergodic theory
and stochastic geometry has been the subject of numerous studies up
to date See for example Baxendale
Carverhill Chappell and
Elworthy Chappell Arnold and San Martin Bougerol
Elworthy
Elworthy and Yor
Elworthy and Rosenberg
As the material would easily ll a small monograph we cannot be self
contained here but will only quote some pertinent results
1

EXAMPLES
We rst deal with the Lyapunov spectrum of under the forward
Markov measure corresponding to dx Corresponding formulas are
given in Sect The calculations and estimates are facilitated by the fact
that for the gradient Brownian ow
mX
j rfjxfjx
Formula
for the sum of the Lyapunov exponents gives
volMZM
mX
jdivfjx dx
volMZM
mX
j ejxdx
from which Chappell s result follows
volMZM trace x dx l dimM
where x TxM TxM TxM is the second fundamental form of M
l is the largest non zero eigenvalue of which is negative and we have
equality in
ifandonlyif ej lejforallj
m
For the top Lyapunov exponent we obtain the formula
ZPMjxsjjxssj RicssdP
where P is an ergodic lift of to P M which exists by compactness if M
has dimension d then P is unique with supp P P M see Arnold
and San Martin p
and Ric u v denotes the Ricci curvature
NowassumeM Sd r fx Rd kxk rgThen
reads
dxt
mX
j ej rhxtejixt dWjt
xuv
rhu vix
rRicuv drhuvil d rand
drd
i e has a negative one point spectrum under Bougerol has shown
that among all hypersurfaces it is only the spheres which possess a one point
spectrum and that the spectrum is simple in all other cases
1

CHAPTER INVARIANT MANIFOLDS
However all this holds for the F measurable invariant measure
limt
t which for the unit sphere will be determined below
Taking into account that the di usion on R of the SDE
also has
generator L
hence t x t R are Brownian motions there is
a second invariant measure which is F measurable and obtained by
lim
t
t
Since L L we also have PL PL hence
and
ForM Sd Rd
d
dr
Interpreting and augmenting the results of Baxendale
within this frame
work gives the following
Theorem Ergodic Theory of Gradient Brownian Flow on
Sd LetM Sd fx Rdkxk g RdandlettheSDE
dxt
mX
j ej rhxtejixt dWjt
generate the gradient Brownian ow on Sd Then
it
G where G is the Lie group of orientation preserving
conformal di eomorphisms of Sd
ii has exactly two invariant measures namely
dx
xdx
xFPfxg
and
dx
xdx
xFPfxg
where dx is the normalized Riemannian volume of Sd In particular
x and x are independent
Further is stable with one point spectrum
d while
is unstable with one point spectrum
d
iii The stable manifold of x is Ms x
Sdnfxg
while the unstable manifold of x is Mu x
Sdnfxg
1

EXAMPLES
Proof Part i and iii are taken from Baxendale For part ii and
say allwehavetoproveisthatforanyf CSd R
lim
tZSdft
xdxfx
Pas
The limit on the left hand side exists by the usual martingale argument
and de nes a invariant measure Crauel Proposition
For d
follows from the fact that for each xed x Sd
lim
t
txx
Pas
For d and however the latter is not true Theorem Prop
erty
nevertheless holds It follows from the considerations prior to
equation
in that for t large enough t
compresses most
of Sd in the sense of Riemannian volume into any given small neigh
borhood of x
which implies
Suppose there were another ergodic invariant measure besides
Then its support C
supp is on the one hand an invariant random
closed set but on the other hand is supported by the stable unstable
manifold of x
which is a contradiction

CHAPTER INVARIANT MANIFOLDS

Chapter
Normal Forms
Summary
Normal form theory is at the heart of the theory of nonlinear deterministic
and random DS Its aim is to simplify ultimately linearize a DS by means
of a smooth coordinate transformation This is fundamental e g in bi
furcation theory However obstructions against transforming away certain
terms called resonances appear so that the simplest possible form is
in general nonlinear
Here we develop normal form theory for RDS which was initiated by
engineers and physicists almost years ago It turns out that the cohomo
logical equations that need to be solved to eliminate terms are now random
a ne di erence or di erential equations so that we can use our results
from Sect Resonances now take the form of integer relations between
Lyapunov exponents
As a preparation we give a brief introduction into deterministic normal
form theory in Sect Then normal form theory for discrete time RDS is
presented in Sect the main statement being Theorem
The RDE
case is dealt with in Sects Theorem for the nonresonant case and
Theorem
for the resonant case In Sect we present the random
analogue of a very successful procedure for simultaneously obtaining the
normal form eliminating the stable variables from the center equations and
determining the center manifold Theorems
and
We apply this
procedure to the noisy Du ng van der Pol oscillator in Subsect
The
SDE case is treated in Sect
where we have to cope with anticipative
data Theorem
1

CHAPTER NORMAL FORMS
Deterministic Prerequisites
In Poincare initiated a technique for simplifying a nonlinear system in
the neighborhood of a reference solution by a smooth change of coordinates
today called theory of normal forms In the deterministic case the reference
solution is almost exclusively assumed to be a xed point sometimes a
periodic solution while in the random case the reference solution can
be an arbitrary invariant measure
For later use we present a brief introduction into the deterministic the
ory for recent presentations see e g Anosov and V I Arnold Vander
bauwhede
Wiggins
or Katok and Hasselblatt
Suppose Rd Rd is a C di eomorphism with
The
normal form problem for is to nd a C di eomorphism h Rd Rd
with h
such that
h
h
is as simple as possible to be speci ed later in a neighborhood of Since
D
Dh
D Dh and D is considered simplest pos
sible if the matrix representation is in Jordan canonical form it is reason
able to assume without loss of generality that D is in Jordan canonical
form and h is near identity i e Dh
id Then D
D for
any such h
Put in terms of the C classi cation of di eomorphisms we look for
that element in the C equivalence class of which is simplest possible
in a neighborhood of The result called normal form of is the natural
generalization of the Jordan canonical form to a di eomorphism at a xed
point
The ultimate aim of normal form theory is the linearization of i e
to nd an h such that D The Hartman Grobman theorem states
that near a hyperbolic xed point a local C hence in particular C
di eomorphism is topologically equivalent to its linear part by means of
a local homeomorphism h The question whether h can be chosen to be
smooth has a negative answer in general Sternberg s theorem see below
The general deterministic procedure along which we develop the random
case is as follows Since and h hence
are C they have Taylor
expansions at of any order and also formal Taylor series expansions
h and which might have radius of convergence equal to since C
functions need not be analytic the reason these power series are called
formal
Deterministic as well as stochastic normal form theory makes statements about germs
of C di eomorphisms or vector elds i e equivalence classes of C di eomorphisms
or vector elds which coincide in a neighborhood of However for ease of presentation
we ignore this point
1

DETERMINISTIC PREREQUISITES
Our rst step is an inductive procedure to determine for each N a
near identity N jet HN such that if we use it as a coordinate transformation
then in the Taylor expansion of x
N x OjxjN theNjet
Nx AxPN
n
n is as simple as possible In applications this
N jet N is the main object of interest
The second step is to determine full formal power series h and which t
into the formal power series relation also called formal equivalence
h
h such that is as simple as possible is then called the formal
normal form of It will turn out that all invariants of formal equivalence
are already determined by the linear part D of the di eomorphism
The third step is to assure that there are indeed C di eomorphisms
h and having the formal power series expansion determined in step
as their formal Taylor series The answer is always a rmative Borel s
lemma
The fourth and nal step is to prove that if two C di eomorphisms
have the same formal Taylor series expansion at they are indeed C
equivalent by a near identity h The main classical result is Chen s theorem
If D is hyperbolic and
formally then
smoothly In
particular we obtain Sternberg s smooth linearization theorem provided
the nonresonance conditions
hold
Let for n
Nd
nf
d ZdjjdX
i
ing
Denote by x
xx
xd
d the scalar monomial in d variables of degree
jj nThen
Hnd HndRd ff XNd
n
xffRdg
is the vector space of homogeneous polynomials of degree n in d variables
withvaluesinRd We also writejj nfor Nd
n Observe that
nd Nd
n
nd
n
so
dimHndR D dimHndRd
d
AbasisF u
ud inRdandthebasis x jjnofHndR R
giveabasis xF xuii djjninHndwhile
HndfX
jjn
dX
ifixuikFfkFfjjnfi RD

CHAPTER NORMAL FORMS
column vectors ordered lexicographically identi es Hn d with RD D
d where kF is the mapping which assigns F coordinates respectively
x F coordinates to an element of Rd respectively Hn d
There is the following canonical choice of a scalar product in Hn d
see Vanderbauwhede
IfFisabasisinRd put
hfgiF X
jjnhfgiFX
jjn kFfkFg S
where xyS Pd
i xiyi is the standard scalar product in Rd and
d This makes Hn d h iF a Euclidean space in which the basis
x F is orthogonal
It will turn out to be very convenient for our purpose to identify Hn d
Hn d Rd with the tensor product of Hn d R and Rd for basic facts see
Sect
HndRd HndR Rd R Rd
where the above choice of bases induces the basis with elements x ui in
Hn d Rd and the isomorphism induced by the coordinate mappings maps
this basis to the standard basis fj ei R Rd The above scalar
product in Hn d can also be obtained by lifting the scalar products hx yi
Pjjn xyinHndR R andhxyiF kFxkFySonRdto
the tensor product see Lemma vii
Let a linear mapping in Rd be represented in the basis F by a d d
matrix A We now de ne the basic cohomological operator on Hn d by
CAnfx Afx fAx thedenitionismotivatedlaterbythe
relation
It is important to note but often not clearly said that
this operator depends on the choice of the basis F in Rd This fact is often
disguised by the assumption that A is in Jordan canonical form in the
standard basis of Rd The point is that the de nition of f Ax f Hn d
depends on the particular matrix representation of the linear operator
If a basis F in Rd is chosen and the above identi cation Hn d R Rd
is made the cohomological operator C A n of a d d matrix A on the space
Hn d is given as the following Kronecker product see Sect
CAnIANAnI
where I and I are
and d d unit matrices The
matrix
NAnisdenedby
NAnNnAAxX
jjnNnAx
In particular the entries of N A n depend nonlinearly on the entries of A
1

DETERMINISTIC PREREQUISITES
Example For d and n
we have
hence D
If F u u is a basis of R the corresponding six basis vectors of
HRHRRRarexeifor
xx eifor
andx eifor
where i
Further
IAAAAA
NA
a
aa
a
aa
aa
aa
aa
a
aa
aA
hence
NAI
BB
B
B
B
B
a
aa
a
a
aa
a
aa
aa
aa
aa
aa
aa
aa
aa
a
aa
a
a
aa
a
C
Lemma LetABbed dmatricesandletn Then
i
kN A nk ckAkn
for the operator and Hilbert Schmidt norm where the constant c is inde
pendent of A
ii
NIn I NABn NBnNAn
hence
NA n NAn forA GldR
iii Withthescalarproducthxyi Pjjn xy inR
NAnNAn
where A A is the transpose of A In particular if U is orthogonal then
soisNUn
iv
NABnINBnINAnI
and
NAnINAnIforAGldR
1

CHAPTER NORMAL FORMS
Proof iBythedenitionofNAn
NnX
jjn
c
dY
ijaij
ij
where
ijjjPd
ij ij ij N andthe matrices andcon
stants c depend only on and but not on A
Consequently
Nn
cX
jjn
dY
ijaij
ij
cX
jjn
dY
ij aij ijnn
cX
jjn
dX
ij
ij
naijn
since the geometric mean is less than or equal to the arithmetic mean Thus
Nn
c
dX
ij aijn ckAkn
HS
and
kN A nkHS ckAkn
HS
Similarly for the operator norm
ii BythedenitionofNABn
ABx X
jjnNAB xX
jjnX
jjnNA NB x
Equating coe cients gives
NAB X
jjnNB NA
iii follows by an elementary calculation see Vanderbauwhede
iv follows from ii and Lemma iii
Remark i In case A diag a
ad
I A diagA A diaga
ad
a
ad
NAn diagajjn NAn I diagaIjjn
where a
a
ad
d
ii IfAisblockdiagonalthensoisNAn I
iii If A is upper or lower triangular then N A n I is lower or
upper triangular

DETERMINISTIC PREREQUISITES
For example for d n
andA diaga a
CAdiagaaaaaaaaaaaaaa
We come back to the equation h
hor
hx
hx
where
and h are C di eomorphisms with
h
andDh id henceA D
D
In this relation is given
and h has to be chosen to make the resulting as simple as possible
Expansion into formal Taylor series at yields
xX
n
nx AxX
n
nx
xX
n
nx AxX
n
nx
and
hxX
nhnx xX
nhnx
where n n hn Hn d Inserting these expansions into equation
and equating coe cients yields the cohomological equations
CAnhn
nknn
where k
and for n
kn Hn d only depends on the terms
nh hn and
n andCAnisthecohomolog
ical operator introduced above Hence a recursive algorithm is possible
for determining h
hn
n from successively solving equation
for n
n
on the basis of data either given at step n or
determined in previous steps
The cohomological equation
is a linear equation in Hn d The
spectrumofCAnis
CAn fi
i
djj ng
where
d are the eigenvalues of A and
d
dIf
C A n equivalently if the nonresonance conditions of order n
i
d
d
id
dZjjdX
jj
1

CHAPTER NORMAL FORMS
are satis ed then the cohomological equation can be uniquely solved for
any right hand side n kn in particular for our favorite choice n
for any kn
Otherwise if
C A n we have resonance of order n In this
case we split the space
Hnd RCAn N
whereRCAn denotestherangeofCAnandNisanycomplementof
R C A n This arbitrariness in the choice of N makes the normal form non
unique Now the choice n is in general not possible anymore Write
kn kR
nkN
n determine hn by solving C A nhn kR
n andput n kN
n
The latter term cannot be eliminated if kn R C A n irrespective of the
choice of N and is part of the normal form of
There is the following canonical choice of N see Vanderbauwhede
Use the scalar product in Hn d introduced above Then C A n
C A n where A A is the transpose of A We hence have the orthogonal
decomposition
Hnd RCAn NCAn
whereNCA n denotesthekernelofCA n
Normal Forms for Random Di eomor
phisms
The Random Cohomological Equation
WefollowArnoldandXu Let beaC RDSinRd withtimeT Z
time one mapping
and with xed point
In contrast to the deterministic case the assumption of a xed point at
is without much loss of generality for an RDS with an invariant measure
see Lemma
Assume throughout that D
satis es the IC of the MET
De nition Random Coordinate Transformation A mea
surable mapping h Rd Rd is called a near identity random coordi
nate transformation if h
DiRdh
and Dh
id
The normal form problem for the RDS is now to determine a random
coordinate transformation h such that the smooth random equivalence
h
h equivalently
h
x
hx
1

NORMAL FORMS FOR RANDOM DIFFEOMORPHISMS
makes as simple as possible with the ultimate aim of smooth lineariza
tion
xD
x
We stress that D
and are generators of cocycles in partic
ular f g Rd f g Rd while h is a static coordinate
transformation in the ber f g Rd This will be important for the inter
pretation of the cohomological equation
Remark Equation
implies that also
nhn
nh
nZ
It is however in general not true that normal forms perpetuate from time
one to arbitrary times n Z So if
is as simple as possible
this might not be the case for
For a discussion and
a deterministic counterexample see Remark of Arnold and Xu
Let
Hn d ff Hn d measurableg
be the vector space of random homogeneous polynomials of degree n
in d variables with values in Rd
We next select and then x a random basis F
u
ud
of Rd adapted to the Oseledets splitting of D
to simplify D
so that in the F F bases the matrix representation A of
D
is block diagonal and the change from the standard basis to the
random basis F is a Lyapunov cohomology see Corollary
The random basis F of Rd gives rise to the random basis
xF jj n onHnd so a randomhomogeneouspolynomialf Hnd
has the coordinate representation
fX
jjn
xfX
jjn
dX
ifi
xui
hence
kFf kFfjjnwithkFfBf
fd CA
The function kF Hn d RD random variables with values in RD
D d given by f kF f is a linear isomorphism and allows
us to identify Hn d and RD Starting with Hn d R Rd we also have
HndRRd
1

CHAPTER NORMAL FORMS
Since the matrix representation of D
inthebasesF andF
is denoted by A
kFD
xAkFx
De nition Random Cohomological Operator and Equa
tion Let F be a random basis of Rd The linear operator
CAn Hnd Hnd
de ned by
fxgxAfxfAxCAnfx
is called the random cohomological operator It depends on the random
basisF andisgiveninHnd R Rdby
kF CAnf
IAkFfNAnIkFf
Here the
matrixNA n NA
isde nedvia A x
PjjnNA x
Any equation of the type C A nf g is called a random cohomological
equation
We stress that the appearance of makes C A n a true bundle mapping i e
wecannotdeneCAn berwise asamappingf CA nf
Introducing the linear operator
UHndHndf Uf f
a succinct form of the cohomological operator suppressing and n is
CAIANAIU
Note that due to their origin the two terms in
are of di erent
character
I A is thegenerator ofa cocycle andmapsfg Hnd to
f g Hnd while
N A n I has its origin in a coordinate transformation and maps
f g Hndtoitself
We now follow the deterministic procedure and expand the three map
pings h and in equation
as formal Taylor series at For this we
write the coordinate transformation h in the basis F F but the

NORMAL FORMS FOR RANDOM DIFFEOMORPHISMS
cocycles
and inthebasesF andF sox x
xd
will be the coordinates in F This yields
xX
n
nxAxX
n
nx
xX
n
nxAxX
n
nx
and
hxX
nhnxxX
nhn x
where n nhn Hnd
Inserting these expansions into equation
equating coe cients
and keeping track of the bers yields analogously to the deterministic case
the following result
Lemma The terms of order n of the formal power series expansion
of are successively given by the terms of order n of and h and lower
order terms More speci cally
A and
CAnhn
nknn
Here
kn
Tn
ASnhSnhSn
where Tn f denotes the terms of order n in the formal Taylor expansion
offandSNf PN
i TnfistheNjetoff
Example For d we have D hence Hn R for all
n For some random basis F
u
xAx
x
x
A
hxxhxhx
xAx
x
x
Hence
k
k
h
Ah
and
CAnhn
AnAnUhn

CHAPTER NORMAL FORMS
As a nal step we remedy the fact that the cohomological operator is not
the generator of a cocycle
Proposition Let C A n be the cohomological operator in Hn d
corresponding to A
iCAncanbewrittenas
CAnNAnIMAnU
where
MAnNA
nA
maps to hence is the generator of a cocycle while N A n I
maps to
ii If A generates the cocycle then N A n generates the cocycle
N nandMAn NA n Ageneratesthecocycle
MnNn
iii Let the Lyapunov spectrum of the cocycle generated by A be
SA
i the Lyapunov exponents listed with their multiplicities and
its splitting be Ei Then the MET holds for N A n and its spectrum
isSNA n f
jj ngwhere
Pd
i i i Further
the MET also holds for M A n which has spectrum
SMAn fi
iSAjjng
and splitting
H
M
ii
iSA SNA nG Ei
SMAn
where G denotes the splitting of N A n
Proof i By Lemma
and the rules for the Kronecker product given
in Lemma omitting the subscript n but indicating the bers of the
mappings
CA
NAI
NAI
IA
U
NAI
NAI
IA
U
NAI
NAA
U
Note that N A in the last line maps to since it is the product of
I whichispartofacocycle andNA
1

NORMAL FORMS FOR RANDOM DIFFEOMORPHISMS
ii By Theorem iii A A generates
if Ai generate i
i
Thus it remains to proof that N A generates N n
Bn ButbyLemma
Bnk
Nnk
Nk
nk
NnkNk
BnkBk
iii In view of Theorem ii and our preparations it remains to
prove the statement on N A
RRIf
AP
BP
is a Lyapunov cohomology then by Lemma i so is
NA
NP
NB
NP
Hence the MET holds for N A in the basis F if it holds in the standard
basis which is again assured by Lemma i
To calculate the spectrum of N A we use singular value decomposi
tion in Rd and R both endowed with the scalar product introduced above
Hence there are orthogonal d d matrices Uk and Vk for which
k
Uk diag k
dk Vk
where k
dk
and
lim
k klogik
ii
d
Thus
k
Vk diag ik Uk
and applying the operation N to the last line and using Remark i
Nk
NUk diag k jjnNVk
By Lemma iii
Nk
NUk diag k jjnNVk
1

CHAPTER NORMAL FORMS
i e we have a singular value decomposition of N k
By the
Furstenberg Kesten theorem the Lyapunov exponents of this cocycle are
lim
kklogk
lim
kk
dX
i
ilogik
dX
i
ii
Example i For d and n
we have
nNd
nf
knkk
ngandD n
If A has simple spectrum
S A f g then the spectrum
SNAnfknkk
ng
is simple but in the spectrum
SMAnfk
nk
l
nlkl
ng
only the exponents for k and l n are simple while the remaining n
exponents have multiplicity put l k
Also hyperbolicity does in general not carry over from A to M A n
Even if
SMAn ifandonlyif
k
nk
for some k
n Further SMAn forsomen ifand
only if
Q
ii If however M A n is hyperbolic for some n
then necessarily
A is hyperbolic since by Proposition iii
niSMAnfor
any i
d
De nition Resonance Nonresonance The linear cocycle
generated by A in Rd is called resonant of order n if S M A n
Otherwise it is called nonresonant of order n
Nonresonant Case
We now formally linearize by determining a formal power series of a
coordinate transformation h for which we can solve
n CAnhn kn
successively for n
inHndforthe choice n
This leads to
the successive cohomological equations
CAnhn kn n
1

NORMAL FORMS FOR RANDOM DIFFEOMORPHISMS
With the representation of C A n from Proposition i this is equiv
alent to
UMAnhnNAnIkngnn
where L A n U M A n is also called cohomological operator and nally
Uhn MAnhn gn n
or berwise
hn
MA nhn
gn
n
Solving for it amounts to looking for a stationary solution of the random
a ne di erence equation in RD
xkMAknxkgnk
For the nonresonant case we can readily use our results from Theorem
i
Let us recall De nition
according to which a random variable f is
called tempered from above with respect to if
lim
t jtjlogkftk
holds P a s We will also make use of Lemma
Proposition If A is nonresonant for some n
andifgn Hnd
is tempered equivalently if kn is tempered then there exists a unique
hn Hn d explicitly given in Theorem i which solves equation
Moreover hn is also tempered
Proof The existence and uniqueness statement is from Theorem i
That the statements about gn and kn being tempered are equivalent
follows by Lemma i which gives
logkgnk logc nlogkA k logkknk
log kknk log c n log kAk log kgnk
and the fact that A and D
are Lyapunov cohomologous
Henceforth we omit the words from above throughout Chap
1

CHAPTER NORMAL FORMS
Corollary Let N
Let A be nonresonant of order n for
n N and suppose that in the Taylor expansion
xAx
PN
n
n x OjxjN
at of order N all n are tempered Then
there is a unique near identity N jet HN x
xPN
nhnxwith
tempered coe cients hn hence is a local analytic di eomorphism such that
is transformed by HN to whose N jet in the Taylor expansion of order
N is linear
xAxOjxjN
Proof All we have to prove is that all hn are tempered provided all n
are But this follows inductively from the formula for kn in Lemma
the fact that sums and products of tempered random variables are tempered
Lemma
and from Proposition
Theorem Formal Linearization of Random Di eomor
phism Suppose that the linear cocycle generated by D
is nonreso
nantofanyorder n ie
i
iSD
jj
and that in the formal random Taylor series of the random di eomorphism
AxPn
n x at all n are tempered Then there
is a random coordinate transformation h with a uniquely determined
formal random Taylor series h x x Pn hn x where all hn are
tempered and h formally linearizes i e the formal random power series
of
h
hisAx
This is clear provided we can solve the third step of the normal form pro
cedure namely nd a random di eomorphism with the prescribed formal
random power series This is assured by the following random version of a
result of Borel
Lemma Given h
hx
xhnHndn
Then
there is a random coordinate transformation h whose formal random Taylor
series expansion has the coe cients Tnh hn
The proof is just an wise version of the deterministic proof given by
Vanderbauwhede
p
and is thus omitted
With the above procedure in nitely at terms of at cannot be
detected as they do not appear in formal power series A random version of
Sternberg s linearization theorem would assert that in the above situation
x and A x are indeed smoothly equivalent by a random coordinate
transformation h x However such a theorem is still lacking
1

NORMAL FORMS FOR RANDOM DIFFEOMORPHISMS
Resonant Case
We now treat the case where we have resonance of order n i e
S M A n In our adapted random basis
MAnBMAn
MAn
MAnCA
whereMAn MAn andMAnaretheunstable stableandneutral
block respectively In this basis the terms of order n of the formal Taylor
series ful ll
n
n
nAMAn MAn MAnAUAhn
hn
hnA gn
gn
gnA
These are three decoupled equations of which we can solve the rst two by
the techniques of the preceding subsection Put n
and n
then
there is a unique hn and hn provided gn and gn are tempered which we
assume
There remains to be considered the neutral part
nMAnUhngnSMAnfg
The corresponding cohomological equation is
Uhn MAnhn gn
n
and the cohomological operator Ln U M A n maps tempered random
variables into tempered random variables hence
LnTn Tn Tn ff Hnd ftemperedg
The only problem hence is to describe Ln Tn This is an extremely
complicated question For the case A I we obtain M A I hence the
classical cohomological equation U I h g which is important in various
contexts and has a rich literature see Katok and Hasselblatt Sects
and
Arnold and Xu have worked out a Hilbert space set up for equations
and
However we prefer to present the cleaner continuous
time version of it in Sect
In practice one need not worry so much about solving equation
We can always satisfy it by choosing hn hence n gn
1

CHAPTER NORMAL FORMS
Normal Forms for Random Di erential
Equations
Letfandgbe vector eldsinRd withf g
They are called
smoothly equivalent f g if the local ows and generated by xt
f xt and xt g xt are smoothly equivalent i e if there is a coordinate
transformation h for which locally
thht
Di erentiating this gives the equivalent in nitesimal form
fhDhg
Deterministic normal form theory for vector elds now seeks an h for which
g is simplest possible
Normal forms for vector elds which are periodic in time have been
worked out using Floquet theory by V I Arnold and Iooss
For
the quasiperiodic case see Chow Lu and Shen
and the references
therein
Analogously if our RDS is generated by an RDE xt f t xt
there are obvious practical reasons for transforming the RDE and not the
resulting cocycle into normal form This problem will be addressed here
based on Arnold and Xu For the SDE case see Sect
The Random Cohomological Equation
We refer to Sect for basic facts about deterministic normal form theory
which we will freely use We add that in the vector eld case the formal
Taylor series have to be inserted into equation
and the deterministic
cohomological operator is
adnA Hnd Hnd hn adnAhnx Ahnx DhnxAx
This operator depends linearly on the entries of A Introducing a basis
FinRdandusingthebasis xjjninHndR givesabasis xF
inHndRd HndR Rd R Rd RDandtheD Dmatrix
representation of ad n A is
adnAIATAnI
where I and I are unit matrices in R and Rd respectively and T A n is
the
matrix describing the linear mapping on Hn d R R given

NORMAL FORMS FOR RDE
by
hX
jjnhxX
jjn
dX
jkhx
xjajkxk TAnh
Example For d n
and D
IA AAAATA
a
a
a
a
a
a
a
aA
hence
TA IBB
B
B
B
B
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
CC
C
C
C
CA
Let now
xtftxt
be anRDEinRdfor whichf
and f
f
Lloc R C By Theorem
generates a local C RDS over
more precisely t D t
R t is a local C di eomorphism
with xed point
We want to classify RDE by smooth equivalence Let
xtgtxt
be a second RDE generating a local RDS We say f and g are smoothly
equivalent f g if the corresponding local RDS and are smoothly
equivalent by a random coordinate transformation h
thht
t
It is crucial to realize that at time t we have to apply the coordinate trans
formation h t
Assuming for the moment that h d dt h t x exists we obtain by
di erentiating
with respect to t the equivalent in nitesimal form
fthtxhtxDhtxgtx
1

CHAPTER NORMAL FORMS
or
gtxDhtxfthtxhtx
As in the mapping case normal form theory for RDE now consists of choos
ing h so as to make g as simple as possible the ultimate aim again being
linearization g x A x
The linear cocycle t
Dt
on T Rd Rd is generated by
the linearized RDE
vtAtvtA
Df
We assume throughout that A L P so that the MET holds for
Now the rst obstacle appears on the linear level As a preparatory
step we try to make
as simple as possible in particular to choose
a random basis adapted to the splitting of which block diagonalizes not
only but also A so that on each of the Oseledets spaces we would have
an RDE which is decoupled from all others
This might however not always be possible But
iifA
A is deterministic A as well as exp tA are in Jordan
canonical form in the same basis
ii if A t is periodic with respect to t then thanks to Floquet the
ory there is a random basis F in which the generator of the cocycle is
constant and in Jordan canonical form see Example
and Iooss
Lemma
iii if has simple spectrum then over a possibly extended DS we can
nd a basis of random eigenvectors which diagonalizes see Corollary
We assume from now on that the best possible choice F for sim
plifying the linear RDE
has been made The nonlinear random
transformation y h x by which we want to simplify
can hence
be near identity i e a random coordinate transformation in the sense of
De nition
The functions f g
and h have formal random Taylor
series expansions at given by
fxX
nfnxAxX
nfn x
gxX
ngnxAxX
ngnx
1

NORMAL FORMS FOR RDE
and
hxX
nhnxxX
nhn x
wherefn gnhn Hnd
Inserting these expansions into equation
and equating coe cients
yields the calculations are as in the deterministic case the following result
Lemma The term of order n
of the formal power series ex
pansion of g is given by
gntxadnAthntxd
dthntxkntx
where
knfn pnfkgkhk k n
and
pnTnfSnfASnhidDSnhidSngAg
is a polynomial of the lower order terms fk gk hk k n
off
g and h respectively Here Tn f denotes the term of order n of f Sn f
is the njet off and adnA is the generator ofa linear cocycle in Hnd
de ned by
and
ForexamplekfkfDhgkffgDhgDhg
etc
De nition Random Cohomological Operator RDE Case
The linear operator
Lnd
dt adnA t
acting on Hn d valued stationary stochastic processes of the form hn t
which are absolutely continuous with respect to time t via
ht
d
dth t
adnAthnt
is called the random cohomological operator in Hn d
With this de nition the system of equations
reads
Lnhnkngnn
called the system of random cohomological equations

CHAPTER NORMAL FORMS
We have to solve
successively for known kn and hn chosen to make
gn simplest possible The most desirable choice of gn is gn
resulting
in the task to solve the random cohomological equation
Lnhn d
dtadnAthntknt
in Hn d equivalently to search for stationary solutions of the hierarchical
system of a ne RDE
hnadnAthnknt
n
Incasegn forallnkn fn Ppqpq nfp hq
In order to apply our basic Theorem ii we need to know the
spectrum S ad n A of the cocycle generated by ad nA in terms of the
spectrum S A of A
Proposition Let A denote the linear cocycle generated by the
RDE vt A t vt with Lyapunov spectrum S A
i the Lyapunov
exponents listed with their multiplicities Then
iThelinearcocycle TAninR generatedby TAnis
TAnNAn
whereNAnisdenedin
Moreover the MET holds for T A n
and
STAnf
jj ng
dX
i
ii
ii The linear cocycle ad nA in RD R Rd generated by ad nA
IATAnIis
adnANAnA
TAn
A
Moreover the MET holds for ad nA and
SadnA fi
i SAjjng
Proof i We have
d
dthAtx
DhAtxd
dtAtx
DhAtxAt
Atx
DhyAtyjy Atx
1

NORMAL FORMS FOR RDE
The matrix version of this states that u N A n solves ut
TA t nutandthusbyuniquenessN An
TAn
The statements about the MET and the spectrum follow from Proposi
tion iii
ii The generator of the cocycle T A n
A is ad nA by Theorem
iii RDE case The statements about the validity of the MET and
the spectrum follow from Proposition iii
Let n We call the matrix A nonresonant of order n if S ad nA
resonant of order n otherwise
Nonresonant Case
We can immediately apply Theorem ii and obtain
Proposition Let A be nonresonant of some order n
and as
sume that kn Hn d is tempered Then the cohomological equation
hnt
adnAthnt knt
has a unique solution hn t which is absolutely continuous with respect to
t explicitly given in Theorem ii In addition hn is also tempered
Corollary Suppose N
and A is nonresonant of order n for
all n N Assume further that in the Taylor expansion f x
AxPN
n fn x O jxjN of order N all fn are tempered Then
there is a unique near identity N jet HN x x PN
nhn xhence
is a local analytic di eomorphism with tempered coe cients hn such that
f is transformed by HN into g given by equation
with a linear N jet
of the Taylor expansion of order N g x A x O jxjN
The proof just repeats the arguments used for Corollary
Theorem Formal Linearization of RDE Suppose A is non
resonant of any order n
and all fn in the formal Taylor series expansion
of f are tempered Then there is a random coordinate transformation with a
uniquely de ned formal random Taylor series h x x Pn hn x
where all hn are tempered and h formally linearizes f i e the formal power
series expansion of g in equation
isA x
The proof follows from the last corollary and Borel s lemma
The full Sternberg linearization which would also transform away in
nitely at terms of f at is not yet available
1

CHAPTER NORMAL FORMS
Resonant Case
In the case of resonance we propose a functional analytic setting for the
random cohomological operator which also allows for duality theory similar
to the deterministic case
For any deterministic or random basis F in Rd and thus
xF jjninHnd weendowthe berHndover withthescalar
product
hfgiFX
jjnkFfkFg
S
wheref Pjjnxfg Pjjnxg xyS Pd
i xiyi is
the standard scalar product in Rd and kF x are the coordinates of x Rd
inthebasisF We can as usualidentifyf Hndh iF
with
kF f jjn RDP S wherethisidenticationisanisom
etry The above random scalar product makes Hn d a non trivial Eu
clidean bundle
We now endow the space Hn d of Hn d valued random variables with
the topology of convergence in probability with distance f g given by
fgjjjfgjjjEjf gjF
jf gjF
This makes Hn d a Frechet space The subset
LndffHndkfk hhffiiEhf fiF g
is a real Hilbert space with scalar product hhf gii Ehf g iF which
is densely and continuously embedded in Hn d with jjjfjjj kfk
Proposition Assume A L P Then
i The random cohomological operator
Lnd
dt adnA t
ht Lnht d
dth t
adnAtht
is a densely de ned linear operator in Ln d
ii The adjointLn ofLninLndis
Lnd
dt adnA t
1

NORMAL FORMS FOR RDE
iehhLnfgii hhfLngiifg DLn DLn
iii We have an orthogonal decomposition
Lnd RLn NLn
Proof i A L P implies ad nA L P The multiplication of h
by ad nA is thus in Ln d for any essentially bounded h The bounded
polynomials are obviously dense in Ln d It thus su ces to prove that
there is a dense set of bounded elements h Ln d for which t h t is
C with all derivatives bounded P a s
Take an arbitrary bounded RD valued random variable h Choose for
eachk NaC bumpfunction k R R withsupp k k k
andRkk ktdt andput
hk Zk
kkthtdt
TheintegralexistsPas sinceh L P khkk khk andhk hin
L For
Ekhk hk EZk
kktht h dt
Zk
kktEkht hkdt
for k
The latter holds since due to the measurablity of t
t the group
of isometries Uth h t is strongly continuous with respect to t R in
L i e continuous in mean square see Appendix A
We now prove that t hk t is C with bounded derivatives of any
order Di erentiating hk t m times with respect to t for xed k and
gives using standard theorems
dm
dtmhkt gmt gmZk
k mdmk
dtmshsds
and gm is bounded
ii We check the formula for Ln for the dense set of test functions f g
of the type constructed in step i
The word esentially will henceforth be supressed in the proof
1

CHAPTER NORMAL FORMS
The scalar product h iF
on Hn d is tailor made to yield
hadnAfgiF
hf adnA g iF see Vander
bauwhede
Taking E on both sides gives
hhad nAf gii hhf ad nA gii
Further by de nition
hhd
dtf gii X
jjnEd
dtftjt g S
X
jjnlim
ttEftfgS
E lim
t lim
t E by L continuity see step i
X
jjnlim
ttEfgt
gS
X
jjnEf d
dtgtjt S
hhf d
dt gii
The fact that D Ln D Ln is proved by standard methods
iii This is a well known fact from Hilbert space theory
Corollary If A L P is nonresonant of order n then N Ln
fg
Proof Clearly f N Ln if and only if f is a stationary solution of
ft
adnA t ft Suppose that S adnA Thenf is the
only stationary solution of this equation see Corollary
By Theorem
iSadnA
S ad nA so A is nonresonant of order n if and
onlyif A is
The converse statement N Ln f g implies S ad nA is false in gen
eral see the examples in Subsect
Also remember that nonresonance
of some order implies hyperbolicity of A
The orthogonal decomposition
Lnd RLn NLn
is marred by the fact that R Ln is in general not closed Furthermore the
term kn Hn d in the equation gn Lnhn kn to be solved is in general

NORMAL FORMS FOR RDE
not in Ln d This makes the concept of normal form de ned in the next
theorem necessary To simplify the situation we restrict ourselves to the
caseSA fgoftotalresonance henceSadnA fgfor alln
Theorem Approximate Normal Form Resonant Case
Consider the RDE
xtftxt
whereinsomebasisA L P andSA fgThenforeach
there
is a random coordinate transformation x h x with formal Taylor
coe cients Tnh Ln d for all n
such that the formal Taylor series of
the right hand side of
xtgtxtDhtxtfthtxthtxt
isin normalformieforeachn thereisagn gR
ngN
n Lnd
with gR
n RLn kgR
nk gN
n N Ln and jjjgn Tngjjj
Proof By Lemma
Tng LnTnh kn where kn was determined
by preceding steps and Tnh can be chosen to make Tng simplest possible
First move the equation from Hn d into Ln d by replacing kn with kn
kR
nkN
n R Ln N Ln Ln d and jjjkn knjjj Approximate kR
n
by kR
n RLn withkkR
nkR
n k and solve LnTnh kR
n in Ln d Finally
put Tng kR
nkR
nkN
n
Having determined all Tnh Ln d we again invoke Borel s lemma
to nd a di eomorphism h
Random Averaging
The idea is to de ne an averaging operator projection on Ln d which
applied to the right hand side of a cohomological equation Lnh f singles
out the part of f that is not in R Ln The ideal situation would be an
orthogonal projection onto N Ln with null space R Ln
The following two possibilities o er themselves as generalizations of the
deterministic case see Vanderbauwhede
fxlim
T TZT
At ft
Atxdt
fxlim
T TZT
AtftAtxdt
The interpretation of
and
is as follows We average in the
ber over all values that we obtain by

CHAPTER NORMAL FORMS
i moving x in the ber with the skew product ow to
t
A t x and t At x respectivelybothinthe t ber
ii and then moving back to the ber by applying the inverse of the
cocycle A and A respectively to the result of i
In the deterministic case averaging works only under special conditions
on A e g semi simplicity and spectrum on the imaginary axis Similarly
in the random case see the examples in Subsect
for illustration
Observe that even in the case where A is deterministic and A A for
which
and
coincide
fxlim
T TZT
e tAft etAxdt
is a random transformation
Examples
Example One Dimensional Case For d we have
D henceHn R R basisxne foralln
and
adnA adnA
nA
Lnd
dtnAtLnd
dtnAt
and
adnA t
exp nZtAsds
AssumingA L P theMETyieldsSA fg EAandSadnA
f n g We have resonance of all orders if and only if
We rst consider the nonresonant case
By Proposition
and
Theorem
we can formally linearize the RDE by a random coordinate
transformation h with tempered coe cients hn provided the coe cients
fn of f are tempered To be more speci c the cohomological equation of
order n
hn
nAthnknt
has a unique tempered stationary solution given by
hn
RexpnRtAsdskntdt
RexpnRtAsdskntdt
1

NORMAL FORMS FOR RDE
We next consider the resonant case
andassumeA L P We
determine
N Ln fstationary solutions of ft n A t ftg
which amounts to nding initial random variables f without loss of
generality by ergodicity for which
ft fexpnZtAsds
orputtingg logfforwhichgt n A t or
gtg
nZtAsds
We have hence arrived at the following basic problem of algebraic ergodic
theory When is the helix H t R t A s ds a coboundary see Remark
It is clear that two solutions of
di er only by a constant
hence either dim N Ln for all n
or foralln
Consequently
NLn fg ifHisnotacoboundary
Rfn
if H is a coboundary
Here f is the solution of
for n unique up to a constant factor
For necessary and su cient conditions on A for the solvability of
see Orey
and Arnold and Wihstutz
A typical case for which
has a solution is when t A t is the derivative of an abso
lutely continuous stationary process A t d dt B t in which case
f eB henceNLn Ren B Wecanusetheapproachdevel
oped in Subsect
if we assume that B L
We now try random averaging for the case A B with B L and
put k f n One easily checks that the Ansatz
gives a projec
tion
with range R
N Ln R k but with the wrong null space
N
RLn k
However
yields
f lim
TTZTftexpnBtBdtkEfk
withR NLn Rk andN k RLn Thus although
isingeneralnotorthogonalf f RLn f NLn andwehave
the in general non orthogonal splitting Ln L P R Ln N Ln
1

CHAPTER NORMAL FORMS
Further is the adjoint of and the projection is orthogonal
if
andonlyifB constifandonlyifA
For a nonrandom A we have A In the nonresonant case A
we can linearize the RDE so the normal form of xt f t xt Axt
Pnfntxn
tisxt Axt IntheresonantcaseA NLn R
and the normal form is deterministic xt g xt Pn E kn xn
t
Note that this example also covers the quite frequently observed case
where A has simple spectrum We can then diagonalize A and thus all ma
trices appearing in our theory by choosing a random basis of eigenvectors
The problem hence splits into D scalar problems of the type just consid
ered
Example A Two Dimensional Resonant Case Assume
that d hence
n
andD n
and
A
A
is deterministic We identify R with C via x
xx
xix
Tx
z sothatTAx iTx WeidentifyHn withthespaceof
C n valued random variables via
fnz
nX
kfk
nznkzk
AsSA fgwehave ofSadnA fgwithmultiplicityD for all
n hence we are in the resonant case
One easily checks that Lnf translates into n
scalar complex
RDE
fk
n
nkifk
nk
n
Hence a stationary solution of such an equation satis es
fk
nt
einktfk
n
k
n
iefk
n is an eigenfunction of the group Utf f t of unitary oper
ators in L P C corresponding to the eigenvalue n k Recall that
the set of eigenvalues of an ergodic Ut is a subgroup of the additive group
R countable if L is separable and that every eigenvalue has multiplicity
the eigenfunctions satisfy jfj const and are mutually orthogonal see
Cornfeld Fomin and Sinai
p
Thus
NLn fnX
kfk
nznkzkfk
n eigenfunction of Ut
corresponding to eigenvalue n k k
ng

NORMAL FORM AND CENTER MANIFOLD
We have nontrivial solutions certainly for n k namely C but
possibly also for other cases as e g t rotation by the angle t in
S shows exercise Such terms do not exist in the deterministic
case This clearly illustrates that random normal form theory depends on
the interplay between the dynamical system and the linear part A of the
RDE
Averaging according to
and
is identical in this case and
yields for the components of fn L P C n
lim
T TZTfk
nteinktdtfk
n
Efk
n
mk
fk
n
mk
ThislimitexistsinL andPas andfk
n has mean zero and satis es
see Doob
p
In this case is indeed the orthogonal
projection onto N Ln with null space R Ln
Simultaneous Normal Form and Center
Manifold Reduction for Random Di er
ential Equations
The desire to simplify engineering and physics DS which are perturbed by
noise has prompted numerous publications see e g Coullet Elphick and
Tirapegui
Nicolis and Nicolis Schoner and Haken
Sri
Namachchivaya and Lin
and the references therein All authors have
worked exclusively in a center stable situation and with a smallness param
eter multiplying the noise terms thus providing a stochastic normal form
as a small perturbation of the deterministic one See Sri Namachchivaya
for a survey which also comments on the older literature and Subsect
We will now follow Arnold and Xu
and connect our general approach
developed in Sects
and which is independent of any smallness as
sumptions with the existing physics and engineering literature by present
ing a random analogue of a very successful procedure proposed by Elphick
et al
for simultaneously obtaining the normal form eliminating the
stable variables from the center equations and determining the center mani
fold Subsect
This connects also with the invariant manifold theory
for RDS see Chap
1

CHAPTER NORMAL FORMS
In Subsect
we will deal with the case of a smallness parameter
where the undisturbed system is allowed to be random In Subsect
we
treat the particular case of a deterministic di erential equation perturbed
by small noise and calculate the normal form of the Du ng van der Pol
oscillator with small noise both for the pitchfork and the Hopf regime
The Reduction Procedure
As in Sect we consider the RDE in Rd
xtftxt
for which f
andf LlocRC ifweonlygouptoa xed
order N
thenf LlocRCN
su ces Then
generates a
local C or CN RDS with xed point x
and the linearized RDE
vtAtvtA
Df
generates the linear cocycle We assume throughout that A L P so
that the MET holds for
We follow the deterministic philosophy and assume that we are in a
center stable situation i e the Lyapunov spectrum of is
r
The general case requires obvious modi cations of our procedure
We de ne the center and stable invariant spaces by Ec
E
EsE
Er
so that we have the invariant splitting
RdEcEs
t Ecs
Ecs t
anddimEcs dcsdc ds d
We next assume that a basis F
Fc Fs
exists which block
diagonalizes the linear part A t of f t x into a center block and a
stable block for conditions see Subsect
and the basis transfer matrix
is a Lyapunov cohomology In this basis
At
Act
Ast
The coordinates in this basis are x
xc
xs and by the coordinate mapping
Ec Es
and Rd Ec
Es
are identi ed with Rdc Rds and
Rd Rdc Rds with the standard bases respectively
1

NORMAL FORM AND CENTER MANIFOLD
The homogeneous polynomials in Ec Es and Rd are written in
the bases xc Fc
xs Fs and x F where is a multi index by which
we identify those polynomials with the corresponding ones in Rdc Rdc
f g Rds f g Rds and Rd Rdc Rds respectively
We now formulate the elimination procedure of order N for which
we need the following Taylor expansions of order N suppressing the argu
ment
fx
Ax
NX
nfnx OjxjN
Acxc
As xs
NX
pqpq
fc
pqxcxs
fs
pq xc xs OjxjN
where fn x is a random homogeneous polynomial of degree n in x with
values in Rd and the fc s
pq xc xs are random homogeneous polynomials
of degree n p q which have degree p f Ng in xc degree
q f NginxsandwithvaluesinRdcs
Similarly for the random near identity polynomial h x x H x of
degree N we have
hx xHx
xc
xs
NX
pqpq
Hc
pqxcxs
Hs
pqxcxs
We replace x in
byhx x Hx andobtainthenewRDE
xtgtxt
where
gtxDhtxfthtxd
dtht x
If f is CN then so is g with corresponding Taylor expansion However
we try to choose h such that g has Taylor expansion
gx Acxc
As xs
PN
ngc
n xc O jxcj jxsjN
PNpqpqqgs
pqxcxs Ojxcj jxsjN
Theorem Normal Form and Center Manifold for RDE
Let N and assume that the coe cients of the Taylor expansion of f
of order N are tempered Then there are random polynomials of degree N
with tempered coe cients namely

CHAPTER NORMAL FORMS
H x withvaluesinRdandH xc xs Ojxcj jxsj
gc xc with values in Ec
Rdc fgandgc xc Ojxcj
gs xcxs withvaluesinEs fg Rds andgs xcxs
O jxsj jxcj jxsj
such that
i The near identity transformation
xhxxHx
xc
xs
Hc xcxs
Hs xcxs
transforms xt f t xt into
xcActxcgct xcOjxcjjxsjN
xsAstxsgst xcxsOjxcjjxsjN
where gc can be made as simple as possible
ii The manifold in Rd given by
McfxcHxc
xc Rdcg
is locally invariant for the truncated normal form equations
xcActxcgctxc
xsAstxsgstxcxs
has the property T Mc
Rdc in particular dim Mc
dc and
Mc
approximates a local center manifold at x
of the original
RDS up to terms of order jxcjN and is locally attracting
Proof i As in Sect and suppressing we replace x in
by
hx x Hx andobtain
fx Hx
x DxcHxc DxsHxs d
dtH xc xs
Now we replace x by g x plug our Taylor expansion for f h and g into
equation
equate coe cients and separate the center and the stable
components For p
and
we recover Acxc Acxc and
As xs As xs For p with p N we obtain for the center component
d
dtHc
p xc AcHc
p xc DxcHc
p xcAcxc gc
pxc Rc
pxc
1

NORMAL FORM AND CENTER MANIFOLD
and for the stable component
d
dtHs
p xc AsHs
p xc DxcHs
p xcAcxc Rs
pxc
For pq with p q Nandq wegetforthecentercomponent
d
dtHc
pq xc xs AcHc
pq xc xs DxcHc
pq xc xs Acxc
DxsHc
pqxcxsAsxs Rc
pqxcxs
and for the stable component
d
dtHs
pq xc xs AsHs
pq xc xs DxcHs
pq xc xs Acxc
DxsHs
pqxcxsAsxs gs
pqxcxs Rs
pqxcxs
Here the Rc s
pq dependonlyonfpq andonHp q gc
pgs
pqforp q
p q The strategy is hence to solve equations
to
step
by step starting with p q and then increasing p q by at each step
We now look at the four equations
to
separately
Equation
This is our well known cohomological equation for
Ac
d
dt adpAc t Hc
pgc
pRc
p
Since we are in the resonant case we could use our theory of normal forms
from Subsect
and choose H c
p such that gc
pNd
dt ad pAc But
we are also free to choose
Hc
p
hence gc
pRc
p
pN
Equation
This is also a cohomological equation corresponding
to the linear cocycle s
p generated by I As T Ac p I having Lyapunov
spectrum f rg i with multiplicity di p dc
p It is thus stable and
equation
has the unique stationary solution
Hs
pZs
pt Rs
ptdt
pN
Equation
This is again a cohomological equation corre
sponding to the linear cocycle c
pqgeneratedbyI I Ac TAcp
I I I TAs q I inthepolynomialspaceHpdcR HqdsR Rdc

CHAPTER NORMAL FORMS
and having Lyapunov spectrum f Pri i i j j qg So all exponents
are positive and the unique stationary solution of
is thus
Hc
pqZc
pqt Rc
pqtdt
pqNq
Equation
Since we do not care about an optimal choice of
gs
pq we simply choose
Hs
pq
hence gs
pq Rs
pq
pqNq
However we could also solve the corresponding cohomological equation to
obtain gs
pq in the non resonant case and an element in the corresponding
kernel in the resonant case hence a nontrivial H s
pq We will do this in
Example
ii We have to show that if t is the local C RDS generated by
the truncated equations
and
then locally
tMc
Mct
More precisely assuming for the moment that we made the trivial choice
Hc
p in equation
we have to show that
t
xc
Hs xc
ct xc
Hstctxc
where c is the local RDS generated by the decoupled center equation
While this is clear for the center component di erentiating the
stable component yields
AsHsxc gsxcHsxc
dHs
dt xc DxcHs Acxc gc xc
Inserting the expansions for H s gs and gc and equating coe cients gives
the cohomological equations
for p N which are satis ed by
our construction
If H c is nontrivial we also recover the cohomological equations
The statement about the tangent space follows from H xc
O jxcj For the claim that Mc approximates a local center manifold of
we use the criterion of Theorem
of Boxler Theorem It
is locally attracting due to the negative spectrum of As
Example Two Dimensional Case Assume d
Hence we can diagonalize the linear part A
A
Ac As
1

NORMAL FORM AND CENTER MANIFOLD
with E Ac
E As All equations are scalar
For
we can make the trivial choice H c
p
or proceed as in
Example
For
we always have a unique solution given by
Hs
pZexpZtpAc u As uduRs
ptdt
For
we have the unique solution
Hc
pqZexpZtp Ac u qAs uduRc
pqtdt
For
we made the trivial choice H s
pq
which made the terms
gs
pq Rs
pq survive However we could do better as follows The corre
sponding cocycle has spectrum
q
qq
We hence keep the choice H s
p or solve the resonant cohomological
equation but choose for q
Hs
pqZexpZtq As u pAc uduRs
pqtdt
which implies gs
pqforqN
The nal truncated equations are thus
xc Acxc
NX
ngc
nxn
c xs Asxs
NX
ngs
nxn
cxs
and the approximate center manifold for the choice H c
n
is the graph
xc PN
nHs
nxn
c
Parametrized RDE
For applications in stochastic bifurcation theory it is of considerable impor
tance to treat the case
xtftxt
1

CHAPTER NORMAL FORMS
of an RDE in Rd which smoothly depends on a parameter Rm We
assume that is in the neighborhood of a critical value c say We
will obtain the approximate normal form and center manifold of
augmented by the trivial equation
as a polynomial in the variables
x RdRm
For simplicity we assume that
f
for all Rm
and that at
we are in a center stable situation i e the cocycle t
generatedby vt A t vt A
Dxf
has Lyapunov spectrum
r These simplifying as
sumptions are basically without loss of generality The general case requires
straightforward modi cations of our procedure
Assume nally that the decoupling of A as described in the last subsec
tion has been successfully carried out We then have the following random
version of a result of Elphick et al
Chap III
Theorem Normal Form and Center Manifold for RDE
with Parameters Let N
and M
Assume that the coe
cients of the Taylor expansion of f in the variable x
oforderNinx
and order M in are tempered Then there are random polynomials of
degree N in x and M in with tempered coe cients namely
H x withvaluesinRd andH xc xs Ojjjxcj jxsj
jxcj jxsj
gc xc
with values in Ec
Rdc fgandgc xc
O j jjxcj jxcj
gs xc xs withvaluesinEs fg Rds andgs xc xs
Ojxsjjj jxcj jxsj
such that
i The near identity transformation
xhx
xHx
xc
xs
Hc xcxs
Hs xcxs
transforms xt f t xt into
xcActxcgct xc OjxcjjxsjNjjM
xsAstxsgst xcxs OjxcjjxsjNjjM
1

NORMAL FORM AND CENTER MANIFOLD
where gc can be made as simple as possible
ii The manifold in Rd Rm given by
Mc fxc
Hxc
xc Rdc Rmg
is locally invariant for the truncated normal form equations
xcActxcgctxc
xsAstxsgstxcxs
has the property T Mc
Rdc Rm in particular dim Mc
dc m andMc
approximates a local center manifold at x
of the original RDS augmented by
up to terms of order
jxcjN j jM and is locally attracting
Proof i Let the corresponding Taylor expansions of f h and gc s with
suppressed be
fx AxXnN
r Mfnrx OjxjN jjM
Hxcxs X
pqN rM
pqr
Hpqr xc xs OjxjN jjM
gcxc X
nN rM
nr
gc
nr xc OjxcjN jjM
gsxcxs X
pqN rM
q
pr
gs
pqrxcxs OjxjN jjM
Here fnr is a random homogeneous polynomial of degree n r which is
homogeneous of degree n in x and of degree r in with values in Rd Hpqr
is a random homogeneous polynomial of degree p q r which is of degree
pqrinxcxs
respectively with values in Rd Rdc Rds gc
nrisof
degree n r but of degree n and r in xc and respectively with values
inRdc fgand nallygs
pqrisofdegreep q rbutofdegreepq rin
xc xs
respectively with values in f g Rds
As in the proof of Theorem
we insert our expansions into equa
tion
equate coe cients and separate center and stable components
We obtain structurally the same equations the di erence being that the

CHAPTER NORMAL FORMS
appearance of gives rise to a factor r n
r for the dimensions of the
corresponding polynomial spaces The result is as follows
Forp r with p Nand r Mweobtain
d
dtHc
prxc AcHc
prxc
DxcHc
prxc Acxc gc
prxc Rc
prxc
and
d
dtHs
prxc AsHs
prxc
DxcHs
prxc Acxc Rs
prxc
Forpqrwith pqNq
rMpqr
we
get
d
dtHc
pqr xc xs AcHc
pqr xc xs
DxcHc
pqr xc xs Acxc DxsHc
pqr xc xs Asxs
Rc
pqr xc xs
and
d
dtHs
pqr xc xs AsHs
pqr xc xs
DxcHs
pqr xc xs Acxc DxsHs
pqr xc xs Asxs
gs
pqrxcxs Rs
pqr xc xs
Here the Rc s
pqrdependonlyonfpqrandonHp qr Hpqr gc
pr gs
pqr
forppqqrrpqrpqr
The step by step strategy to solve these equations is as follows Start
with r Then equations
to
reduce to equations
to
which have been solved above This determines Hpq gc
p and
gs
pq Notethatfor pq
and
we recover the linear
parts In the next step put r
and solve all equations for p q N
andsoforthtor M
For r
the above equations have the same structure as those for
r
so hpqr gc
pr and gs
pqr will have the same structure as hpq gc
p and
gs
pq The presence of amounts to having r n
r
equations with the
same cohomological operator instead of one for r
Note nally that due to our assumption
Rr forallr
which results in the fact that we do not have terms of the type
rin
H and gs and of type r in gc Also as gs only appears in equation
for which q there are no terms of the form gs
pr
ii See the proof of the corresponding fact of Theorem

NORMAL FORM AND CENTER MANIFOLD
Remark New Terms in Parameter Case For use in Subsect
we look at those equations among
to
that do not
appear for
For r
the lowest possible triples are
and
These will lead to corrections of order in of the linear parts
of the transformation h so it will be near identity only for
as well
as of the linear parts of the resulting equations More precisely the linear
part of h has the form
xcPimiHcs
ixs
xsPimiHsc
ixc
while the RDE start with
xcActXim
iRci t Axc
xsAstXim
iRs
i tAxs
and the center manifold starts with xc P i m iH sc
ixc
Small Noise A Case Study
By small noise we mean that the parametrized random vector eld
f x intheRDE
is deterministic at the critical value
This is of course a particular case and we will not reiterate our procedure
Let us stress that in the small noise case the linearized equation
is also deterministic Consequently all cohomological operators appearing
in our procedure are of the form d
dt B where B is a deterministic linear
operator But note that the corresponding cohomological equations are
of the form d
dt B h t f t hence have stationary processes as
solutions
The prototypical Du ng van der Pol oscillator
yyyyyy
under the in uence of parametric and additive noise has been the subject of
numerous investigations see Arnold Sri Namachchivaya and Schenk Hoppe
Schenk Hoppe
and the references therein For
xed and the bifurcation parameter the system
exhibits a Hopf
bifurcation for
For
xed and the bifurcation parameter it
undergoes a pitchfork bifurcation at
1

CHAPTER NORMAL FORMS
To reduce complexity we will treat a particular case of parametric noise
Let the parameter be replaced by
t where t
t isa
zero mean stationary stochastic process and is an intensity parameter
Withx y
y the perturbed version of
is
xt
xt
t
xt
xxx
The linearization of
atx is
vt
vt
t
vt
in particular for
vt
vt
The Pitchfork Scenario Under Small Random Perturbations
Put for simplicity
Then the eigenvalues of
are
p
We treat
as a small two dimensional parameter and
will determine the simultaneous stochastic normal form and center manifold
reduction of
for small x and
As a rst step we diagonalize the linear part of
at
yielding writing again x for the new coordinates
xt
xfxfxfxfx
fxfxxfxx
ft xt
where putting b p
fbf
tbfpbfptb
f
bf pbf
b
fpqkl denoting the coe cient of xpxq k l Note that here d
dc
ds
xcxxsxAcAs
We seek the transformation
depending on time and chance
xxHtx
xcHctxcxs
xsHstxcxs
1

NORMAL FORM AND CENTER MANIFOLD
which tranforms
into
xcgctxc
OjxcjjxsjN jjjjM
xs
xsgstxcxs
OjxcjjxsjN jjjjM
The result for N and M is The transformation H Hc
Hs has the
form
Hc
pHc
Hc
Hcpxs
HcHcpHc
pxs
Hc Hc
Hc
xc xs
pHcpHc
pHc
xc xs
Hs
Hs
HspHs
pxc
pHs
Hs
pHspxcxs
pHsHspHs
pxc
Hs
Hs
Hs
xs
The truncated stochastic normal forms are
xc gc
gc
gc
xc
gc
gc
gc
xc
xs
xs gs
gs
gs
xs
gs
gs
gs
xc xs
In equation
the coe cients are as follows
gc
t
tp Hs
t
gc
t pHs
t
t
1

CHAPTER NORMAL FORMS
gc
t
t pHs
t
gc
t
t pHs
t pHc
t
pHs
t pHs
t
gc
tptHs
t
Hc
tHs
t
pHs
tptHc
t
pHs
t
Hs
t
where
HsZespsds
HcZesspds
HsZesps
Hs
sds
HsZesps
Hs
sds
Hs Zes
sHs
sds
The approximate stochastic center manifold is the graph
RRxc
xsmcxc
R
given by
mc xc
Hs
Hs
pHs
pxc
pHs
Hs
pHs
pxc
The computational e ort for these results is enormous and could only be ac
complished by using the computer algebra program MAPLE There are
cohomological equations to be solved to determine the coe cients These
cohomological equations ll pages and can be found in an appendix of

NORMAL FORM AND CENTER MANIFOLD
a reprint version of The very complicated explicit expressions of the
coe cients H s and H s as functions of lower order terms can also be
found there
The scalar center equation
can now be utilized for stochastic
bifurcation theory of the original two dimensional system
similarly
asinXu
See Subsect
The Hopf Scenario Under Small Random Perturbations
We now assume that
is xed and is the bifurcation parameter which
varies in the interval j j p so that the damped eigenfrequency
dp
is well de ned We follow
and obtain the normal form for small x
andThe eigenvalues of
are i d which are purely imaginary for
Wehaved dc
while the stable component is not present
All cohomological equations are thus resonant in the random sense i e all
Lyapunov exponents vanish Our recipe for their solution is as follows If
in a cohomological equation d
dt BH G RRisarandomprocess
we choose H and G R while if R is nonrandom we search for a
nonrandom H we solve BH G R for which G is as simple as
possible by means of deterministic normal form theory see Sect
The truncated stochastic normal form for N and M in polar
coordinates r
x
rcos x
r sin is after formidable computa
tional e orts which we omit as follows
rt
rt
rt
rt
d
rt
rt
d
dsint
rt
dsintrtcostt
t
d
drt
d
rt
d
rt
d
cos t
rt
dcos t rt
dsintt
For
we recover the deterministic truncated normal form from which
the Hopf bifurcation at
can be read o
1

CHAPTER NORMAL FORMS
Normal Forms for Stochastic Di eren
tial Equations
Normal form theory for SDE is confronted with the technical problem of
the appearance of terms which at time t are not measurable with respect to
the information available at that moment but rather anticipate it These
terms originate as stationary solutions of a ne SDE with a hyperbolic linear
part and a possibly anticipative additive part see Theorem
for the
simplest of such cases but such SDE as well as their solutions are ruled
out in classical stochastic analysis
This fact has been clearly seen but not rigorously handled by the pio
neers of stochastic normal form theory see Coullet Elphick and Tirapegui
Nicolis and Nicolis
Schoner and Haken
Sri Na
machchivaya and Lin
We follow Arnold and Imkeller
and present
a rigorous treatment
The Random Cohomological Equation
We consider the SDE
dxt
mX
jfjxt dWjt fj
in Rd where as usual dWt stands for dt and where fj C for j m
Then
uniquely generates a local C RDS Theorem
and
thedomainDt andrangeRt of t Dt
Rt are
neighborhoods of How many di erent such local RDS exist modulo a
smooth conjugacy
The linear cocycle t
t
Dt
onTRd Rdis
generated by the linearized SDE
dvt
mX
j Ajvt dWjt Aj Dfj
The MET always holds for
giving the Lyapunov spectrum
Sf
dg
We now conjugate the local RDS generated by
with another local
RDS by means of a near identity random coordinate transformation
see De nition
thht
t locally
1

NORMAL FORMS FOR SDE
where is generated by an SDE
dxt
mX
jgjtxtdWjt
and h is chosen such that the SDE
makes sense and its coe cients
gj are as simple as possible the ultimate aim being linearization i e
dxt
mX
j Ajxt dWjt
Although it will turn out that the transformation h t x h t x
to be applied at time t is in general not adapted to the ltration of W at t
we proceed formally and justify later Applying the Stratonovich lemma
to
gives
dtdhtDhttdt
where dht denotes the t di erential of h t x Inserting the di erentials
of tand tinto
yields
mX
jfjhtxtdWjtdht
mX
jDhtxtgjtxtdWjt
equivalently
dxt
mX
jgjtxtdWjt
Dhtxtdht
mX
jDhtxtfjhtxt dWjt
This is an equation for h and the gj where the choice of h is made such
that the gj are the simplest possible
We now make the simplifying assumption that we choose the canonical
basis in Rd In other words we leave
untouched and refrain from
making it as simple as possible by choosing an appropriate non adapted
random basis cf Corollary
for the diagonalization of a linear SDE
in case of a simple spectrum
We again make the formal Taylor series Ansatz
fjx AjxX
nfjnxj
m
1

CHAPTER NORMAL FORMS
gj x AjxX
ngjnxj
m
hxxX
nhn x
wherefjn Hnd whilegjn hn Hnd
Plugging this into equation
and equating coe cients yields an
identity for the linear part and for n
mX
jgjntdWjt
mX
j adnAjhn t dWjt dhn t
mX
jkjntdWjt
where ad nAj Hn d Hn d is the linear operator de ned by hn
adnAj hn x Ajhn x Dhn x Ajx in matrix form adnAj I
Aj TAjn IonHndRd HndR Rdseeequations
and
and
kjn fjn Pnfjkgjkhk k n
j
m
Here Pn is a deterministic polynomial of the lower order terms of fj gj and
h which is independent of j and given explicitly in Lemma
We now de ne the random cohomological operator SDE case by
dLn hn dhn
mX
j adnAj hn dWjt
acting on those Hn d valued stationary stochastic processes hn t for
which the expression in
makes sense With this de nition
turns into the system of random cohomological equations suppressing the
t argument
dLn hn
mX
j kjn gjn dWjt dKn dGn
which have to be solved successively for n
for dKn known from
previous steps and hn choosen to make dGn the simplest possible The
most desirable choice is again dGn
resulting in the task of solving the
cohomological equations
dLnhn dKn n
1

NORMAL FORMS FOR SDE
in Hn d or equivalently looking for stationary solutions of the a ne SDE
dhn
mX
j adnAjhn kjn t dWjt n
where kj n t depends on the solutions of
of lower order k
n We have kj fj deterministic but for n kj n is typically
random and not adapted to the ltration of W
Let n be the linear cocycle on Hn d generated by the SDE
dvt
mX
j adnAj vt dWjt n
The MET holds for n without additional assumptions and gives the
spectrum S n fi
iSjjng
This follows from Theorem iii SDE case and ii and from Propo
sition
We know from looking at Theorem
that we have a chance of nding
a unique stationary solution of
provided the linear SDE
is hyperbolic We call the linear cocycle
generated by
nonresonant of order n if this is the case i e if S n resonant of
order n otherwise
Nonresonant Case
Theorem Formal Linearization of SDE Given the SDE
and assume that the linear cocycle generated by
is nonreso
nant of any order n Then there is a random coordinate transformation
h whose formal Taylor series h x x Pn hn x is uniquely de
termined and has tempered coe cients hn such that h formally linearizes
equation
Remark It is quite remarkable that at the beginning and at the
end of the above procedure we have two bona de classical SDE while the
transformation converting the solutions of the rst into the solutions of the
second is anticipative
The proof of Theorem
is quite complicated and is divided into three
steps the rst two of which are of technical nature while the third one
on invariant measures of a ne SDE is of independent interest and also
nishes the proof of Theorem for the white noise case
1

CHAPTER NORMAL FORMS
Step Boundedness of Moments of Solutions of a Hierarchical
System of A ne SDE
Our main task here will consist in proving that all processes in our hierar
chical system of a ne SDE obtained by solving the cohomological equations
step by step for xed initial conditions satisfy the conditions of the following
lemma
Lemma Let u
utx t
x Rd be an Rk valued stochastic
process which for xed x is P a s continuous with respect to t is adapted
and satis es the following two conditions
Conditions C For any p and any compact set K Rd there
exist constants cp and cp K R such that
E sup
tjutjp
cp
E sup
tjutx utyjp
cpKjx yjp forallxy K
Then u is P a s jointly continuous with respect to t x and for p d and
any compact set K Rd there exist constants Cp R and q such that
E sup
xKsup
tjutxjp CpdiamKq
Proof The joint continuity of u follows from C by Kolmogorov s con
tinuity criterion applied to the C valued process x
uxwith
p d is a well known implication of the fundamental continuity
lemma of Garsia Rodemich and Rumsey see for example Barlow and Yor
formula b or Arnold and Imkeller
We now prove that conditions C are passed on from u to processes ob
tained from u by reasonable operations
Lemma Let u satisfy conditions C and let
vtxZt
usxdWs wtxZt
usxds
where W is a scalar Wiener process Then v and w satisfy conditions C
Proof This is an immediate consequence of the Burkholder and Holder
inequalities

NORMAL FORMS FOR SDE
The following considerations will be crucial for the algorithm by which we
solve our hierarchical system of a ne SDE Let d d N and for i
msupposethat uitx t
x Rd is a parametrized semimartingale
with decomposition
uitxZt
wisxds
mX
jZt
vjisxdWjs
with values in Rd Let moreover B B Bm be d d matrices and
denote by t t R the linear ow in Rd generated by the linear SDE
dyt
mX
j Bjyt dWjt
We now consider the SDE
dxt
mX
jBjxtujtxdWjtxxRd
Let tx t Rbethe owgeneratedby
the value of which at
x Rd willbewrittenas tx x
Lemma Letui uitx t
x Rd begivenby
As
sume that wi and vji hence ui
i m j m satisfy conditions
C Let
tx xZt
sxxds
mX
jZtjsx xdWjs
be the semimartingale decomposition of the solution ow of
Then
and j j m satisfy conditions C
Proof i We prove only the second of the conditions C for
Let us start by writing the Ito form of
We have
dyt
mX
j Bjyt ujtx dWjt
B
mX
j Bjyt
mX
j vjjtxAdt
1

CHAPTER NORMAL FORMS
Now xp andacompactsetK Rd Rd Thenforany x x
y y Rd Rd Burkholder s and Jensen s inequalities as well as the
hypothesis yield with suitable constants cK and with the abbreviation
ftEsup
stjsxx
syyjpp
f cKjxx yyjZftdt
To
we have to apply Gronwall s lemma to obtain with a suitable
constant cp K R
E sup
tjtxx
tyyjp
cpKjx x yyjp
This is the second of the conditions C for
ii Next observe that according to
for any t
and
xxRdRd
jtxxBjtxx ujtx
txx
B
mX
j BjAtx x
mX
j vjjtx
Hence by our hypotheses and
j and satisfy the conditions C
as well
Next we show that conditions C are inherited from u and its characteris
tics to a polynomial of u and its characteristics
Lemma Let
utx
mX
jZt
vjsxdWjsZt
wsxds t
xRd
take values in Rd and assume that u vj and w satisfy conditions C Let
p be a polynomial in the variable y Rd If
putx
mX
jZtqjsxdWjsZt
rsxds
then p u qj and r also satisfy conditions C
1

NORMAL FORMS FOR SDE
Proof First note that by Ito s formula qj and r are of the same structure
as p u Hence it su ces to prove the conditions for p u
Due to Holder s inequality it is evidently enough to consider the case
ofarealvalueduandapolynomialpoftheformpy ylforsomel N
Thenfory y R
pypyyylX
kykylk
Now observe that by conditions C for any x Rd
E sup
t jutxjp
cp
for a suitable constant cp R Then
and an application of
Holder s inequality obviously allow us to deduce conditions C for p u
from
and the condition C for u
Lemma Letu utx t
x Rd with values in Rd satisfy con
ditions C let t t R be the linear ow generated by
ford d
and let W be a scalar Wiener process Then the processes
vtxZt
s usxdWs wtxZt
s usxds
satisfy conditions C
Proof t satis es the SDE
dt
mX
j tBjdWjt
I
Hence for any p we have see Remark
E sup
tktkp
Then it is clear that Burkholder s inequality Holder s inequality and
imply that v and w satisfy conditions C
We nally come to the announced hierarchical system of a ne SDE Let
dn nNbeasequenceofintegers For j mandeachn Nlet

CHAPTER NORMAL FORMS
Aj n be a dn dn matrix and let pj n be a polynomial in the variables
x
xn
Rd
Rdn with pj bj Rd a xed vector
Then our hierarchical system of a ne SDE is as follows
dxt
mX
jAjxtpjdWjtxxRd
dxt
mX
jAjxtpjxtdWjtxxRd
dxn
t
mX
j Ajnxn
t pjnxt
xn
t
dW jt
xn xn Rdn
The algorithm for successively solving this system is as follows Let
n t t R be the linear cocycle in Rdn generated by the linear SDE
dyn
t
mX
j Ajnyn
tdWjtynynRdn
We rst solve the rst a ne SDE
Denote the resulting a ne
cocycle by t t R which by the variation of constants formula is given
by
tx
tx
mX
jZt
s pj dWjsA
Then we insert the cocycle t x into the second equation
in place of xt which gives a non autonomous a ne SDE whose solu
tion owisdenotedby tx etc Atstepninsert tx
ntx
xn xn into the nth equation
which gives a non
autonomous a ne SDE
dxn
tmX
j Ajnxn
tpjntx
ntx
xn xn dWjt
with initial conditions xn xn Rdn Denote the solution ow of this
equation by n t x
xn
1

NORMAL FORMS FOR SDE
Note that the rst n a ne SDE considered as one equation generate the
C RDS
tx
xn
tx txx
ntx
xn xn
Lemma For n N represent the solution of the nth equation
as
ntx
xn xnZt
vnsx
xn ds
mX
jZt
un
jsx
xn dWjs
Thenforanyn N naswellasvnandun
j j m satisfy conditions
C
Proof We use induction on n
The assertion holds for n
This is an immediate consequence of
Lemma
choosing uj bj
Let us now assume that the assertion holds for
n and their
characteristics Consequently Lemma
yields that all the components
ofpjn
tx
ntx
xn xn and their semimartingale
characteristics satisfy conditions C Hence Lemma
applies and gives
condition C for n as well as for unj and vn
Lemma Let for t R
Xt mX
jZt
s bj dWjs
and for n
and x
xn Rd
Rdn let
Xntx
xn
mX
jZt
ns pjn tx
ntx
xn xn dWjs
Then Xn satis es conditions C
Proof This is an immediate consequence of Lemmas
and

CHAPTER NORMAL FORMS
Here is our nal and main result of step
Proposition Consider the hierarchical system of a ne SDE in
troduced in
to
Then for any n
andp d
dn
and any compact set K Rd
Rdn there exist constants cn p R
and q such that
E
sup
x xn Ksup
t jXntx
xn jp
cnpdiamKq
where Xn is de ned by
Proof Combine Lemma
with Lemma
Step Inheritance of Temperedness
We rst study the inheritance of temperedness from above De nition
in case tempered vectors are inserted into random elds Property
will play a crucial role hereby as indicated by the following lemma
Lemma Let Xy y Rd be aPas continuous random eld
with values in Rd for which the following condition holds For p d
and any compact set K Rd there exist cp R and q such that
E sup
y KjXyjp
cpdiamK q
Let Y be a random vector with values in Rd and for
m Nlet
Amf jYtjmexpttRg
Then there exists a constant c m R such that for any n Z
EAmsup
tnnjXYtjp
c mexpnq
where p and q are related by
Proof For Amandt nn
we have
jY tj mexpt mexp expn
Hence due to
EAmsup
tnnjXYtjp
EAmsup
yBmnjXyjp
cpdiamB mn q
1

NORMAL FORMS FOR SDE
where
Bmnfy Rdjyjmexp expng
But diam B m n cd m exp n with a constant depending just on d
and m Hence the desired inequality follows readily from
Proposition Let X y y Rd be a P a s continuous random
eld with values in Rd for which the following condition holds For p d
and any compact set K Rd there exist cp R and q such that
E sup
y KjXyjp
cpdiamK q
Let Y be a tempered random vector with values in Rd Then the Rd valued
random vector X Y is tempered
Proof DeneAmasinLemma
We shall prove that
lim
t tlogjXY tj
remarking that the behavior for t
can be treated similarly Since Y
is tempered A m
m
Pasforany
Let
be given We have to prove that there exists a
such
thatforanym Nwehave
A mlimsup
t tlogjXY tj Pas
will indeed imply temperedness since A m P a s for m
and any
To prove
let
m n Nbegiven Thenby
and
Lemma
PAmfsup
t nn tlogjXY tj g
PAmfsup
tnnjXY tjexpng
EAmexpnp sup
tnnjXYtjp
cm expnq np cm expnq p
Now choose
pq Then for all m N the Borel Cantelli lemma yields
A mlimsup
n
sup
tnn tlogjXYtj
This clearly implies
and completes the proof

CHAPTER NORMAL FORMS
As a second issue we study the inheritance of temperedness by geometric
series
Lemma Suppose X is an Rd valued tempered random vector Let
Tn n N be a sequence of random linear operators in Rd with negative Lya
punov index i e for some deterministic
lim sup
n
n log kTnk
Then
YX
nTnXn
is absolutely and geometrically convergent and tempered
Proof Choose
such that
Then by the assumptions from
a certain index n on kTnk X n
exp
n from which the
convergence statements follow As to the temperedness of Y there exists
a random variable R such that for t R
jYtjX
n kTnkjX n tj
RX
n
expn nexpnjtj
R exp
exp jtj
which clearly implies that Y is tempered
Step Invariant Measures of a Hierarchical System of A ne
SDE
We return to the setting of a hierarchical system of a ne SDE introduced
above and determine its invariant measures
So let n t t R be the linear cocycle in Rdn generated by
The MET holds for n giving its Lyapunov spectrum S n Having in
mind our cohomological equations in the nonresonant case we assume that
all these cocycles are hyperbolic i e S n for all n N Denote by u
n
s
n the projection whose range is the unstable stable space and whose
kernel is the stable unstable space of n For convenience we recall the
following facts see Sect
1

NORMAL FORMS FOR SDE
Lemma Assume that the cocycle n is hyperbolic Then there
exists a constant n
such that
lim sup
k klogku
nnkkn
and
lim sup
k klogks
nnkkn
We rst consider the a ne SDE of order one thereby proving part iii of
Theorem
Theorem Consider the a ne SDE in Rd
dxt
mX
jAjxtbjdWjt
and assume that the corresponding linear cocycle generated by dyt
Pm
j Aj yt dW jt is hyperbolic Then the a ne cocycle generated by
has a unique invariant measure namely the random Dirac measure
ie wehave t
t Here
s
u with
sX
k
s
kX
k
uX
k
ukXk
where
XmX
jZtbjdWjtX mX
jZtbjdWjt
The sequences in
and
converge P a s absolutely and ge
ometrically and the limits s and u as well as
s
u are tempered
random vectors
Proof i We rst prove the statements on temperedness X and X
are tempered by Proposition
since a constant vector is tempered
Hence Lemmas
and
yield that s and u exist and are tem
pered Thus is tempered
1

CHAPTER NORMAL FORMS
ii We now check that is invariant i e that t
t
This is equivalent with s u t t
s u t Using the variation
of constants representation of and the fact that for k Z
mX
jZk
k
s bj dWjs
kXk
we obtain after some rather lengthy but elementary manipulations
tu
lim
N
t
NX
k
ukXk
ut
s
mX
jZt
s bj dWjs
A similar argument holds for the stable component
iii The uniqueness of the invariant measure is a consequence of hyper
bolicity and was proved in Theorem
Now suppose that
n are tempered random vectors with values in
Rd Rdn respectively such that i
i i n istheunique
invariant measure of the ith equation of our hierarchical system of a ne
SDE We now study the nth SDE
dxn
t
mX
j Ajnxn
tpjnt
nt
n
n dWjt
in Rdn Note the fundamental fact that the validity of Lemma implies
the substitution rule for Stratonovich integrals see Arnold and Imkeller
Corollary If is any random variable then u t though nonadapted
is Stratonovich integrable and
Zt
us dWsZt
usxdWs
x
Hence the anticipative a ne SDE
makes sense and it generates an
anecocycle nt
n We now determine its invariant measure
Theorem Suppose all equations of the hierarchical system of
a ne SDE have a hyperbolic linear part Then each of the equations has a
unique invariant measure This measure is a random Dirac measure sta
tionary solution whose supporting random variable can be obtained step by

NORMAL FORMS FOR SDE
step by determining according to Theorem
inserting t into
the second SDE etc where the stable and unstable component of n of the
nth equation
are given by
s
nX
k
s
n
nkXn
k
u
nX
k
u
n
nkXnk
where
Xn
mX
jZnt pjn t
nt
n
n dWjt
Xn
mX
jZnt pjn t
nt
n
n dWjt
The sequences in
and
converge P a s absolutely and ge
ometrically and the limits s
nandu
naswellas n
s
n
u
n are tempered
random vectors
Proof Since Proposition
holds and since
n isatem
pered vector Proposition
applies thus Xn and Xn are tempered
Hence by Lemmas
and
the series in equations
and
have the convergence properties claimed and the limits s u
n are
tempered Hence nally n is tempered
Invariance of n is proved as in Theorem
and uniqueness as in
Theorem
Remark i By means of the substitution rule
can
also be written as
s
mX
jZt stbjdWjt
u
mX
jZt utbjdWjt
where the integrals exist as the P a s limits of the non adapted
Stratonovich integrals R T and R T for T
Similarly for n
ii Again by the substitution rule
the cocycle
evalu
ated at
n is equal to the solution of the rst n equations
with non adapted initial value and is the unique stationary solution of
these equations t
t

CHAPTER NORMAL FORMS
This completes the proof of Theorem
We close with an example
Example One Dimensional Case This is the white noise
version of Example
For d Hn R choose the basis xne
for all n
The linearization of the SDE dx Pm
jfjx dWjis
dv Pm
j Aj v dW j whose explicit solution is
t
expAt mX
j AjWjt
Hence A is the Lyapunov exponent Further ad nAj
nAj
andS n f n gWehavenonresonanceforallnifandonlyif
The unique stationary solution of the nth cohomological SDE
dhn
mX
j adnAjhn kjn t dWjt
is
hn Pm
jRexpnPm
l AlWltkjn t dWjt
Pm
jRexpnPm
l AlWltkjn tt dWjt
Small Noise Case
Simultaneous Normal Form and Center Manifold Reduction
We are now in a position to carry over the procedure of Sect from
the RDE to the SDE case We will however not repeat details and just
emphasize that all nonresonant anticipative Stratonovich cohomological
equations in the SDE analogues of Theorems
and are solved as
in the nonresonant case of Subsect
We make the convention that for
all resonant equations we make the trivial choice Hpq
and Hpqrs
This mathematically rigorous procedure nally justi es earlier important
work on normal forms for SDE mentioned at the beginning of this section
Small Noise on Parametrized SDE
In stochastic bifurcation theory one needs to study an SDE
dxtfxt dt mX
jfjxt dWjt
1

NORMAL FORMS FOR SDE
in Rd where the smooth vector elds fj x smoothly depend on a pa
rameter Rm and R is a small intensity parameter We assume
that fj
for all Rm so that the linearization of
is
dvt A vtdt mX
j Aj vtdWjtAj
Dxfj x jx
jm
Assume further that for
the deterministic equation
vtAvtADxfxjx
is in a center stable situation and has been brought into Jordan canonical
form by a deterministic coordinate transformation so that Rd Rdc Rds
is the invariant splitting for A
Ac
As
We treat
Rm as a small parameter and seek the normal
form and center manifold reduction for small x
All cohomological
operators hence all linear cocycles are now deterministic and of the form
dLn hn dhn ad nA hndt
We add the trivial equations d
and d
The result is given by
the we hope rather obvious white noise analogue of Theorem
The
truncated center equation is
dxc Ac xcdt mX
jgcjtxc
dWjt d
d
and the approximate center manifold is the graph
Rdc Rm
xc
mc xc
Hs xc
Rds
where gcj and mc are random polynomials of order N in xc and M in
Example Du ng van der Pol Oscillator with White
Noise We now consider the white noise version of the example in
Subsect
to which we refer for notations
Replacing by
Wt W symbolically stands as usual for white
noise and putting
iny y y y yygivestheSDEinR
dxt
xt
xxxdt
xt dWt
1

CHAPTER NORMAL FORMS
with linearization at x given by
dvt
vt dt
vt dWt
The calculations are formally the same as in the RDE case We introduce
the auxiliary processes U V X and Y which are the stationary solutions
of the scalar SDE
dUt Utdt dWt dVt Vtdt dWt
and
dXt Xtdt dUt dYt Ytdt Ut dWt
Note that V anticipates the future of W The analogues of the RDE ex
pressions given above are now
WHspUHcpV
Hs pUpXHspY
gc
UW gc
UWgc
UW
gc
UVYUU
UXVW
and
gc
UXV
W
Inserting these expressions into the center equation up to terms of order
inxcand in
yields
dxc
U xcdt
U
xc dWt
U
UVYUU
UXV
xc dt
UXV
xc dWt
Note that this is an anticipative SDE since V is anticipative
The approximate center manifold is the graph xc
mc xc
xs given by
mc xc
pUpZp
pUXpxc
ppUX
HspHspxc
1

NORMAL FORMS FOR SDE
where H s and H s are certain anticipative random variables
The scalar center SDE
can now be utilized for stochastic bifur
cation theory of the original two dimensional SDE
in the neighbor
hood of
see Subsect

CHAPTER NORMAL FORMS

Chapter
Bifurcation Theory
Summary
In this chapter we investigate qualitative changes in parametrized families
of RDS and call these studies stochastic bifurcation theory
The rst problem is to develop a mathematical formalization called
D bifurcation of qualitative change which is connected to the stability
of an RDS under an invariant measure i e to the Lyapunov exponents
and reduces to the deterministic de nition of bifurcation in the absence of
noise Subsect
We propose as a rst task to study the branching of
new invariant measures from a family of reference measures at a parameter
value at which the reference measure has a vanishing Lyapunov exponent
We also discuss an older concept on the level of densities in state space
called P bifurcation and discuss its relation with D bifurcation Subsects
and
As the theory of stochastic bifurcation is still in its infancy we proceed
mainly by way of instructive examples
Sect treats explicitly solvable one dimensional examples of stochas
tic transcritical and pitchfork bifurcation In Subsect
we prove a
general criterion for a pitchfork bifurcation in dimension one using the the
ory of random attractors to which we give a brief introduction
Sect is devoted to the study of the prototypical noisy Du ng van
der Pol oscillator We report on the state of the art as far as rigorous results
are concerned and also present numerical ndings supporting conjectures
about the correct scenario of stochastic Hopf bifurcation
Sect is devoted to the only available general condition due to Bax
endale for a D bifurcation out of the xed point x and an associated
P bifurcation of the new branch of measures in a family of SDE in Rd
1

CHAPTER BIFURCATION THEORY
Introduction
Despite of its rapid development in the last decade and the fact that it is
probably the most relevant part of the whole theory of RDS bifurcation
theory of RDS is still in its infancy There are few rigorous general theorems
and criteria and many phenomena have been proved only by computer
simulations or for particular models Nevertheless this chapter was in
cluded although it is less self contained and less mathematically rigorous
than the rest of the book
The interest in understanding the changes encountered in a determin
istic bifurcation scenario in the presence of noise is historically at the
beginning of the theory of RDS and has been one of its strong driving
forces
This interest arose in the s and prompted numerous investigations
by engineers S T Ariaratnam F Kozin Y K Lin N Sri Namachchivaya
C W S To W Wedig and many others physicists and chemists W
Ebeling P Coullet R Graham H Haken K H Ho mann W Horsthem
ke R Lefever M Lucke F Moss G Nicolis E Tirapegui K Wiesenfeld
and many others and mathematicians L Arnold P Baxendale R Khas
minskii W Kliemann P Kloeden G Papanicolaou M Scheutzow among
others
In particular bifurcations delayed or advanced by noise bifurca
tion intervals and the disappearance of branches from the deterministic
bifurcation diagram were observed numerically experimentally and by ap
proximations based on averaging and scaling methods
The literature is so vast that the reader is referred to the book by Horst
hemke and Lefever for the phenomenological approach see Subsect
and to the review papers by Arnold and Boxler for the literature
prior to
Sri Namachchivaya
Wedig for the literature
prior to
andToetal
and the references therein
What is stochastic bifurcation theory or more precisely bifurcation
theory of RDS As with other notions for RDS it should be an extension
and generalization of deterministic bifurcation theory to which it should
reduce in the absence of noise
We hence ask what bifurcation theory of deterministic dynamical sys
tems is about Of the numerous books on this subject several provide
excellent background including the now classical book by Guckenheimer
and Holmes
and those of V I Arnold Chow and Hale Hale
and Kocak
Ruelle
and Wiggins
The general term bifurcation is used to describe qualitative changes
in the phase portrait orbit structure of parametrized families
of

INTRODUCTION
dynamical systems e g those generated by a family of di erential equations
xtfxt
Rk
which occur when the parameter is slowly varied The vague notion
of qualitative change has been successfully formalized by using the con
cepts of topological equivalence and structural stability A parameter value
is called an abstract bifurcation point of the family if the family is
not structurally stable at i e if in any neighborhood of there are
parameter values such that is not topologically equivalent to
If we replace topological equivalence by local topological equivalence
for example in the neighborhood of a particular point we arrive at the
de nition of a local abstract bifurcation point
This abstract de nition of qualitative change via structural instability
does not indicate which qualitative change actually happens at a bifur
cation point of a family For example when there is bifurcation in the
narrow sense of the word i e when some invariant object splits into several
such objects after all bifurcation means splitting into two then we
drop the word abstract and just speak of a bifurcation point
We now turn to the most elementary and classical local bifurcations
namely those of equilibria xed points in particular the saddle node tran
scritical pitchfork and Hopf bifurcation The modest aim of this chapter
is to discover their stochastic analogues
To begin with assume from now on that f
for all Rk and
that f depends smoothly on How do we detect the bifurcation of the
family of reference xed points x
at meaning the coexistence of
with at least one other small solution x e g another xed point
limit cycle etc in a neighborhood of
It is futil to search for this phenomenon at parameter values for
which x is a hyperbolic xed point of i e for which the linearized
system t D t generated by
vtAvtADf
Rk
is hyperbolic for
The reason is that hyperbolic xed points are
structurally stable for any family
Since for all close to by
the local Hartman Grobman theorem
and are locally topologically
equivalent in a neighborhood U Rd of the family
is locally
structurally stable at
Moreover x
cannot bifurcate at in the narrow sense either
Since in the case of hyperbolicity the only bounded solution of
is
the trivial solution the local Hartman Grobman theorem implies that the

CHAPTER BIFURCATION THEORY
only solution of
which stays in U for all time is also the trivial
solution t
Consequently a necessary but not su cient condition for the bifurca
tion of small solutions out of the xed point x
at
is that the
matrix A is not hyperbolic i e has eigenvalues on the imaginary axis If
in addition A is hyperbolic for some in each neighborhood of then
is a local abstract bifurcation point which may also be a local bifurcation
point
The simplest possible cases for a scalar parameter R and at
here are the following As increases from negative to positive values
a either a simple real eigenvalue
of A increases from negative
to positive values with
andd dj
while the rest of
the spectrum of A remains in the left hand side of the complex plane C
allowing for the transcritical and the pitchfork bifurcations
b or a pair of complex conjugate eigenvalues
iof
A moves from the left hand side to the right hand side of C crossing the
imaginary axis at
with positive speed d d j
allowing
for the Hopf bifurcation
There are very e cient means of further reducing the search for bifur
cations of the trivial solution at
i Center manifold theory tells us that it su ces to investigate the
system on the center manifold which is decribed by a di erential equation
in the center space of A with dimension in case a and dimension in
case b
ii This low dimensional system can be simpli ed further by normal
form theory in fact steps i and ii can be carried out simultaneously
see Elphick et al
iii A persistence theorem assures that without loss of generality the
normal form can be truncated beyond low order nonlinear x terms from
which the bifurcation behavior can be often easily read o
The simplest possible truncated normal forms for the above mentioned
elementary bifurcation scenarios of the xed point x
are for a scalar
parameter R and at
saddle node bifurcation x
x inR
transcritical bifurcation x
x xinR
pitchfork bifurcation x
x xinR
Hopf bifurcation r
rr
ar in polar coordinates of R
In all scenarios with a trivial reference solution bifurcation is accompa
nied by an exchange of stability The reference solution loses its stability
at the bifurcation point while the bifurcating solutions are stable
Here x is a xed point of only for the parameter value
1

WHAT IS STOCHASTIC BIFURCATION
An analogous situation holds for families of mappings where the period
doubling or ip bifurcation has to be added to the catalogue of elementary
bifurcations of a xed point whose truncated normal form is x
x
x x in R the second iterate of which has two stable xed points
p for
What is Stochastic Bifurcation
De nition of a Stochastic Bifurcation Point
In complete analogy to the deterministic case stochastic bifurcation theory
studies qualitative changes in parametrized families of RDS e g those
generated by a family of SDE
dxt fxtdt mX
jfjxtdWjt
Rk
How can we formalize qualitative change in this situation
In order to mimic deterministic reasoning we need the de nition of lo
cal topological equivalence of RDS introduced in connection with the Lo
cal Hartman Grobman Theorems in Chap For a systematic study of
topological dynamics of RDS see Nguyen Dinh Cong
As in the deterministic case we add the word abstract if the bifurca
tion point of a family of RDS is a point of structural instability and omit
it if we have a bifurcation point in the narrow sense see below
De nition Abstract Bifurcation Point i Let
Rk be
a family of continuous RDS on Rd A parameter value is called an
abstract bifurcation point of the family if the family is not structurally
stable at i e if in any neighborhood of there are parameter values
such that and are not topologically equivalent
ii Let
Rk be a family of local continuous RDS on Rd with xed
point A parameter value is called a local abstract bifurcation point
of the family if the family is not locally structurally stable at in a
neighborhood of i e if in any neighborhood of there are parameter
values such that and are not locally topologically equivalent in a
neighborhood of
We now turn to stochastic bifurcation in the narrow sense As invariant
measures are the random analogues of deterministic xed points we should
start our investigations by nding necessary and su cient conditions for the
bifurcation of a family of invariant measures What objects do we expect

CHAPTER BIFURCATION THEORY
to bifurcate out of at a local bifurcation point Again analogous to
the deterministic case the simplest possible choice is a new family
of small invariant measures
De nition D Bifurcation Point Let
Rk be a family of
local C RDS in Rd with a respective family of ergodic invariant measures
A parameter value D is called a local dynamical or D bifurcation
point of
Rk if in each neighborhood of D there is an for
which there is an invariant measure
with
Dinthe
topology of weak convergence as
D One could also require that
converges to D in the Hausdor sense i e that in addition to the above
lim
Ddsupp jsupp D
wheredAjB supx AdxB isthe
Hausdor semi metric
See Fig
Figure D bifurcation point of a family of measures
We further simplify the situation by assuming that
Rk is a family
of local C RDS all having the xed point x
The family
Rk is thus a family of ergodic reference measures
We linearize the local RDS at x
and obtain the family of linear
cocycles t
Dt
We assume that the IC of the MET are
satis ed by for all providing us with the Lyapunov spectrum
S
fidii
pg
and the splitting
RdE
Ep
One basic di culty is that even if we assume that depends smoothly on
the Lyapunov exponents their multiplicities and the Oseledets spaces
will in general depend only measurably on Perhaps the most extreme
expression for this is a result by Arnold and Nguyen Dinh Cong
On the other hand in many situations in which small perturbations of
a given cocycle contain enough randomness the Lyapunov exponents
their multiplicities and even in a certain sense the Oseledets spaces de
pend continuously on the size of the perturbation The most general result
supporting this is one by Ochs generalizing an earlier result of Ledrap
pier and Young
and such continuity is found in many applications
see the examples in this chapter
See the next subsection for the reason for using of the word dynamical
1

WHAT IS STOCHASTIC BIFURCATION
We now prove that a parameter value for which the linearization
is hyperbolic can never be a local D bifurcation point
To be more speci c consider a family
RkofC RDSwithdis
crete or continuous two sided time T and xed point such that x
t x is C and the derivatives are continuous with respect to t x
We mimic the deterministic procedure and study the augmented RDS
onRd Rkdenedby
tx
tx
x RdRktT
The RDS is C it has xed points
and its linearization at
is
given by
t
Dxt
t idRk
where t
Dxt
satis es the IC of the MET if and only
if does which we assumed The spectrum and splitting of can be
obtained from that of in an obvious manner by increasing the possibly
zero multiplicity of a vanishing Lyapunov exponent by k and augmenting
the possibly trivial center space Ec of to Ec Rk
IfUisaset wecallameasure Usmallifsupp U
Theorem Necessary Condition for D Bifurcation Point
Assume the situation just described Assume further that satis es the Lo
cal Hartman Grobman Theorem discrete time or continuous
time at
respectively Suppose there exists a family dx
dx
of invariant measures such that the invariant measure dx d
dx
d isUT
small for some and T
Then necessarily is non hyperbolic i e
is among its Lyapunov
exponents
Proof All invariant measures of clearly have the form dx d
dx
d where dx is invariant If dx
d
dx
d is UT small for some then cannot be hyperbolic
for the following reason In the hyperbolic case the only ergodic invariant
measures of are of the form dx
d By the local Hartman
Grobman theorem this carries over to UT small measures of
This theorem convinces us that our approach towards stochastic bifurcation
is correct in the sense that
UT is the neighborhood of
de ned in the text following the Hartman
Grobman Theorems
1

CHAPTER BIFURCATION THEORY
it reduces to the deterministic concept in the absence of noise and
it ties the phenomenon of local branching of a family of invariant
reference measures at a parameter value D to the stability of the reference
measure in a neighborhood of D
Hence we will have to search for parameter values for which one typi
cally the largest for the stochastic Hopf bifurcation also the second largest
Lyapunov exponent changes its sign
The Phenomenological Approach
We now present an alternative and older approach towards stochastic bi
furcation
Scientists have been observing in numerous experimental numerical and
analytic studies qualitative changes of probability densities p which are
stationary for the forward Markov semigroup corresponding to the SDE
i e which are solutions of the Fokker Planck equation
Lp
LfmX
jfj
where L is the formal adjoint of the generator L here we have written
L in Hormander form
For example the stationary density can exhibit transition from a uni
modal one peak to a bimodal two peaks or crater like form see Fig
Also the number and location of extrema of p as functions of and their
bifurcations have been carefully studied For a systematic treatment and
survey see Horsthemke and Lefever who call such phenomena noise
induced transitions Arnold and Boxler
and Sri Namachchivaya
Many case studies can also be found in the proceedings volume
edited
by Horsthemke and Kondepudi
This concept can be formalized based on the ideas of Zeeman
according to which two probability densities are called equivalent if there
are two di eomorphisms
such that p
q
This gives rise
to a notion of structural stability based on this equivalence and thus of a
point of structural instability of families p of densities which captures the
above mentioned observations For example the transition point P from
a unimodal density p to a bimodal density is such a point of structural
instability
We call such a phenomenon phenomenological or P bifurcation and the
corresponding parameter value P a phenomenological or P bifurcation
point in distinction to the concept introduced in De nition
which

WHAT IS STOCHASTIC BIFURCATION
we called dynamical or D bifurcation and denote the corresponding pa
rameter value by D
Figure P bifurcation point of a family of densities
Besides being basically restricted to the white or Markovian noise case
the concept of P bifurcation has the following severe drawbacks
i First the concept is static since the stationary density p x dx
measures the asymptotic proportion of time spent by a typical solution
t xof
in the volume element dx Hence p has guratively
forgotten any dynamics of
ii Second p is an object determined by the one point motions t
t x of and can thus in principle not be related to the stability of
which is a property of the two point motions t
txty
forx y ininnitesimalformofD t xv henceisapropertyofthe
Lyapunov exponents
iii The underlying concept of invariant measure is too narrow Sta
tionary measures for the forward Markov semigroup are in a one to one
correspondence to invariant measures which only depend on the past for
ward Markov measures see Sect
but there are typically many more
invariant measures Restricting ourselves to stationary measures leads to
missing branches in bifurcation diagrams representing non Markovian
invariant measures and prevents us from understanding stochastic bifur
cation
Is there a connection beween the two types of bifurcation In general
there is not We will next present an example of a family of RDS undergoing
a P bifurcation but not a D bifurcation and then an example where the
converse occurs
In Sect however we will reveal a surprising connection between the
two concepts
Additive Noise P Bifurcation but not D Bifurcation
If x t is a solution of a di erential equation and if this equation is dis
turbed by noise we call the noise multiplicative with respect to x t if
x t also solves the disturbed equation Otherwise the noise is called ad
ditive with respect to this solution It is just called additive if it is additive
The term dynamical bifurcation has also been used in a di erent context namely
for phenomena obtained by letting the bifurcation parameter
t deterministically
depend on time for example by augmenting xt f xt by t for small See
Beno t for a survey
Zeeman
himself says at the end of his paper It seems a pity to have to
represent a dynamical system by a static picture
1

CHAPTER BIFURCATION THEORY
with respect to any solution of the undisturbed equation For example the
noise t in xt
t xt xt is multiplicative with respect to x
but additive with respect to x p for
Consider the scalar SDE
dxt fxtdt gxt dWt fxt gxtgxtdt gxtdWt
with smooth coe cients f and g Assume further that the conditions of
Theorem
are satis ed which ensure that are non exit boundaries
and hence that
generates a local smooth RDS which is forward
hence strictly forward complete The generator of the forward Markov
semigroup of
is
Lfx gxgxd
dx gxd
dx
so the Fokker Planck equation is
Lpfggp
gp
We assume for simplicity that g x for all x R ellipticity implying
that there is at most one stationary probability and also assume positive
recurrence implying that there is exactly one stationary probability with
smooth density given by
pxN
gx expZxfy
gy dy
where N is a normalization constant
We can explicitly calculate the Lyapunov exponent of the RDS gen
erated by
under the stationary measure with density p more pre
cisely under the invariant measure which corresponds to p but is
not needed for the calculation For this we solve the linearized SDE
dvtfxtvtdtgxtvtdWt
coupled to
to obtain
vt v expZtf gg xsdsZtgxsdWs
and nally for example under the assumption that g is bounded and
f gg LpxdxZRfx gxgxpxdx
1

WHAT IS STOCHASTIC BIFURCATION
Integrating the last expression by parts and using the explicit expression
for p we nally arrive at formula
of the following theorem
Theorem Scalar RDS with Additive White Noise is Al
ways Stable Assume that the Fokker Planck equation
of the
SDE
with g x all x R has a necessarily unique solution
dx p x dx with density
Then
i The Lyapunov exponent of is given by
ZR fx
gx pxdx
Hence the RDS generated by
is always exponentially stable under
ii The invariant forward Markov measure limt
t
corresponding to is a Dirac measure
a
andA fa g
is the random attractor of in R in a universe of sets which contains all
bounded deterministic sets see De nition
iii The RDS has no other invariant measure besides
iv Foranyinitialmeasure PR and t RRPtx dx
lim
t
t
EaPfag
Proof ii The measure is ergodic since is Theorem
and is
a Dirac measure by Theorem
The remaining statement is proved by
Crauel and Flandoli Corollaries
and See also Subsect
iii By a result of Crauel Corollary all invariant measures of
a continuous RDS having a random attractor in the universe of bounded
deterministic sets are supported by this attractor Since the attractor is a
one point set it can carry only a Dirac measure
iv By dominated convergence and the fact that by the de nition of
the attractor t t x
a forallx Rforanyf CbR
lim
t
tfZRE lim
tfttxdxEfa
The fact that is globally stable in the sense that it attracts arbitrary initial
measures has been observed much earlier see Horsthemke and Lefever
Sect
by using the relative entropy
VtZlogd
dtxtdx
1

CHAPTER BIFURCATION THEORY
as a Lyapunov function and proving that dV t
dt
always and if and
only if t
Example E ect of Additive Noise on a Pitchfork Bifurca
tion For
dxt
xt xtdt dWt
R
the unique stationary measure has density
pxNexp
xx
For xed
the density is unimodal for
and bimodal for
Hence the family p
R undergoes a P bifurcation at P for each
The IC of the MET is satis ed and the Lyapunov exponent of the RDS
underp is ZRxp xdx
which by formula
is strictly negative for all values of R and
The P bifurcation of the densities p at P is what remains of
the pitchfork bifurcation at
present for
However there is
no D bifurcation since the measure
a
corresponding to p
is the unique invariant measure of and always has negative Lyapunov
exponent One can thus say that additive noise destroys a pitchfork bifur
cation which is the title of the paper by Crauel and Flandoli
D Bifurcation but not P Bifurcation
Baxendale
considers the following family of SDE on the two
dimensional torus M T S S
dxt X
jfjxtdWjt
where withx x x M
fx
sinxcos x sin x
fx
cosxcos x sin x
1

DIMENSION ONE
fx
sinxsinxcosx
fx
cosxsinxcosx
and
For all we have L
hence all of the one point
motions are Brownian motions and the Fokker Planck equation does not
depend on Its unique solution is the normalized Lebesgue measure on
M However the smooth global RDS generated by
does depend
on For example the sum of the two Lyapunov exponents of under
is
cos
and the top exponent satis es
or or ifandonlyif or
or
Hence D
is an abstract bifurcation point Computer
simulations indicate that it is a D bifurcation point Thus through this
example it is shown that a system can exhibit a D bifurcation without any
phenomenological change in the invariant measure for all values of
Dimension One
We continue the study of the example treated in Subsect
and Example
from the point of view of stochastic bifurcation It su ces to treat
the prototypical cases N and N see Arnold and Boxler
which
amounts to investigating the e ect of white noise added to the bifurcation
parameter in the truncated normal forms xt
xt xt andxt
xt xt
of the transcritical and pitchfork bifurcations respectively Then we will
consider the saddle node case xt
xt and the e ect of real noise
Transcritical Bifurcation
For N we have
dxt
xt xtdt xtdWt
which is solved by
tx
xetWt
xRtesWsds
The ergodic invariant measures are
i
for all
1

CHAPTER BIFURCATION THEORY
ii
whereRetWtdt
RetWtdt
We have E
Solving the forward Fokker Planck equation
Lpx
x
xxpx
xpx
yieldsi p
for all
ii for
qx
Nx
expxx
x
where
N
We do not nd a solution for
corresponding to E dx q x dx
This is due to the fact that for
is F measurable we could
however nd by solving the backward Fokker Planck equation
The family of marginal densities q
of
clearly undergoes a
P bifurcation as shown in Fig at the parameter value P
While
q is decreasing in x with pole at x for
P it is a unimodal
function with q
for
P
A similar phenomenon holds for the marginal densities for
at
P
Figure P bifurcation of q at P
The Lyapunov exponent of the linearized SDE
dvt
xt vtdt vt dWt
with respect to the two invariant measures is see Example
i For
ii for
E
We hence have a D bifurcation of the trivial reference measure of the
family of local RDS
Rat D
The nal bifurcation diagram of

DIMENSION ONE
the stochastic transcritical bifurcation is given in Fig
Figure Bifurcation diagram of the transcritical bifurcation
For each of the invariant measures we can also easily determine the
invariant manifolds
i For
the stable manifold for
is the random interval
Ms
which is F measurable the center manifold for
is Mc
R and the unstable manifold for
is Mu
which is F measurable For
the orbit behavior is
very interesting While for any x
ttx
ast
which
can be obtained e g for x from
ttxRteWsds
and Lemma
we have again for x
lim inf
t
tx
lim sup
t
tx
since the corresponding di usion process on
is null recurrent For
details see Remark
ii For the second measure the unstable manifold for
is
Nu
and the stable manifold for
is Ns
Why do we have a P bifurcation of q
atP
This turns out
to be a large deviations phenomenon which is explained in Subsect
De ne the p th moment Lyapunov exponent of a linear RDS by
gp lim
t tlogEkt vkp pR
In our case using the explicit solution of the linearized SDE at x
gppp
It has been shown by Baxendale
that the marginal distributions
q
of the new branch
of invariant measures undergo a P
bifurcation in a neighborhood of x
at that parameter value P for
whichthesecondzeroofp g p isatp dimR
hence at
P see Fig
The intuitive reason is as follows Although the
trivial solution x becomes unstable at the D bifurcation point D
the repulsion from or the attraction by are too weak for small
1

CHAPTER BIFURCATION THEORY
to move the most probable value of a typical trajectory of the nonlinear
system away from The new branch is nally freed from the old
reference branch for
P
Figure The p th moment Lyapunov exponent
Pitchfork Bifurcation
For N we have
dxt
xt xtdt xtdWt
which is solved by
tx
xetWt
xRtesWsds
The ergodic invariant measures are
i For
the only invariant measure is
ii For
we have the three invariant forward Markov measures
and
where
ZetWtdt
We have E
Solving the forward Fokker Planck equation
Lpx
x
xxpx
xpx
yieldsi p
for all
ii for
qx
Nx
exp x
x
x
andqxqx
where N
Naturally the invariant measures
are those correspond
ing to the stationary measures q Hence all invariant measures are Markov
measures
1

DIMENSION ONE
The two families of densities q
clearly undergo a P bifurcation
at the parameter value P
which is the same value as for the
transcritical case since the SDE linearized at x is the same in both
cases
The Lyapunov exponent of the linearized SDE
dvt
xt vtdt vt dWt
with respect to the three measures is see Example
i For
ii for
E
Hence we have a D bifurcation of the trivial reference measure at
D
and a P bifurcation of at P
The nal bifurcation dia
gram of the stochastic pitchfork bifurcation is given in Fig
Figure Bifurcation diagram of the pitchfork bifurcation
The invariant manifolds are as follows
i For
the stable manifold for
is Ms
R the
center manifold for
is Mc
R for the orbit behavior see the
transcritical case and Remark
and the unstable manifold for
is
the random interval Mu
which is F measurable
ii For the two other measures
which exist for
the stable
manifolds are Ns
and Ns
respectively
Saddle Node Case
The perturbed version of xt
xt is
dxt
xt dt dWt
R
We put formally x u
u andusex
Wt x to arive at the undamped
stochastic linear oscillator u
Wtu
For z
z
z
u
uwe
obtain the SDE
dzt
ztdt
zt dWt
Explosion of
is equivalent to u z becoming or the projection
of
onto S
dt
sin t
cos tdt cos t dWt
1

CHAPTER BIFURCATION THEORY
crossing the angles
arctan u
u
Linear SDE of the type
and their projections onto S have been extensively investigated see e g
Arnold Oeljeklaus and Pardoux
It follows that in our case
is positive recurrent on S equivalently all solutions t x of
explodePas ieE
for all R
Why does the saddle node bifurcation disappear after perturbation
by white noise The general intuitive reason is that even if is small the
perturbed parameter Wt can become big hence has a global e ect
on the behavior The resulting behavior of the perturbed system at some
xed value is a nontrivial mixture of the deterministic behavior for
the various frozen parameter values In the saddle node case white noise
always pushes the system to negative parameter values where it explodes
We can avoid this by using real noise properly scaled down with see
Subsect
A General Criterion for Pitchfork Bifurcation
As we will now use the theory of random attractors we rst give a brief
introduction to this subject
The investigation of random sets which attract many other determin
istic or random sets is an important new line of research and part of the
topological dynamics of RDS As in the deterministic case random attrac
tors encapsulate information about the long term behavior of the system
and the study of their qualitative change is part of bifurcation theory
Random attractors have been introduced and studied simultaneously by
Crauel and Flandoli
and Schmalfu
For further
results see also Arnold and Schmalfu Crauel Crauel Debussche
and Flandoli
Debussche
and Flandoli and Schmalfu
A useful synopsis and applications to the bifurcation theory of the noisy
Du ng van der Pol oscillator see Sect
are given by Schenk Hoppe
Chap and
We follow the approach of Schmalfu and de ne a random attractor
only relative to a universe of sets which are attracted This more exible
notion allows us to capture local phenomena by separating disjoint invariant
setsThe following de nitions are not the most general but are adapted to
our purposes
De nition Universe Absorbing Set Attractor Domain of
Attraction Let be a continuous two sided local RDS on a Polish space
X
1

DIMENSION ONE
A universe D is a collection of families D
of non empty sub
sets of X which is closed with respect to set inclusion i e if D D
and D
D forall thenD D
A set B D is called absorbing in D for the RDS if B absorbs all
setsinDieforanyD Dthereexistsatimet t D such
that
ttDtBforalltt
A global random attractor of in D is a random compact set
A D with the following properties
aAisstrictlyinvariant t A A t fort T
bAattractsallsetsinDieforallD D
lim
tdttDtjA
where d AjB supx A d x B is the Hausdor semi metric The
universe D is called a domain of attraction of A
We stress that the de nitions of absorbing and attracting are based on
moving points from time t to time and not from to t This enables
us to study the asymptotic behavior as t
in the xed ber at time
Note that b implies
lim
tdtDjAt
in probability
In general this cannot be improved to P a s convergence see Remark
Here are some basic results of Schmalfu
and Schenk Hoppe
Proposition Existence and Uniqueness of Random At
tractor Let be a continuous local two sided RDS on the Polish space
X and let D be a universe If there exists a random compact absorb
ing set B D which is strictly forward invariant i e which satis es
tB
Btfort
then there exists a random attractor A
with domain of attraction D given by
A
tttBt
n
nnBn
Furthermore
i A E E being the set of never exploding initial values of
and D
E
t Dt foranyD DwhereDt denotes
the domain of t
1

CHAPTER BIFURCATION THEORY
ii If C D is strictly invariant then C A In particular the
attractorisuniqueinDandifE D thenA E for all
iii If B is connected for any then so is A
iv If is invariant and if for any
there exists a D D for
which D
then A
v If t t isF measurablefor allt andBisF measurable
then A is F measurable too
Theorem Su cient Condition for Stochastic Pitchfork Bi
furcation Consider the scalar SDE
dxt
xt xtgxtdt xtdWt
with parameter R where g is smooth and satis es
Cx xgx Cx forsomeC C andallx R
Then
generates for any a smooth local RDS which is strictly
forward complete Moreover the family
R undergoes a stochastic
pitchfork bifurcation at D More precisely
i The family
of invariant measures has Lyapunov exponent
hence loses its stability at D
For
is the unique
invariant measure and A f g is the random attractor of on R
in the universe of all families C
of subsets of R
ii For
there are exactly two additional invariant measures
for
with
and
with
Both measures are F measurable and have negative Lyapunov exponents
Further A
f g is the random attractor of on the state
space
in the universe of all families C
of subsets of
which are tempered from below in particular containing all deterministic
sets bounded away from
Similarly A
f g is the random attractor of on the
state space
in the universe of all families C
of subsets of
which are tempered from below
We call a family C
of subsets of
tempered from below if
there exists a random variable r
which is tempered from below and
C
r
for all similarly for
As before a family
C
of subsets of Rd is called tempered from above if it is a subset
of a random ball Br fx Rd kxk r g with a radius r tempered
from above see Subsect
Remark i Condition
means that g represents the higher
order terms In particular it implies that g
g
g
1

DIMENSION ONE
ii In contrast to the deterministic case where one only needs local
conditions on g in a neighborhood of we have to impose the global con
dition
We prove this theorem in two steps
i We rst prove the existence of the random attractors
ii Then we show that they are one point sets
Lemma Consider the scalar SDE
dxt axt fxt dt xt dWt
where f is smooth and satis es
xfx bx cforallxRwhereab
c
Theni
generates a smooth local RDS which is strictly forward
complete
ii possesses an attractor A a a in the universe of random
compact subsets of R tempered from above
Proof i Uniqueness of the solution follows from the local Lipschitz
property of the coe cients Forward completeness hence in R strict
forward completeness follows from
xax
xfx
xKjxj
see e g Gihman and Skorohod Chap x Remark
ii We construct an absorbing set as follows Observe that x satis es
dxt
axt xtfxt dt xt dWt
Since ax xf x
a b x c and by the comparison theorem Ikeda
and Watanabe Chap we have
jtxjtjxj
where is the RDS generated by
dzt
abztcdt ztdWt
The last SDE is a ne with stable linear part hence its unique invariant
measure is the Dirac measure supported by
cZe abt Wtdt
1

CHAPTER BIFURCATION THEORY
and attracts all points with exponential speed see Subsect
It follows that in case c which is without loss of generality
B
is an absorbing set in the universe of random compact sets which are tem
pered from above
Proof of Theorem
We basically follow
i We use Lemma
For
put a
andfx
xgx
Then xf x
x xgx
Cx
so that condition
holdswithb c buta b For
choose any a
and put
fx
ax x gx Sincexfx
ax
C x there exists
aconstantc caC suchthatxfx c For
choose a
and
fx
x x gx chenceb butab
The result is that for all R
is strictly forward complete and
has an attractor in the universe of random compact sets which are tempered
from above
We will now show that A actually attracts any family C Assume
x
and use the estimate x g x
C x to compare
with the explicitly solvable SDE
dyt yt
Cytdt ytdWt
to obtain see
jtxj
xetWt
CxRtes Wsds
etWt
CRtesWsds
Hence
jttxj
CRtesWsds rt
It follows from the fact that the right hand side of
is independent
of x that maps any set after positive time t into a subset of the
compact interval r t r t
For
rt
exponentially fast as t
thus any set is
shrunk to A f g exponentially fast
For
rt
ast
by Lemma
1

DIMENSION ONE
ii For
rtb
CRetWtdt
so that by the comparison theorem
A
a
a
bb
Furthermore b b is strictly forward invariant under
since
tb
tb
b t where is the dominating RDS
Let now
and x We have just proved that b is strictly
forward invariant
On the other hand applying the transformation y x
is
transformed into
dyt
yt
hyt dt yt dWt
where h y
gxxygy
C where we have used
the left hand side of condition
Also
C
C
Comparing
with the corresponding stable a ne SDE
dzt
zt
Cdt ztdWt
we conclude that the interval
b
CZetWtdt
is strictly forward invariant for
hence
b
CRetWtdt
is strictly forward invariant for Finally the intersection b b of the
two intervals is non void since C C strictly forward invariant and
obviously F measurable
Again by the comparison theorem B b b is an absorbing set
for all families tempered from below Hence there exists an attractor A
a a in this universe which is F measurable
Similarly for x
The following result of Crauel and Flandoli implies that the above
attractors A and A
are one point sets Since by Proposition
the attractor supports all invariant measures the Dirac measure on
the attractor is the unique invariant measure It is a Markov measure For
the Lyapunov exponent of the two non trivial measures is negative
by Theorem
This concludes the proof of Theorem
1

CHAPTER BIFURCATION THEORY
Proposition Suppose is a continuous local RDS generated by
an SDE on an interval I R such that the associated Markov semigroup
has a unique stationary probability measure Suppose further that C is a
strictly invariant random compact set which is F measurable Then C is
a one point set
Proof Since C is a random compact set which is F measurable
x
maxC and x
min C
x
are F measurable
choose a countable dense set of F measurable selections see Proposition
Continuity of t
t and injectivity of t for all t and
imply that t is order preserving Thus by the strict invariance of C
tx
xt
andtx
xt
for all t R Hence the two invariant measures
x
are Markov
measures Their expectations give rise to two stationary measures which
by assumption have to be identical This would be contradicted by Pf
x
xgHencexx
xandCfxg
Remark Caution For and g say the attractor A
f g is not attracting forward in time In fact and are both natural
repelling boundaries for the di usion process on
sinceR R x dx
andRRxdx
where
Rx expZxay
bydy C
xexpx
andax
x x andbx
x are derived from the Ito form of
see Gihman and Skorohod Chap x Theorem Hence
the di usion process is null recurrent in
andforanyx Pa s
lim inf
t
tx
lim sup
t
tx
However limt
ttx
which implies that t x
in
probability This supports the sensibleness of our de nition of an attractor
Exercise Criterion for Stochastic Transcritical Bifurca
tion By similar arguments we can establish a transcritical bifurcation
at
for the family
dxt
xt xtgxtdt xtdWt
1

DIMENSION ONE
where g is smooth and satis es
CxgxCx
xRCC
For
we nd a one point attractor in
and for
we nda
one point repellor an attractor under inverse time in
The details
are left to the reader as an exercise
Real Noise Case
Transcritical and Pitchfork Bifurcation
If the parameters and in xt
xt
xN
tN
are perturbed by
real noise we obtain the RDE
xt
txt
txN
t
We assume that
LE
and P
The RDE
can be explicitly solved by the same technique as
its white noise version see Subsect
and the local C RDS it
generates is
x
tx
xetSt
NxNRtseN
sSsdsN
where we use the abbreviation
St Zt
sds
We can read o from the expression
the explosion time the do
main and the range of t the set E and the invariant measures
We can also calculate their Lyapunov exponents and nally determine the
bifurcation behavior of the family
R
For a thorough study of the transcritical case N and pitchfork case
N seeXu
If we con ne ourselves here to the case
hence to the RDE
xt
txtxN
tE
we obtain exactly the same results as for the SDE
with Wt replaced
with St
The corresponding result for
is not so easy to obtain but follows
from the following lemma which also is of independent interest
1

CHAPTER BIFURCATION THEORY
Lemma Let L P withE
Then
ZeSt dt
Pas whereSt Zt
sds
Stochastic Saddle Node Bifurcation
The saddle node scenario of xt
xt disappears if is perturbed by
white noise see Subsect
If however we consider
xt
t
xt
where
aba
the situation is di erent Putting again x u
u
we obtain the undamped random linear oscillator u
tu
whose rst order form is
zt
t
ztz
u
u
with angular part
t
sin t
tcost
Explosion of
is equivalent to crossing the angles It follows
that E for
andthatE fg
If we assume that f where is an elliptic di usion process on a
compact manifold and f is smooth and not constant we can use the results
of Arnold Kliemann and Oeljeklaus Example
saying that
has a simple spectrum
Then
Eddfor
where
d tan
and are the angles of the two di erent Oseledets spaces of
The
two ergodic invariant measures of
for
are
d
with
E tan
Since d
as
we can say that
exhibits a stochastic saddle
node bifurcation with the familiar bifurcation diagram
1

NOISY DUFFING VAN DER POL OSCILLATOR
Discrete Time
In discrete time T Z the rst task should be to study the e ect of noise
on the bifurcation scenarios of the families of di erence equations xn
xn where
xx
x xN in particular N transcritical
N pitchfork
x
x x x period doubling and x
x
x saddle node Obvious choices would be additive noise or
parametric noise in the sense that the deterministic bifurcation parameter
is replaced by
n
n a zero mean stationary sequence Unfortunately
there is no explicit expression for the nth iterate of such a mapping here
There is an abundance of case studies on the e ect of parametric noise
on the logistic map i e of the random di erence equation
xn
rnxn
xn
where rn is a stationary sequence of which we only quote the following
In a survey by Lucke Chap VI the case rn r
n with small
is studied around r deterministic transcritical bifurcation of a new
xed point x
r
r and r period doubling bifurcation of a limit
cycle from the xed point x Lucke gives physicist s style expansions of
all interesting quantities in powers of the noise intensity Additive noise
on the logistic map is also discussed
Hess Kiwi Markus and Rossler
study the case where rn is a two
state Markov chain or a periodic function with states A and B and explore
the incredibly complex behavior in the A B plane See also Lasota and
Mackey Chap and the references therein
Dimension Two The Noisy Du ng van
der Pol Oscillator
Introduction Completeness Linearization
The deterministic nonlinear Du ng van der Pol equation
yyyyyyy
R
has become a paradigm for mathematicians physicists and engineers
There are numerous physical and engineering problems whose dynamics
are described by
for some parameter values some of these problems
have been listed in
In particular for
we obtain the van der Pol equation and for
we obtain the Du ng equation
1

CHAPTER BIFURCATION THEORY
The bifurcation behavior of this equation with parameter
R
which exhibits pitchfork Hopf and global bifurcations was investigated by
Holmes and Rand
By putting for simplicity
and
i e by considering
yyyyyy
R
and perturbing the remaining parameters and the right hand side of
by real or white noise we arrive at the noisy more speci cally the random
for real noise or stochastic for white noise Du ng van der Pol equation
respectively
y
ty
tyyyy
t
where
and are intensity parameters t
tttis
a stationary stochastic process or white noise and
R are bifurcation
parameters
As a rst order system for x x
x
y
y
takes the form
x
x
x
tx
txxxx
t
The bifurcation behavior of this and related equations has received consid
erable attention since the early s Of the earlier work on P bifurcation
we mention just Ebeling Herzel Richert and Schimansky Geier for
the case
and white noise and the references therein and for
more recent reviews Arnold Sri Namachchivaya and Schenk Hoppe
and Schenk Hoppe
Strict Completeness of the Noisy Du ng van der Pol Equation
We rst will study the possible explosive behavior of the local RDS
generated by
and recall the notion of strict forward backward
completeness from De nition
We take the following results from
Schenk Hoppe
Proposition Random Du ng van der Pol Equation Sup
pose the stationary stochastic process t
t t t has locally
integrable trajectories Then the random Du ng van der Pol equation
generates a local C RDS which has the following properties
i is strictly forward complete for any
ieDt
Rforallt
ii Assume there exist constants c c
such that
PAPfjtjjtjjtjctcg
1

NOISY DUFFING VAN DER POL OSCILLATOR
Then for arbitrary and is not backward complete i e R t
R
for all t
Proof i The idea of the pathwise proof is as follows For a given RDE
of the type xt f t xt with locally integrable
wetryto nd
a Lyapunov function V Rd R such that V x
as kxk and
for which
dV xt
dthDVxtft xticcktkcVxt
with constants c c c
Then the Gronwall Bellman Lemma B
applied to the integrated form of the last inequality in the time interval
T yields that xt exists up to the arbitrary time T thus establishing
strict forward completeness
In our case the Lyapunov function V x x
x can be used The
details are left to the reader
ii The reasoning is based on the following deterministic argument
Proposition in
Given the di erential equation xt f t xt
in the sense of Caratheodory possessing a unique local continuous ow
assume that there exists an unbounded domain G Rd and a function
V C G R such that a G is forward invariant under the local ow
andbLVxt hDVxftxi forallx Gandt
Then
xVx
for all x G where x is the explosion time of the
solution with initial value x at t
Indeed de ne n infft kxtk ng Then n
andbyb
Vxtn VxZtnLVxssds t n
Letting n
the positivity of V gives V x
t forallt
hence V x
We apply this result to the random Du ng van der Pol equation with
reversed time t replaced by t as follows Choose G fx R x
cx xgwhere
isarbitrarybutxedc cc c
is
su ciently large and V x
x The result is that
Pf x cgPA
forallx G
The details are once again left to the reader
Proposition Stochastic Du ng van der Pol Equation
The stochastic Du ng van der Pol equation
dx
xdt
dx
x
xxxxdt
x dWt
x dWt
dWt

CHAPTER BIFURCATION THEORY
generates a local C RDS which has the following properties
i is forward complete for any
ii If
or if
then for any value of the remaining
parameters is strictly forward complete but is not backward complete
Proof i We apply the following generator condition of Schenk Hoppe
Theorem for the completeness of a dissipative second order SDE
Consider a d dimensional second order stochastic Ito di erential equation
formally written as
yfyy gyyW
where f Rd Rd Rd and g Rd Rd Rd m are locally Lipschitz con
tinuous and W is an m dimensional Brownian motion We rewrite
asa ddimensionalItoSDEforx x x yy as
dx
xdt
dx fxdt gxdWt
which has the in nitesimal generator
L thx xihfx xi dX
ijgxgxijxixj
LetVRdRd R Vx x Ex x hkxkbeaCfunction
whereEx x
forallx x Rd Rdandhisanonnegative
increasing function with h t
ast
Assume that there exist
constants c c c
and a non negative increasing function h R
R suchthatforall x x Rd Rd
LVx x
ccVxx
chkxk
and that for all su ciently large xed c
lim
t hct
ht
Then the maximal solution of
is complete
The proof of this theorem consists in applying Dynkin s formula and
theestimateforLVtoVxt n xt n where n infft
kx t k ng and then the Gronwall Bellman Lemma to arrive at
limn Pfn Tg forallT
We apply this theorem to our SDE
whose Ito form is obtained
by replacing by
WechooseEx x
xhkxk kxk
1

NOISY DUFFING VAN DER POL OSCILLATOR
h
Then the estimate for LV is satis ed for c j j
c jjjj
and c
The details are left to the reader
ii The proof of strict forward completeness is accomplished by con
verting the SDE into an RDE and then by proving the strict forward com
pleteness for the RDE as follows
Consider the case
Then the Stratonovich SDE
can be
written as a Lienard equation
dx
x
x
xdt
dx
xxdtxdWt
dWt
wherey x andy x
x
x whose Ito Stratonovich correc
tion term is zero Taking
for simplicity and applying the
transformation
zt xtxtWtWt
we convert
into the pathwise nonautonomous ordinary di erential
equation
x
zxWtWt
x
x
z
xx
zxWtWt
x
xWt
We apply the idea of the proof of Proposition i to
using the
Lyapunov function V x z x
z Details are again left to the reader
Similarly if
by using the transformation z t
exp Wt x t theSDE
can be converted into an RDE whose
strict forward completeness can be checked
We nally have to prove that
is not backward complete This
can be seen for the case
by reversing time in
and applying
the proof of Proposition ii Here
PA PfjWtjjWtjcforallt cg
holds for any c c
GfxzRx czxg
where
andcisbigenoughandVx z
x The result is that
Pfc
xg PA
forallx x z G

CHAPTER BIFURCATION THEORY
Remark i It is an open problem as to whether or not the general
stochastic Du ng van der Pol equation is strictly forward complete
ii Noisy Du ng equation By omitting the term y y in equation
we obtain the random stochastic Du ng equation
y
ty
tyy
t
In the random case if
are locally integrable then equation
is strictly forward complete and strictly backward complete for all param
eter values
In the stochastic case
is forward and backward complete for
arbitrary parameter values Moreover it is strictly forward complete and
strictly backward complete if
or if
iii Noisy van der Pol equation By omitting the term y in equation
we obtain the random stochastic van der Pol equation
y
ty
tyyy
t
In the random case if
are locally integrable then equation
is strictly forward complete but not backward complete for all parameter
values
In the stochastic case
is forward complete for arbitrary param
eter values Moreover it is strictly forward complete but not backward
complete if
or if
Assumption To reduce complexity we will henceforth treat only the case
and write
assumed locally integrable and
W W Thus for the remainder of this section our equations become
x
xxx
x
t
x
for the real noise case and
dx
xxxdt
xdt
x dWt
for the white noise case
Both systems generate a local C RDS which is strictly forward
complete but not backward complete
1

NOISY DUFFING VAN DER POL OSCILLATOR
Linear Analysis
The linearized RDS D is generated by the linearization of the above equa
tions namely in the real noise case by
v
x
xx
x
v
v
t
v
and in the white noise case by
dv
x
xx
x
vdt
vdt
v dWt
For any invariant measure the trace formula gives
E x In particular for
we obtain
We also need the eigenvalues of the linearization of the deterministic
system
atx
vt
vt
which are
r
For
we have two real eigenvalues while for
we have
a pair of complex conjugate eigenvalues
id
dr
where d represents the damped eigenfrequency of the linear system y
yy
We recall the following well known facts for the deterministic Du ng
van der Pol equation
For the frozen parameter
and the bifurcation parameter
increasing across
the trivial solution x
undergoes a Hopf bi
furcation at
with a stable limit cycle for
bifurcating out of
x For the frozen parameter
and the bifurcation parameter
increasing across
it undergoes a pitchfork bifurcation at
with
two stable steady states p for
bifurcating out of x
In the remaining part of this section we want to study the changes
occuring in these scenarios when the noise is switched on
1

CHAPTER BIFURCATION THEORY
Hopf Bifurcation
What is Stochastic Hopf Bifurcation
Consider for example equation
for
frozen and with in
creasing across
On the phenomenological level of the Fokker Planck equation computer
simulations Schenk Hoppe
show that for su ciently negative the
Dirac measure is the only stationary measure Increasing we witness
the premature birth of a nontrivial stationary density p at
D
This density has a pole at x
If we increase even further this den
sity undergoes a P bifurcation at
P where the pole disappears and
from which on it becomes crater like the parameter interval D P was
called bifurcation interval by Ebeling et al see Fig for which
D
and P
Figure Stationary density in the Hopf regime for
and
As will be explained in Subsect
P is that parameter value for
which the moment Lyapunov exponent
gp lim
t tlogEk t vkp
has its second zero at p
dim R By Proposition
applied to
our case
P see also Example
What happens on the dynamical level Extensive simulations of the
RDS its random attractors and invariant measures
lead us to
believe in the following D Hopf bifurcation scenario
It is a general observation that noise splits deterministic multiplicities
of eigenvalues For
which we assume and
the determin
istic linear system has two complex conjugate eigenvalues i d which
amounts to just one Lyapunov exponent
with multiplicity
For
however the linearized SDE
dvt
vt dt
vt dWt
has two di erent simple Lyapunov exponents i see Arnold Oel
jeklaus and Pardoux
which satisfy
For small
we can use the asymptotic expansions given by Pardoux and Wihstutz
Theorem
and others namely
Theorem Small Noise Expansion of Lyapunov Exponent
1

NOISY DUFFING VAN DER POL OSCILLATOR
Case of Complex Eigenvalues Let
dxt
ab
baxtdtmX
j Ajxt dWjt
whereab Rb andAj ajkl j
m Then as
the
top Lyapunov exponent
satis es
a
mX
jajaj
aj aj
O
and the rotation number satis es
bcO
where c is some constant such that bc
For our case this yields
dO
and for the rotation number
dc
dO
for some c
For the real noise case a similar expansion holds For example if t
f t where t is an elliptic di usion on a compact manifold and f is
smooth and not constant then the spectrum of the linear RDE is simple
Arnold Kliemann and Oeljeklaus
and for E
dSdO
where S is the spectral density of t Ariaratnam and Sri Na
machchivaya
Expression
matches the result
for the
white noise case where S u
Furthermore the rotation number ex
pands as
dc
dTdO
where T is the sine spectral density of t see Example
1

CHAPTER BIFURCATION THEORY
Consequently at the deterministic bifurcation point
we have for
the white noise case neglecting O terms
d
d
ieat
the top exponent has already crossed and is positive
It follows from
that the top Lyapunov exponent changes sign
atD
O
and the second Lyapunov exponent changes sign
atD
O
For
D is stable it is the unique invariant measure and
A fgisthe attractor oftheRDS inR
We have a rst bifurcation from at
D
of a stable ergodic
measure
x
x
which is a convex combination
of two Dirac measures sitting on the two boundary points of the one
dimensional unstable manifold of x
which is a saddle point This situ
ation persists for
D D Asshownabove at P
DD
undergoes a P bifurcation
and are the only invariant measures
and both are Markov measures Hence E solves the Fokker Planck
equation The attractor A in the universe of tempered sets of R is
the closure of the unstable manifold of x
the two boundary points
supporting the measure In the punctured plane R n f g the attrac
tor A in the universe of simply connected tempered sets consists of the
two point set supp and A A where here means the boundary of
a submanifold
At
D
we have a second bifurcation from of a measure
x
x
which is again a convex combination of
two Dirac measures is a saddle point i e has a positive and a negative
Lyapunov exponent while has two positive exponents
For
D the stable measure is supported by the boundary of the
unstable manifold of The closure of this unstable manifold is an invari
ant topological circle around x random limit cycle and supports
both measures
and On this circle we have hyperbolic dynamics
is attracting and is repelling in contrast to the deterministic case
where the dynamics on the limit cycle is just rotation and the invariant
measure is unique The interior of the circle is the two dimensional unsta
ble manifold of Its closure is the attractor A in R It carries all three
invariant measures In the punctured plane R n f g however the attrac
tor A is the invariant circle on which we have two invariant measures in
particular the unique Markov measure Again A A See Fig
where
and
1

NOISY DUFFING VAN DER POL OSCILLATOR
Figure Lyapunov exponents invariant measures and attractors in the
Hopf regime
The next four Figs
to
show the deformation of a grid of
points discrete Lebesgue measure by the mapping x
t
x
t txast with
and time increment
Figure Supports of invariant measures and attractors in the Hopf
regime for
D
Figure
Supports of invariant measures and attractors in the Hopf
regime for D
P
Figure
Supports of invariant measures and attractors in the Hopf
regime for
P
Figure
Supports of invariant measures and attractors in the Hopf
regime for
D
The whole scenario see Fig
has been very well established nu
merically for real and white noise Schenk Hoppe
Arnold
Sri Namachchivaya and Schenk Hoppe
The task is now to prove it
mathematically but so far this has been only partly accomplished
Figure
The full stochastic Hopf bifurcation diagram
Attractors and Invariant Measures
We rst present a general theorem on the existence of a random attractor
in the real noise case obtained by Schenk Hoppe
Theorem Existence of Attractor Real Noise Case Con
sider the local C RDS generated by the random Du ng van der Pol
equation
where m Ej t j Then for arbitrary parameter
values
R and j j m possesses a random attractor A in
R in the universe of all families C
of subsets of R which are
tempered from above Moreover the attractor is measurable with respect to
the past F and supports all invariant measures
Proof By Proposition
all we have to construct is an absorbing
set which is tempered from above We follow the procedure described and
carried out in
1

CHAPTER BIFURCATION THEORY
i We rst nd a Lyapunov function V R R which is con
tinuous surjective has the property that pre images of bounded sets are
bounded and satis es a di erential inequality of the form
dVtx
dt
tVtx
t
where E t
andEj tj
In our case we can use
Vx
xxxx
x
After some elementary estimates exercise we obtain that
dVtx
dt
jjjtjV t x
c
where c c
is a positive constant
ii The a ne RDS t generated by the RDE
zt
jjjtjzt c
associated with
satis es
Vtx
tVx
and has a unique stationary solution with initial value
r
cZetRt jjj sjdsdt
which is exponentially stable hence is the attractor of in the universe of
sets in R or R which are tempered from above Further for any
the set
r is absorbing exercise for in this universe
iii We now lift these ndings to the original RDS By our assump
tions on V ful lled for our particular choice
B
V
r
is a compact random set which is absorbing in the universe of pre images
D V D of sets D in R tempered from above This universe con
tains in particular all sets in R tempered from above since their image
under the polynomial V is tempered from above
Hence we can apply Proposition
and the proof is complete

NOISY DUFFING VAN DER POL OSCILLATOR
Remark i The proof furnishes the estimate
AV
r
ii There is so far no analogue of the above theorem for the white noise
case The existence of an attractor was established for the purely additive
white noise case
of
by Schenk Hoppe
Remark Invariant Measures Under the conditions of The
orem
all invariant measures are supported by A There exists at
least one invariant measure supported by A Theorem
and since A
is F measurable there is at least one invariant forward Markov measure
supported by A Theorem
Further since is a local homeomorphism the boundary A of A
is a random see Schenk Hoppe Lemma
compact invariant set
which also supports an invariant measure Since A is F measurable it
also supports an invariant forward Markov measure
We now give partial answers to the question of the structure of A in the
rst regime
D and in the third regime
D again taken from
Unfortunately as yet there are no rigorous results for the intermediate
regime D
D
Theorem Attractor for
D Consider the local C RDS
generated by the random Du ng van der Pol equation
where
m Ej t j Take for simplicity
Then for arbitrary param
eter values
and which satisfy
jj jjm jj
jj
the attractor A of in the universe of all subsets of R tempered from
above is A f g and is the unique invariant measure of
Moreover all orbits t t x and t x tend to zero exponen
tially fast as t
Proof i We need the extra condition on and for bounds on the
Lyapunov exponents
of the linearized RDE
vt
t
vt
1

CHAPTER BIFURCATION THEORY
which is the random harmonic oscillator This becomes clear through the
estimates
where
mjj jj
jj
which hold for any
and
To prove
we use the Lyapunov function V v v
vvv
for which V v
since
andVv
ifandonlyifv
This yields
dVvt
dt
Vvt
vtvt vt t
Since
jjkvk Vv jjkvk
andjvv
vj jjkvkweobtain
mjtjVvt dVvt
dt
mjtjVvt
Comparing V v t with the solutions of the exponentially stable linear
RDE dominating from above and below and using
gives
lim inf
t tlogkvtk limsup
t tlogkvtk
for any random possibly non adapted initial value v This proves
ii We now prove that the attractor which exists by Theorem
is equal to A f g By step i and conditions
both Lyapunov
exponents of the invariant measure are negative
Now we use the non negative Lyapunov function
Vx
x
x
xx
x
for the nonlinear equation which yields
dV xt
dt
mjtjVxt
We gave conditions under which
at the beginning of this subsection see
1

NOISY DUFFING VAN DER POL OSCILLATOR
since V x
jjkxk for
Comparing with the associated linear RDE we see that x is expo
nentially stable with rate
It is much harder to show that in the third regime
D there exists
an attractor which does not contain x
Theorem Attractor for
D Consider the local C RDS
generated by the random Du ng van der Pol equation
where
m Ej t j Take for simplicity
Then for arbitrary param
eter values
and which satisfy
jjm
there exists an attractor A for restricted to the punctured plane R nf g
in the universe of all tempered subsets of R n f g
Furthermore A supports all invariant measures of in R n f g
A family C
of subsets of Rd nf g is called tempered if it is contained
in an annulusfx Rdnfg r kxk R gsuchthatristempered
from below and R are tempered from above
Proof In order to avoid unnecessary repetition we will just sketch the
proof and refer the reader to
for details
i By using an appropriate Lyapunov function V and comparing with
a dominating exponentially stable a ne RDE we obtain an absorbing set
BV
R tempered from above This ensures the existence
of an attractor in the sets of R tempered from above
ii We now use another Lyapunov function V and apply the procedure
ofstepitoWxt where
W V RnfgRnfg
We arrive at an exponentially stable a ne RDE dominating W xt hence
at a non compact forward invariant set for the original V xt given by
R which absorbs all sets tempered from below Hence B
V
R
is an absorbing set for all sets tempered from below
iii Finally the intersection B B B is a non empty absorbing set
for all sets in R n f g tempered from below and from above That B is
non void follows from the fact that both B and B absorb xed points
By comparing the Lyapunov functions V and V it can be shown that
there are values of
for which B is C di eomorphic to a
random annulus around

CHAPTER BIFURCATION THEORY
Normal Form Real Noise Case
Applying the linear transformation
T
d
d
dr
to
linearizing at x
and writing the linearized RDE in polar
coordinates v r cos v r sin gives
rt
dsinttrt
t
d
d
costt
We immediately obtain the estimates
mj jd
mjjd m Ejtj
which for
are sharper than the estimate
In Subsect
we calculated the truncated normal form of the ran
dom Du ng van der Pol equation
for xed
and small x
in polar coordinates
In order to obtain a more tractable simpler RDE we omit further higher
order terms from the normal form of Subsect
and arrive at the sim
pli ed normal form equations
rt
dsinttrt
rt
t
d
d
costt
drt
These equations were derived by a scaling argument in
where extensive
numerical simulations were also made It was shown that the bifurcation
behavior of the simpli ed normal form equations
and
is
qualitatively the same as that of the full equation
To demonstrate that equations
and
are indeed more
tractable than the original equations after all they di er from the lin
earized equations
by only one term we study their attractor
We write
t
dj tj
and note that by
ifE
and
ifE
The radial part obviously satis es the estimates
trt
rt rt
trt
rt
1

NOISY DUFFING VAN DER POL OSCILLATOR
Using the right hand side it follows from Subsect
thatfgisthe
attractor provided E
which by
assures that both Lyapunov
exponents of are less than or equal to zero
Further the stochastic process y r satis es the RDE y
rr
which yields
tyt
yt
tyt
Solving the corresponding dominating a ne RDE and transforming back
to r we see that the attractor A of the normal form
and
is contained in the compact forward invariant set
rR eRt
s dsdt kxk rR eRt
s dsdt
where
and the right hand side respectively the left hand side of
this estimate is omitted if E
respectively if E
This leads to the following theorem see Schenk Hoppe Sect
Theorem Attractor for Normal Form Suppose L
Then the following holds
i The truncated normal form equations
and
generate
a local C RDS which is strictly forward complete
ii has a random attractor A which attracts all families of subsets
of R Further A E E the set of never exploding initial values of
and A supports all invariant measures We have
AfgE
BE
where B is the random ball with radius de ned by the right hand side of
Moreover for E
the upper bound on the attractor tends to
zero iii If E
the RDS has an attractor A in the subsets of the
punctured plane R n f g which are tempered from below The attractor is
contained in the annulus de ned by
andA A
Proof i Strict forward completeness can be proved as above using the
Lyapunov function V r r
ii The ball whose radius is the right hand side of
is tempered
from above and attracts all families tempered from above Hence there is
a unique attractor which is tempered from above
To prove that A E it su ces to show that E belongs to the
corresponding universe i e is tempered from above Due to strict forward

CHAPTER BIFURCATION THEORY
completeness E
E
t Dt Usingr jjt r
r we see that E is contained in the set of never exploding initial values
of the dominating RDE The latter set hence the former is tempered from
above
It follows that A E attracts R hence all other families of subsets
of Riii The annulus
is an absorbing set which is tempered in the
punctured plane R n f g
Computer simulations show that the RDS behaves exactly as described
at the beginning of Subsect
The attractor is A f g for
it is a one dimensional set if
and nally for
A is a topological random ball around x
while A is a topological
circle with A A We are however not able to prove this yet
Pitchfork Bifurcation
What is Stochastic Pitchfork Bifurcation
Consider now for example the SDE
for
frozen while
moves up across
Denote the corresponding RDS by
On the phenomenological level of the Fokker Planck equation computer
simulations
show that for up to and to some extent beyond
the Dirac measure is the only stationary measure Increasing well
beyond we witness the delayed birth of a non trivial stationary density
pat
D
This density has a pole at x which persists for
all
D and two pronounced peaks shifted away from as increases
see Fig
for which D
Figure
Stationary density in the pitchfork regime for
and
On the dynamical level the trivial measure is the only invariant mea
sure for
D andA fgistheattractoroftheRDS inR
loses its stability at
D where the top Lyapunov exponent
becomes positive and we witness the bifurcation of one
new ergodic
invariant measure
x
x
which is stable It is supported
by the two boundary points of the one dimensional unstable manifold of
x
which with its boundary forms the attractor A in R The at
tractor in the punctured plane R nf g in the universe of simply connected
tempered sets is A f x g and A A where here means

NOISY DUFFING VAN DER POL OSCILLATOR
the boundary of a submanifold see Fig
where
and
Figure
Lyapunov exponents invariant measures and attractors in the
pitchfork regime
The deformation of a grid of points by x
t
ast
is
shown in Figs
and with
and time increment
Figure Supports of invariant measures and attractors in the pitchfork
regime for
D
Figure Supports of invariant measures and attractors in the pitchfork
regime for
D
Why does only one measure bifurcate in contrast to the bifurcation of
two new steady states p in the deterministic case The intuitive
reason is as follows Assume
so that the eigenvalues of the
deterministic system are real and distinct It might however happen that
t
with positive probability which is always the case for
white noise t Wt irrespective of the size of This means that the
noisy system is pushed into parameter regions where there are complex
conjugate eigenvalues hence it picks up rotation which makes it impossible
to distinguish between x and x This will not occur if we assume
that the real noise t satis es t
in which case we have
the bifurcation of two new stable Dirac measures x This phenomenon
demonstrates again that stochastic bifurcation is typically not local with
respect to parameters
What can we say about the location of D The two di erent Lyapunov
exponents of the linearized SDE
satisfy
Their second order asymptotic expansion for
which we assume
for small was obtained by Pardoux and Wihstutz Theorem
and
others
Theorem Small Noise Expansion of Lyapunov Exponent
Case of Real Eigenvalues Let
dxt
a
a
xtdt mX
j Ajxt dWjt
where a
a andAj ajkl j
m Then as
the top

CHAPTER BIFURCATION THEORY
Lyapunov exponent
satis es
a
mX
jajajO
In our case r
O
For the real noise case a similar expansion holds For example if t
f t where t is an elliptic di usion on a compact manifold and f is
not constant then the spectrum is simple and for E
r Shypp
O
whereShyp a R Cte atdt andCt E t isthecorrelation
function of t see Lucke and Schank
and others
Note that
with F denoting the spectral distribution of t
ZCte atdt aZa dF
matches the white noise result
where Shyp a
Consequently at the deterministic bifurcation point
we still have
for the white noise case
O
and the top Lya
punov exponent changes sign at
D
O
while the second exponent remains always less than
We now search for a P bifurcation point P of p i e for a parameter
value at which for the moment Lyapunov exponent g
Using
Proposition see Example
such a value does not exist since
g Unfortunately not much of the above numerically observed bifurcation
scenario has been rigorously proved
Normal Forms Real Noise
In Subsect
we have derived the truncated scalar center RDE for the
pitchfork scenario under small real noise perturbations for the parameter

NOISY DUFFING VAN DER POL OSCILLATOR
value
In a new coordinate system again denoted by x x
given by the eigenvectors v
andv p
corresponding
to the eigenvalues and of the deterministic linearized equation at the
bifurcation point
the truncated RDE for the center variable z x
is
zt
t
t
tzt
tzt
we have skipped for simplicity the
and terms in the coe cient of
z where L we want to use spectral theory of stationary processes
and E
tptttpttt
tpt
and
tpetZt
es sds
The scalar RDE
can now be utilized to study the bifurcation be
havior of the original two dimensional random Du ng van der Pol equation
We rst calculate the Lyapunov exponent c
of
for
We have
E
Shyp ShypaZCteatdt Ct Et
E
and E
Hence
c
Shyp hot
which completely agrees with the expansion of
given in
for
In particular we recover c
ifD Shyp O
The nonlinear scalar equation
has the general structure zt
t zt t zt hence can be explicitly solved see Subsect for details
In particular assume j t j M Then all the i are also bounded
and there is a constant C such that whenever j j j j C
t
so that the original system cannot pick up rotation and
t
t so that the RDE
is strictly forward
complete Then we have the stochastic pitchfork scenario for
In
particular for
D there are two new stable Dirac measures
z By a persistence argument the original random Du ng van der
Pol equation undergoes a stochastic pitchfork bifurcation at D and the
new Dirac measures bifurcating out of x
are supported by points x
which can be represented in the deterministic x x
xc xs coordinate
system as
x
z
mcz
where mc is the center manifold determined in Subsect
A treatment
of this situation including computer simulations was done by Xu
1

CHAPTER BIFURCATION THEORY
Normal Forms White Noise
In Example
we have derived the truncated scalar center SDE
for the pitchfork scenario of the stochastic Du ng van der Pol equation for
and small and In the coordinate system described in the real
noise case after dropping the
and terms in the z coe cient we
obtain
dzt
Ut ztdt
Ut zt dWt
Ut ztdt zt dWt
Here Ut is the stationary Ornstein Uhlenbeck process solving dUt
Ut dt dWt which is adapted
Equation
can now be utilized for stochastic bifurcation theory
of the original two dimensional SDE
Linearizing the SDE
at z gives
dvt
Ut vtdt
Ut vt dWt
which yields the Lyapunov exponent
c
lim
t tZtUs dWs
where we have used
ZtUs dWs ZtUsdWs t and lim
t tZtUsdWs
This is in full agreement with the asymptotic formula
for the top
Lyapunov exponent
In particular c
ifD
O
The SDE
can be explicitly solved by converting it by means of
y
z into an a ne SDE see Subsect
However the presence of
the term z dW entails that all forward orbits explode and E f g
We believe that this re ects the inevitable presence of rotations
The Noisy Du ng Equation
Beginning already at an early stage of the subject great e orts were devoted
to the study of the bifurcation behavior of the parameter driven damped
anharmonic oscillator also known as the Du ng equation
y
tyyy
which is
for the particular case
and t
t
locally integrable real or white noise
1

GENERAL DIMENSION FURTHER STUDIES
By Remark ii the RDS generated by
is global in both
cases for all parameter values
For
frozen
and the bifurcation parameter
undergoes a pitchfork bifurcation at
where the two new steady states
p bifurcate out of x For
the linear analysis is the same
as for the noisy Du ng van der Pol equation Physicist s style small noise
expansions for the stability threshold D the bifurcating solutions
and their moments are given by Lucke and Schank
and Lucke
Ariaratnam and Xie give for the white noise case an exact formula in
terms of the stationary measure on S and a small noise expansion of the
top Lyapunov exponent and investigate for
D the bifurcated solution
of the Fokker Planck equation by the method of stochastic averaging
The top Lyapunov exponent as a function of and for
and
the bifurcation diagram with respect to the parameter was numerically
calculated for the white noise case by Wedig
and investigated
analytically by Baxendale Example
General Dimension Further Studies
Baxendale s Su cient Conditions for
D Bifurcation and Associated P Bifurcation
We will now present a general criterion for a D bifurcation of a branch of
forward Markov measures out of the xed point for a family of SDE in Rd
and for a subsequent P bifurcation of the new branch in a neighborhood
of These results were obtained by Baxendale in two profound papers
on which this subsection is based As the matter is very technical
and since most proofs in the above papers are based on earlier results from
and we have decided not to reproduce the proofs here but rather
to give a complete and self contained formulation of the results with some
intuitive explanations
Let
dxt
mX
jfjxtdWjtfxt
mX
jDfjfjxtdt mX
jfjxtdWjt
dWt dt be a family of SDE in Rd with R which is only chosen to
simplify presentation all results are true for in an open set of some Rk
forwhich x fj x isC andwhereDfgx Pd
ifixg
xix
Hence
generates for each a local C RDS
1

CHAPTER BIFURCATION THEORY
The associated possibly explosive Markov di usion process xt x
t x t R inRdhasgenerator
LfmX
jfj
dX
ibi
xi
dX
kl akl
xk xl
where
bixfxi
mX
j
dX
kfjxkfji
xkx
and
akl x
mX
jfjxkfjxl
We now assume that
ff
fm
for all R
so that
is an invariant measure for for all R This branch of
trivial measures will serve as a reference branch from which we will study
the bifurcation of a new branch of measures supported by Rd n f g at
the parameter value D at which the reference measure becomes unstable
To this end we linearize the SDE
atx
and obtain the linear
SDE
dvt A vtdt mX
j Ajvt dWjt A
mX
jAjvtdtmX
j AjvtdWjt
where Aj Dfj is the Jacobian of fj at x
The SDE generates
the linear RDS t
Dt
The di usion process vt v
t v t R hasgeneratorDL
say given by
DL dX
i
dX
k
bi
xkvkvi
dX
kl
dX
pq
akl
xpxq vpvq vkvl
Note that DL only depends on the values of the coe cients a and b
of L hence on the values of the vector elds fj in an arbitrarily small
neighborhood of x
We refer to Sect for a detailed treatment of In particular the
MET holds for
without further integrability conditions and under the
Lie algebra or hypoellipticity condition
dimLAh hm s d foralls Pd
1

GENERAL DIMENSION FURTHER STUDIES
where hj s Aj s hAj s sis on P d are the projected vector elds
the top Lyapunov exponent
satis es
lim
t tlogk t vkforallv Pas
and the moment Lyapunov exponent
gp lim
ttlogEktvkppR
is well de ned i e the limit exists and is independent of v
It is
well known that g is a convex analytic function of p satisfying g
and dg dp
see Fig
Clearly and g p control the
P a s and p th moment stability of the linearized RDS In addition
g is the Legendre transform of the rate function for large deviations of
t log k t vk away from see Arnold Oeljeklaus and Pardoux
Arnold and Kliemann Stroock
Baxendale
and Baxendale
and Stroock
It turns out that and g also control the behavior near of the
nonlinear process t x For the following theorem see Baxendale
Theorem
Theorem Assume that for some xed R the following con
ditions on the SDE
are satis ed
H Condition at in nity There exist functions f g C Rd with
g
positive constants c and R and a neighborhood N N of
such that for each N the Markov process t x t is complete
andthere existsf C Rd satisfying f f Lf g c and
Lfx gx forkxkR
H Condition between in nity and zero For all r
and x
there exists T
suchthatP TxBr
where P denotes the
Markov transition probability with generator L and Br fx Rd
kxk rg
H Condition at zero Let LA A
Am v Rdforallv
Then the following hold
o There exists an open interval U
such that for all
U and g are well de ned through
and
is
continuous in U and g p is continuous in U for each p R
iIf
for some U then the local RDS satis es
P lim sup
t tlogktxk
for all x
For a weaker version of condition H see Baxendale Sect
1

CHAPTER BIFURCATION THEORY
In particular there is no other stationary measure in Rd nf g for the Markov
process generated by L and the stable manifold of x is dense in Rd
ii If
for some U then there exists a unique probability
measure on Rd n f g such that
P
lim
T TZT
u t xdtZRdnfguy dy
for all bounded measurable functions u Rd n f g R and all x
In
particular is the unique stationary measure on Rd n f g for the Markov
process generated by L
Finally there exists a unique
such that g
and
constants
and K
for which
Kr
Brnfg Kr for
r
iii If
for some U then there exists a nite but not
nite measure on Rd n f g unique up to a multiplicative constant such
that
P
lim
TRTutxdt
RTv t xdt
RRdnfguy dy
RRdnfgv y dy
for all bounded measurable integrable functions u v Rd n f g R with
Rvd
and for all x
In particular is the unique up to
a multiplicative constant stationary measure on Rd n f g for the Markov
process generated by L Moreover Rd n Br for all r
and
there exists a C
such that
lim
r
RdnBr
jlogrj C
Figure
The p th moment Lyapunov exponent
Suppose that in addition to the assumptions H H and H the
SDE
is non degenerate in Rd n f g for example in the sense that
dimLAf fm x d forallx
R
Then Theorem
gives a complete classi cation of the di usion process
t x on Rd n f g as positive recurrent null recurrent and transient
according to being positive zero or negative
For d the statements of the theorem apply to
restricted to one
of the invariant sets
and
1

GENERAL DIMENSION FURTHER STUDIES
D Bifurcation
We will now specify a certain scenario in which actually bifurcates out
of the trivial reference measure The following is a reformulation of part of
Baxendale s Theorem in
Theorem Su cient Condition for D Bifurcation Suppose
that there exists an D R for which the conditions H H and H
of Theorem
are satis ed and that the function g in H satis es
gx
as kxk Then the following holds
Assume that D
and that there is an open interval
D
contained in the interval U of Theorem o such that
for
D and
for
D Then lim
D
and if
for
D denotes the unique stationary measure of Theorem ii
limD
D
in the topology of weak convergence of probability measures in Rd Hence the
family undergoes a D bifurcation of the new branch of forward Markov
measures from the branch of trivial measures
at the parameter
value D where the trivial measure loses its stability
Further for
D the nontrivial zero
of g guring in
Theorem iii ful lls
limD
Remark Assume the situation of the last theorem Why can the
generator L of the one point motions t x feel that v becomes
unstable for the linearized RDS at
D and give birth to a non
trivial solution of the Fokker Planck equation L
for
D
The principal reason is that the quantities
and g are determined by
the law of the one point motions t v which is determined by DL
given by
andhencebyL infactbythecoe cientsofL inan
arbitrarily small neighborhood of The smaller Lyapunov exponents are
however not determined by the one point motions of See Arnold
for a discussion of one point versus ow quantities
Remark Rate of Convergence to Dirac Measure Assume
the situation of Theorem
Baxendale Theorem
also deter
mined the rate at which approaches the Dirac measure in
Let
D denote the unique nite stationary measure of D on Rd n f g for
which the constant in
is
CD VD
whereVD dgD
dp
1

CHAPTER BIFURCATION THEORY
Then
limD
D
in the sense that
limDZux dxZux Ddx
for all continuous functions u Rd n f g R satisfying u x g x
as
kxk anduxkxkp
asx forsomep
This result incorporates the nite stationary measure D into the
bifurcation scenario at
D
It is still unknown whether the measures
D
are stable i e
have negative top Lyapunov exponent
P Bifurcation
Assume again the situation of Theorem
Then
holds for all
D We now compare the mass assigned by to Br n f g with
that assigned by the Lebesgue measure
LebBrnfg LebB rd
where Leb B is the volume of the unit ball B in Rd Theorem ii
yields possibly with a di erent constant K
Krd
Brnfg
LebBrnfg Kr dfor
r
implying that
lim
r
Brnfg
Leb Brnfg
ifd
ifd
while the ratio remains bounded away from and for
d Hence
Baxendale s result allows the following interpretation where we use the
term P bifurcation in the sense of the qualitative change of in a neigh
borhood of just described for a discussion of the concept see Subsect
Theorem Su cient Condition for P Bifurcation Assume
the situation of Theorem
Assume further that there exists an P
D forwhich P dbut
d for
P and
d for
P and let be the unique stationary measure of in Rd n f g
Then the family of undergoes a P bifurcation at
P in the sense
expressed by
1

GENERAL DIMENSION FURTHER STUDIES
Baxendale s ndings are collected in Fig
Figure
D bifurcation and P bifurcation
There is a simple criterion for g d
for which we recall the
following facts
Consider the linear SDE
dxt A xtdt mX
j Ajxt dWjt
inRdandassumethatdimLAh hm s d foralls Pd
where hj s Aj s hAs sis is the projection of the linear vector eld
x Aj x onto P d Then the p th moment Lyapunov exponent g p
limt
tlogEk t vkp p R is wellde nedandindependentofv
Further the determinant t
det t satis es the scalar SDE
d t traceA tdt mX
j traceAj t dWjt
which has the solution Liouville s equation
det t
exp traceA t mX
j traceAj Wjt A
and the p th moment Lyapunov exponent
hp lim
t tlogEjdet t jp traceA p p mX
j trace Aj
Proposition Criterion for g d
Let the above Lie alge
bra condition be satis ed Then
gdh
trace A
mX
j trace Aj
hence
gd
trace A
mX
j trace Aj
Proof For equation
see Baxendale and Stroock Corollary
i

CHAPTER BIFURCATION THEORY
Remark P Bifurcation on the Level of Densities Let
have a smooth density p in Rd n f g Then under certain circumstances
it follows from Theorem
that the family of densities p undergoes a P
bifurcation at
P in a neighborhood of in the sense of Subsect
For D
P p has an integrable pole at x p P approaches a
nite positive limit as x
and for P
limxpx see
Fig
Remark Local Dimension of at x
The qualitative
change of at P can also be predicted by looking at the local dimension
of atx denedby
lim
r log Brnfg
log r
and comparing it to the local dimension d of the Lebesgue measure
Remark Why is There a P Bifurcation at P Consider
the scenario assumed in Theorem
When passes through D from
left to right the top Lyapunov exponent of the reference measure
becomes positive hence x becomes repelling for t x Since
by condition H the Markov process cannot explode it is bound to build
up a stationary measure in Rd n f g For
D
small hence
small however the estimates show that t x still spends a
large amount of time in small punctured neighborhoods of hence the
mass of the occupation measure is large around compared to the mass of
the Lebesgue measure
Only if
P hence d delicate estimates show that the percent
age of time spent in a small punctured neighborhood of becomes small
compared to its volume and the new stationary measure of the nonlinear
RDS is nally freed from the control by the linear RDS
We nd it quite remarkable that the condition for the P bifurcation of
is a condition on the one point motions of the linear RDS obtained
by linearizing at x
The sequence of two bifurcations at
D and at some point
P
was observed by physicists and the interval D P was called bifurcation
interval see e g Ebeling et al
A su cient condition under which has a smooth density in Rd n f g is that L is
hypoelliptic which is implied by the Lie algebra condition dim LA f fm x d
for all x
1

GENERAL DIMENSION FURTHER STUDIES
Examples
Example Stochastic Pitchfork and Transcritical Bifurca
tion in Dimension One We can verify the assumptions and statements
of this subsection for the SDE
dxt
xt xtdt xtdWt
which is treated in great detail in Subsect
Let us only add that
gives P
and the nite measure at D
to which
q x dx converges in the sense of Remark as
has density
qx
xexp x
on
andqxqxas
for all x
with the analogous
situation on
Remember that for
the di usion is null recur
rent on
but nevertheless lim q
and f g is the attractor in
R For the analogous results for the transcritical case see Subsect
dxt
xt xtdt xtdWt
we restrict our considerations to the state space
Example Hopf and Pitchfork Bifurcation for the Stochas
tic Du ng van der Pol Equation We brie y return to the bifurcation
scenarios of the stochastic Du ng van der Pol equation dealt with at length
in Subsects Hopf bifurcation and pitchfork bifurcation Due
to the lack of rigorous results we have to rely on a small noise analysis of
to determine the rst and second D bifurcation point The point of
P bifurcation can however be calculated explicitly by means of
Since here m
trace A
and trace A
we obtain h p
p
and h
We thus have a P bifurcation at P in
the Hopf case and no P bifurcation in the neighborhood of x in the
pitchfork case since g
h
hence d
for
all values of
D
Example Averaged Du ng van der Pol Equation in the
Hopf Scenario Starting with the random Du ng van der Pol equation
with
xed and the bifurcation parameter assuming that
tsatisesE t
and certain strong mixing conditions see Khasminskii
or Freidlin and Wentzell Chap x scaling the state the
noise intensity and the dissipation as x
x
and
1

CHAPTER BIFURCATION THEORY
respectively the method of stochastic averaging states that the solution of
the so transformed original RDE
can be approximated in distribu
tion on a time interval of length O by the solution of an averaged SDE
which in polar coordinates is
drt
Rrt dt rt dWt
dt
a Srtdt
dWt
for details see Arnold Sri Namachchivaya and Schenk Hoppe
The
parameters here are
R
S
da
d
d
d
dS
dT
where the sine and cosine spectral densities are de ned respectively as
TzZCtsinztdt SzZCtcosztdt
C t E t being the covariance function of t The case a
S
and
was treated by Baxendale Example
The system
can be explicitly solved by rst solving
via the transformation u r and then inserting this solution into
This permits a complete explicit analysis of the averaged system
The only invariant measures are the following two Markov measures
iForall R
with Lyapunov exponents
andgp p p Hence
D
P
The rotation number is
a
ii For
D
drd
drd
where
qRRexp t Wtdt
1

GENERAL DIMENSION FURTHER STUDIES
which is the disintegration of the solution of the Fokker Planck equation
with density in Euclidean coordinates given by
pxx
Rxx
expRxx
The density has an integrable pole at x for D
P the nite
maximum R at the origin for P vanishes at zero and has a maximum
value away from zero for
P as predicted The nite stationary
measure at D has density
pDxx
xx
expRxx
The linearization of the averaged equations in R n f g R can also
be explicitly solved the details are left as an exercise see
This yields
the Lyapunov exponents
since angular distances are preserved
and
The overall picture is thus the following
i For
D all trajectories converge to with exponential speed
and rotation number a The attractor in R is f g For
D the
attractor is still f g
ii For
D the attractor in the punctured plane R n f g for
sets tempered from below is the random circle with radius
and the
rotation number of the nonlinear system is
lim
t
t
t
aS
R
this rate vanishes for
R
S a Further E
R The nal bifurcation
diagram for the averaged system is show in Fig
Figure
Bifurcation diagram of the averaged system
While the original random Du ng van der Pol equation in the Hopf
regime undergoes a sequence of two D bifurcations out of the trivial solu
tion at points where the two di erent Lyapunov exponents of change
their sign the averaged equation only has one D bifurcation since due to
averaging has a one point Lyapunov spectrum

CHAPTER BIFURCATION THEORY
Further Studies
The Stochastic Lorenz Equation
The deterministic Lorenz equation
x
sx sx
x
rxxxx
x
bx xx
in R with parameters s the Prandtl number r the Rayleigh
number and b a geometric factor is extremely well investigated see
e g Guckenheimer and Holmes
Sparrow
or almost any other
book on dynamical systems and the references therein The equation is
dissipative hence forward complete and has a global attractor in R which
for certain parameter values e g for s
r
and b
is
exceedingly complicated often called strange attractor
For s and b xed and r the bifurcation parameter we have the following
elementary local bifurcations in
For
r the origin is the global attractor
Atr
the origin undergoes a pitchfork bifurcation and two non
trivial xed points
x pbr pbr
r
x pbr pbr
r
are born
Atr ssb
s b weassumes
b the two nontrivial xed points
x become unstable and undergo a subcritical Hopf bifurcation
We now look at the case where the bifurcation parameter r is perturbed
by white noise say i e we study the SDE
dx
ss
r
b Axdt
xx
x xAdt Ax dWt
The bifurcation theory of this SDE was studied by Sri Namachchivaya
and Keller
Keller proved the existence of a stationary measure
and of a random attractor also for more general stochastic perturbations
and made extensive numerical studies
Let us calculate the stochastic D bifurcation point corresponding to the
deterministic pitchfork bifurcation point r The linearization of
atx is
dvt
ss
r
bAvtdt Avt dWt
1

GENERAL DIMENSION FURTHER STUDIES
The deterministic eigenvalues are a b and
a
sps
sr
The third equation in
is always stable and decoupled from the rst
two equations which after transformation of the drift matrix to diagonal
form reads
dut
a
a
utdt
s
aa
ut dWt
For small intensity parameter Theorem
yields for the top Lyapunov
exponent of
and of
r
a
s
s
s
rO
For r
s
s
i e the origin is still exponentially
stable and the pitchfork bifurcation is delayed to the parameter value rD
for which rD
which is using
rD
s
s
O
Since the moment Lyapunov exponent function g p of
satis es
g
trace A s
there will be no parameter value r
rD for which we have a P bifurcation of the new stationary measure in a
neighborhood of x
The Stochastic Brusselator
After the discovery of oscillatory chemical reactions by Belousov and
Zhabotinskii the Brussels school see Nicolis and Prigogine
Chap
proposed the following simplest possible di erential equation for the
concentrations x and x of reactants of a chemical reaction scheme which
undergoes a Hopf bifurcation
x
abxxx
x
bx xx
Here a b are parameters of which b is the bifurcation parameter This
model became known as the Brusselator and has been one of the paradigms
of nonequilibrium dynamics
1

CHAPTER BIFURCATION THEORY
The system
has a unique xed point at x
a ba which is
stable for b a
and unstable for b a
Atb a
the
xed point x undergoes a Hopf bifurcation
If we perturb the bifurcation parameter b by white noise say we obtain
the SDE
dx
abxxx
bx xx
x
x
dWt
Note that the noise on b is additive noise with respect to the xed point x
see Subsect
for the de nition of additive and multiplicative noise
hence the mathematically rigorous methods developed so far cannot be
applied
The e ect of noise on the Hopf scenario of the Brusselator was studied by
Lefever and Turner
using asymptotic methods and numerically
by Krebs
A systematic study including random attractors is still
pending
1

Part IV
Appendices
1

Appendix A
Measurable Dynamical
Systems
A Ergodic Theory
This is a summary of some well known facts on ergodic theory which can be
found in most of the many excellent books on the subject e g in the books
by Cornfeld Fomin and Sinai
Krengel
Mane
Petersen
Rudolph
Sinai
Walters
Probability Space F P Let
be an abstract set F a algebra
of subsets of and P a probability measure on F The pair F is called a
measurable space and the triple F P a probability space A probability
space is said to be complete if the algebra F contains all subsets of sets
of probability In this book we do in general not assume completeness of
our probability space unless explicitly stated
A algebra F is called countably generated if there exists a countable
family E An n N F such that the algebra E generated by it
equals F A probability space F P is said to be countably generated if
there exists a countable family E F such that for each A F and
thereexistsanA EforwhichPAMA
The latter is equivalent to
Lp F P being separable for all p
If F is countably generated
then F P is countably generated for any P
Aset FwithP
is called a set of full measure and a set
N FwithPN
is called an exceptional set or null set A function
f
of two measurable spaces F and F is called
measurable if f F F It is called a bimeasurable bijection if it is
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
measurable and measurably invertible
Time T We will study families of mappings indexed by the elements t of a
set T called time The most general time could be a measurable semi group
T i e T is endowed with a algebra T rendering the semi group oper
ations measurable However in this book T exclusively stands for the
following additive topological semi groups
T R two sided continuous time
T R sometimes T R
R one sided continuous time
TZf
g two sided discrete time
T Z f gsometimesT Z
ZorTN
f g one sided discrete time
T is always endowed with its Borel algebra B T
Measurable Dynamical Systems A family t t T of mappings of
F into itself such self mappings are also called transformations is
called a measurable dynamical system with time T or measurable semi ow
if T is a semi group if it satis es the following three conditions
t
t is measurable
id identity on if T
Semi ow property s t
s tforallst T
It follows from condition that t
is measurable for all t T
as a section of a measurable mapping for discrete time this is equivalent
to condition If T is a group conditions and imply that all t are
measurably invertible with t
t
If T is discrete then n
nn Twhere
is the time
one mapping In this case the measurable DS consists of the iterates of
a measurable mapping in case T Z or N or a bimeasurable bijection
in case T Z Conversely every such mapping generates through its
iterates a measurable DS
For continuous time T R or R we often use the less clumsy notation
t instead of t
Dynamical System s is henceforth abbreviated as DS
algebras with respect to which measurability is to be understood are not mentioned
in case they are clear from the context e g in product spaces we take the product
algebra and in topological spaces we take the Borel algebra generated by the open
sets
means composition
1

A ERGODIC THEORY
Measure Preserving Transformations Let be a measurable map
pingof FP to F ThemeasurePonFdenedby
PA Pf AgA F istheimageofPwithrespectto
The mapping f f
UfisanisometryofLp F P to
Lp FP
p
andRf dP Rfd P Ameasurable
mapping of F P to F P with P P iscalledahomomor
phism of the corresponding probability spaces It is called an isomorphism
if in addition it is measurably invertible A homomorphism of F P
to itself i e a measurable map with P P is called an endomorphism
and P is said to be invariant with respect to An endomorphism which
is measurably invertible is called an automorphism For an automorphism
wealsohave P P
A measurable DS t t T on a probability space F P for which
each t is an endomorphism is called a measure preserving or metric DS
and is denoted by
FP ttTorforshortby or IfTis
discrete P is invariant with respect to t t T if and only if it is invariant
with respect to
Let be a metric DS with continuous time Then the semi group
Utt TofisometriesofLp FP
p
denedbyf Utf
f t is strongly continuous with respect to t T Krengel Theorem
As a consequence for each measurable f the measurable stationary
stochastic process see Appendix A t f t is continuous in probabil
ity
Invariant Functions and Sets Let F be a measurable space A
function f is called invariant with respect to the measurable self mapping
of F withrespecttothemeasurableDS t t T iff f
for all
f t f forallt Tandall
A set is called
invariantwithrespectto or t t T if Aisinvariant ie if A A
or t A A for all t T An invariant set of a measurable DS consists
of whole orbits or trajectories i e t t T A if A The family of
measurable invariant sets of or t t T forms a sub algebra I F A
set A is called forward invariant with respect to or t t T if A
A
orA t Aforallt T whereT T R equivalently A A
or t A A for all t T An invariant set is clearly forward invariant
Notions Mod P If for two measurable functions f f on a probability
space FPthesetf f f g Fhasfullmeasurewesay
Here and at many other occasions we use the following elementary relations
f fA Aff A Af fA Aiffisinjective andff A Aif
f is surjective
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
that f f mod P A measurable function f is called invariant mod P
with respect to the endomorphism the metric DS F P t t T if
f fmodP f t fmodPforallt TwheretheexceptionalsetNt
can depend on t T A set A F is called invariant mod P with respect to
the endomorphism the metric DS F P t t T if A is invariant
modPequivalentlyifPAM A ifPAM t A
for all
t T The family IP of sets invariant mod P is a algebra depending on
P with I IP F where IP and I are related as follows
IP fB F thereisanA IwithPAMB g
For the proof of this relation one needs the fact that to each real valued
measurable function f invariant mod P with respect to the metric DS
FP t t T thereisafunctionf fmodPwhichisinvariant
In particular to each set A F invariant mod P with respect to a metric
DS there exists a set A A mod P which is invariant In case of discrete
time take f lim supn f n and in the case of continuous time set
f lim
t ess supff s
stg
the ess sup taken for xed with respect to Lebesgue measure on T
A forward invariant set is clearly invariant mod P for any P
Ergodic Theorems Let F t t T be a measurable DS and let f
be a real valued measurable function De ne
iforT Z
fflim
n
n
nX
kfk fexistsg
ii forT Z
fflim
n
n
nX
kfk f
lim
n
n
nX
k f k f bothexistand
ff
fg
Then f Iandfisinvarianton f
iii forT R Let
FZftdtFZjftjdt
1

A ERGODIC THEORY
SnZnftdtnX
kFk
and
f f f locally integrable lim
n
nFn
lim
n
n Sn f existsg
Then f F is forward invariant f is invariant on f and for all
f
lim
t tZtfsdsf
iv for T R Let with the notations of the case T R
f f f is locally integrable lim
n
nFn
lim
n
nSn f and lim
n
nSn f
both exist and f f
fg
Then f Ifisinvarianton fandforall
f
lim
t tZtfsdslim
t tZtfsdsf
The Birkho Chintchin ergodic theorem states that if t t T is a mea
surable DS on F then for any invariant probability P on F and
anyf L FP
Pf
f de nedoutside fbyf
isaversionofEfjI forT Z
T Z andT R andaversionofEfjIP forT R
Iff Lpfor some p
then f Lp and convergence to f also
holds in Lp
We also need the following extended version of the ergodic theorem If
onlyf L thenstillP f
provided we allow f R f g
Butweonlyhavef L
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
Ergodic Dynamical Systems A metric DS F P t t T is called
ergodic if all sets in I equivalently all sets in IP have probability or
Equivalently a metric DS is ergodic if and only if all invariant functions
invariant functions mod P are constant mod P As a result the limit f
in the ergodic theorem will be constant on an invariant set invariant set
mod P only for T R of full measure if the system is ergodic
Let for a measurable DS F t t T I denote the convex set of
invariant probabilities and let E denote the ergodic measures in I
Then two elements P P I are either equal or di er already on the
algebra I In particular di erent elements of E are mutually singular
and E coincides with the extreme points of the convex set I
We will often restrict ourselves to the ergodic case on the grounds of
the existence of a decomposition of an arbitrary invariant measure into
ergodic components For example we have the following ergodic decom
position theorem see Deuschel and Stroock
Theorem
Let
F t t T be a measurable DS Assume in addition that is a Polish
space and that for each xed t T t
is continuous Then for
each P I there is a probability P on E for which
PZEQPdQ
An ergodic decomposition theorem for compact metric and measurable
is given by Mane Chap II For Lebesgue spaces see Rokhlin
Isomorphisms of Metric DS Two probability spaces i Fi Pi i
are said to be metrically isomorphic if there exist sets i of full Pi
measure and an isomorphism from
FPto
FP
TwometricDS iFiPi ittT i
with the same time are
metrically isomorphic if there exists a metric isomorphism of the corre
sponding probability spaces with sets i of full measure which are forward
invariant for one sided time and invariant for two sided time such that on
t
talltT
Factors and Extensions The probability space F P is a factor of
F P and the latter is called an extension of the former if there
exist sets
of full measure and a homomorphism of
FP
to
FP
GivenametricDS FP t t T Itiscalledafactorofthemetric
DS
F P t t T and the latter is called an extension of the
former if this holds for the corresponding probability spaces with sets
1

A ERGODIC THEORY
of full measure which are forward invariant for one sided time and
invariant for two sided time and a homomorphism
satisfying
on
t
t
allt T
We say that
covers
Two metric DS are isomorphic if and only
if they are factors of each other Factors of metric DS on Lebesgue spaces
are determined by invariant sub algebras and a metric DS with a factor
allows a skew product representation
Natural Extension In the theory of random dynamical systems it is
very advantageous to have a model of the noise which has two sided time
Two sided time can be achieved basically without loss of generality by ex
tending a metric DS with one sided time This procedure is known as
natural extension and was carried out for discrete time and a Lebesgue
space F P by Rokhlin x See also Cornfeld Fomin and Sinai
p
and Sinai
p
LetT ZorR putT Z orR andlet
FP
ttT
be a metric DS with one sided time The metric DS
FP ttT
with two sided time is called a natural extension of if
i is a factor of the DS restricted to T
ii the algebra G
F is exhaustive in the following sense
t G t T F mod P Here is the homomorphism occuring in
i A natural extension exists and is unique up to isomorphisms for discrete
time and F a standard measurable space or F P a Lebesgue
space Further is ergodic if and only if is and and have the
same entropy see Rokhlin
pp and
If
FP ttTis atwosidedmetricDSthen
F P t t T is a one sided metric DS and is the natural extension
of with associated homomorphism id A new DS with one sided
timeon canbedenedbyasub algebraG Fwith t G G
fort T Now
GP ttT isafactorof againwith
associated homomorphism id Its natural extension is the two sided
DS with F replaced with G
tG t Z whichisafactorof
For the canonical metric DS on path space with shift corresponding to a
stationary stochastic process there is a more direct extension from one sided
to two sided time see Appendix A
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
Lebesgue Spaces A probability space F P is called Lebesgue space
see the pioneering work
of Rokhlin if it is metrically isomorphic
to a probability space which is the disjoint union of an at most countable
possibly empty set fx x g of points each of positive measure and the
space
s L possibly absent where L is the algebra of Lebesgue
measurable subsets of the interval s and is Lebesgue measure Here
s P pn pn measure of the point xn
As ergodic theory becomes more satisfactory for Lebesgue spaces and
since this class covers many applications some books on ergodic theory deal
exclusively with Lebesgue spaces e g Petersen
and Rudolph
Standard Measurable Spaces A measurable space F is called
standard measurable space sometimes also called Borel space if it is iso
morphic by means of a bimeasurable bijection to a Borel subset of a Polish
space see Zimmer Appendix A or Parthasarathy
In particu
lar the algebra F of a standard measurable space is countably generated
and countably separated For any probability P on a standard space F
de ne FP to be the P completion of F Then FP P is a Lebesgue space
Almost Periodic Functions and DS A subset of R is called relatively
dense if there is an L such that each interval of length L contains an
element of the set For a function f R Rd and
a number R
iscalledan almostperiodoff ifjft ftj forallt R
A function f R Rd is called almost periodic in the sense of H Bohr
if i f is continuous and ii for each
the set of almost periods of
f is relatively dense For details see Dunford and Schwartz IV
An almost periodic function is bounded and uniformly continuous Let
Hf ff t t Rgbethehull ie thesetofalltranslatesof
f Cb R Rd Then f is almost periodic if and only if the closure H f
of H f is compact If this is the case H f consists of almost periodic
functions and has the structure of a compact Abelian Polish group G with
unit e f The group operation is de ned as follows For g f t and
hfsputghfstforglimftnandhlimfsn
putg h limf tn sn
We associate to an almost periodic function f the following canonical
metric DS Let T R
G Hf FtheBorel algebraofG
t
t the translation of by t Then t
t is continuous
and hence measurable The normalized Haar measure of G is the unique
invariant probability Under P is ergodic Particular cases i G T
torus for f periodic ii G T n n torus for f quasi periodic
H f is compact if and only if f is periodic exercise
1

A STOCHASTIC PROCESSES AND DYNAMICAL SYSTEMS
Noise modeled by this DS is of particular interest as it is in the intersec
tion of topological dynamics and ergodic theory and is a rst step beyond
periodic excitation
A Stochastic Processes and Dynamical
Systems
Several classes of stochastic processes e g stationary processes processes
with stationary increments in particular Brownian motion white noise
which occupy strategic positions in probability theory are linked to metric
DS which will be brie y surveyed in this section Our motivation is that
metric DS constructed in this section enter as noise into parameters of
di erence or di erential equations this way generating a random dynamical
system see Chap
There is an abundance of good texts from which we quote freely and
of which we only mention Bauer Breiman Doob
Dudley
Durrett Ganssler and Stute Gihman and Skorohod
Karlin and Taylor
Meyn and Tweedie
Wentzell
Basic De nitions Let F P be a probability space E E a mea
surable space and T a set A family
t t T of random variables
t
E is called a stochastic or random process with parameter set
or time T and state space E In this book time T is equal to one of the
additive semi groups N Z R Z R Z R For xed the function
given by t t is called a sample function trajectory path The
mapping given by
into ET ET where
ETY
tTEETO
tTE
is measurable Note that ET is generated by the algebra of cylinder sets in
ETie setsoftheform
ZASY
tTnSEfET t
tr ASg
where S ft
trg P T the family of non void nite subsets of
TandAS ES PutP
P The stochastic process t given by the
coordinate functions t
t on FP ETETP iscalledthe
canonical realization of We have P P in which case the two stochastic
processes and are called equivalent Two processes are equivalent if and

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
only if their nite dimensional distributions i e the distributions PS of S
on ESES arethesameforanyS PT
For two processes
on the same probability space is called a version
ormodicationof ifP t
t for all t T They are called indis
tinguishable if f t
t forsomet Tg NwithN Fand
PN
Indistinguishable processes have P a s the same trajectories
while versions can have disjoint sets of trajectories Versions are indistin
guishable for discrete time and also in the case of continuous time if E is
a Hausdor space and the processes have P a s continuous or right left
continuous trajectories
Let PS S P T be a projective family of nite dimensional distribu
tions projective meaning that if S S then PS
S
SPS S
S the
canonical projection from ES to ES Then Kolmogorov s celebrated
fundamental theorem states that there is a stochastic process
ttT
on some probability space F P whose nite dimensional distributions
are the prescribed ones provided E E is a Polish space with its Borel
algebra or more generally E E is a standard space
Canonical Realization on Path Space Shift For any stochastic
process we can switch to the equivalent canonical realization F P
ETETPandt
t
De ne the transformations t ET ET t T by
ts
ts stT
The mapping t is ET ET measurable More precisely If
Ets
usutst
then uEtsEtu
susothatuisEtu
s u Ets measurable for all s t
is thus ltered with respect to the ltration Ets Further
id if
Tst
s t from which for two sided time t
t
follows The transformations are called the unilateral for one sided time
or bilateral for two sided time shift transformations
If E E is a topological space with its Borel algebra then the uni
lateral shift
t is continuous for each xed t T in the product
topology of ET and the bilateral shift is even a homeomorphism Note
however that in the uncountable case E T B ET
For discrete time T N Z Z the mapping t
t is trivially
measurable hence ET ET t t T is a measurable DS
For continuous time T R or R the canonical realization has certain
defects which make the model useless for our purposes e g
1

A STOCHASTIC PROCESSES AND DYNAMICAL SYSTEMS
i many interesting sets namely those whose description depends on
an uncountable number of t s are not in ET
ii the mapping t
t is not B T ET ET measurable when
ever E contains a non trivial set
The following is a procedure for obtaining an equivalent realization for
continuous time without the above de ciencies If ET but P
where
P
inffP A
AAETg
istheoutermeasureof wecangodownfrom FP ETETP to
the space
ET P where P A P A is the unique probabil
ity measure on ET which has the same nite dimensional distributions
as P The coordinate functions t
t on both spaces de ne equiv
alent stochastic processes but all trajectories of the second process are in
We will now describe some important cases for which there are also
convenient criteria for P
Canonical Spaces C R Rm and C R Rm First consider the case
TRLet
CRRm RmR Rm couldbereplacedbyaPolish
space E Clearly Bm R Endow with the compact open topology
given by the complete metric
dX
n
nk kn
kknkkn
sup
ntnjt
tj
This makes a Polish space in fact a Frechet space The Borel algebra
F is the trace in of the product algebra Bm R so F
ttR
The set Rm R is invariant with respect to the group of shifts t t R
For xed t T t is a homeomorphism and t
t is contin
uous thus see Dudley
Proposition
measurable Therefore
F t t R is a measurable DS We have the natural ltration
Fts
usutst
withuFtsFtu
s u hence is ltered with respect to Fts
A criterion for P
is Kolmogorov s criterion see e g Kunita
p
For T R analogous statements hold for C R Rm with obvious
changes
Canonical Spaces D R Rm and D R Rm Many important classes
of possibly discontinuous stochastic processes like semi martingales
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
Markov processes processes with independent increments have trajecto
ries which are cadlag or cadlag after the French continu a droite avec
des limites a gauche The set of cadlag functions on T R with values
in Rm which could be replaced by a Polish space E is de ned by
DRRm
f RmR Forallt Rlim
sts
t lim
sts
t existg
Clearly
Bm R Each
is measurable bounded on bounded
intervals and has only at most countably many discontinuities of the rst
kind jumps It is regularized at jumps to be right continuous
D R Rm can be made a Polish space the so called Skorokhod
space for details see e g Billingsley Chap Jacod and Shiryaev
Chap IV or Ethier and Kurtz Chap The Borel algebra
F is the trace in of the product algebra Bm R so F
ttR
C R Rm is continuously imbedded in D R Rm and the relative Skorokhod
topology in C is the C topology
The set
Rm R is invariant with respect to the group of shifts
ttRForxedtRt
is a homeomorphism and t
t is continuous thus measurable Therefore F t t R is a mea
surable DS We again have the natural ltration Fts
usut
stwithuFtsFtu
s u so that is ltered with respect to Fts
For criteria for P
see e g Billingsley Chap or Ganssler
and Stute
pp
For T R analogous statements hold for D R Rm with obvi
ous changes
A Stationary Processes
Canonical DS Corresponding to a Stationary Process A stochas
tic process with time T and state space E E is called stationary if for all
tt
tr TwehavePt t tr t Pt tr Anequivalentbutmore
concise way of saying this is that on ET ET P
tP P forallt T
i e the shifts are measure preserving For discrete time a stationary
stochastic process gives rise to the canonical metric DS
ETETP ttT
For continuous time this model is inadequate see above However in
many relevant cases the stationary process can be realized on C R Rm
1

A STATIONARY PROCESSES
CR Rm orDRRm DR Rm andinthosecasesweconsiderthe
canonical metric DS on the respective space introduced above
Conversely if is a metric DS and f F E E is measurable
then t f t is a measurable stationary stochastic process A
stochastic process is called measurable if the mapping t
tis
measurable
Canonical DS for Processes with Stationary Increments A pro
cess with continuous time T R or R state space Rm and with
is
said to have stationary increments if for any t
tr the distribution
oftttt
tr t tr t isindependentoft T Incase can
berealizedinCRRm orCR Rm DRRm DR Rm weconsider
the Borel set
CRRmfCRRm
g
with its relative topology and Borel algebra F We rede ne the shift so
that it leaves invariant by
t
ts
st
tstR
Then t is a homeomorphism for each t and t
t is continuous
hence measurable With the natural ltration now de ned as
Fts
u
vsuvt
wehave u Fts Ft u
s u so is ltered with respect to Fts
SimilarlyforthecasesC R Rm D RRm andD R Rm
has stationary increments if and only if its distribution P on is
shift invariant The corresponding metric DS is the canonical one for
The canonical process t
t is a helix over i e satis es identically
ts
s
ts
Every stochastically continuous process with stationary independent in
crements and
can be realized on D R Rm and gives thus rise to
a canonical metric DS see Ganssler and Stute
p
Since by the
zero one law the tail algebra T is trivial mod P the DS is ergodic see
below
Canonical DS for Brownian Motion Wiener Process White
Noise The only processes with stationary and independent increments
which can be realized on C R Rm have the form
ttGt
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
where Rm and Gt is a Gaussian process with stationary independent
incrementsG andGt Gs N jt sj with Rm m non
negative de nite
A standard Brownian motion or Wiener process Wt t R i e with two
sided time in Rm is a process with W
and stationary independent
increments satisfying Wt Ws N jt sjI The corresponding measure
Pon F where
C R Rm and F is the uncompleted Borel
algebra on is called Wiener measure the probability space F P
is called Wiener space The corresponding canonical ergodic metric DS
describes Brownian motion or Gaussian white noise as a metric DS
We stress that t
t is not B FP FP measurable where FP
is the completion of F with respect to Wiener measure
This canonical DS describing Brownian motion Wiener process white
noise is one of the fundamental objects of this book since it drives stochastic
di erential equations
Invariant and Tail Algebra Let
FP ttTbeoneofthe
canonical DS introduced above with the canonical ltration Fts s t Let
T
t TFt
and for two sided time T
t TFt
be the tail algebras T remote past T remote future These
algebras are invariant t T T t T T Recall that
I IP F are the algebras of invariant sets and mod P invariant sets of
the DS Then
iifTisonesided I T henceIP T modP
ii ifTistwosided IP T modPandIP T modP
Consequently if one of the tail algebras is trivial mod P is ergodic
For example if T is discrete and P T is a product measure or if P
corresponds to a process with stationary independent increments then T
is trivial mod P by Kolmogorov s zero one law
Natural Extension of Canonical DS We can always assume that the
canonical DS described above have two sided time thanks to the existence of
the following natural extension from one to two sided time for the general
method see Appendix A
The mapping t
t is however B F P FP measurable where is
any nite measure on B e g Lebesgue measure See Kager
Lemma
A BmodPmeansthatforeachA AthereisaB BwithPAMB
1

A MARKOV PROCESSES
i Discrete time case Suppose
EZEZP
nnZisa
canonical one sided time metric DS where E E is a standard measurable
space and n
k
k n is the one sided shift De ne a invariant
probability P on the measurable DS EZ EZ n n Z with two sided time
as follows On a typical cylinder put
PfnA
nr Arg
PfnNA
nrNArg
where we have chosen N
sobigthatni N fori
r
This projective family of nite dimensional distributions uniquely extends
by Kolmogorov s theorem to a probability P on EZ which by construc
tion is invariant The resulting two sided canonical metric DS
EZ EZ P n n Z is the natural extension of with homomorphism
EZ EZ given by
The algebra G
EZ
n n Z is exhaustive since
nGnZ
nnZEZ
ii Continuous time case Now our one sided metric DS is one of the
canonical DS desribed above on
CRRmDRRmCRRm
or D R Rm By the same procedure as for discrete time using Kol
mogorov s theorem we can naturally extend it to a canonical metric DS
on
C R Rm etc Here the homomorphism
is the
restriction of a function from R to R
A Markov Processes
Basic De nitions Let T Z or R let E E be a standard space
andlet Pt t Tbeatransitionfunctionon EE ie Pt xB isakernel
foreach xedt Pt B ismeasurableforeach xedB EandPt x is
aprobabilityonEforeach xedx E withP xB xB and
PstxBZEPsyBPtxdy forallst T
Chapman Kolmogorov equation Let be a probability on E Then there
exists a unique probability P on the canonical space F ET ET
such that the coordinate process t
t is a homogeneous Markov
process with respect to the natural ltration Ft
s
stwith
transition function Pt t T and initial distribution i e it satis es for any
s t and any measurable f
EftjFs PtsfXsPas
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS
where Ptf x RE f y Pt x dy is the semigroup of positive linear oper
ators corresponding to the kernel Pt and P is given on cylinders by
PtB
tr Br
ZBZBrdxPttxdx Ptrtr xrdxr
where t t
tr andB Br EandextendedtoETby
Kolmogorov s theorem for T Z Ionescu Tulcea s theorem could be
used which works for any measurable space E E
With Px P x the Markov property on the canonical space reads For
all t and measurable f
EftjFt EtfPas
In particular for any x E t and measurable f
Exf tjFt EtfPxas
i e F Px x E t t T forms a Markov family with transition function
PttT
Stationary Measures Feller Processes The measure P is t in
variant i e the Markov process is stationary if and only if is Pt invariant
orstationary ie ZEPt x dx forall t
t su ces if T Z When does Pt admit a stationary
Suppose E E is a Polish space with its Borel algebra and let Cb E
be the Banach space of bounded continuous real valued functions on E If
iPtCbE CbEforallt
ii limt kPtf fk foreveryf CbE onlyforT R
the semigroup Pt and the associated Markov process are called Feller
A Feller transition function Pt admits a stationary if and only if the
sequence
x
T
TPT
kPkxTZ
TRTPtxdt T R
It is important to distinguish between Pt invariant measures and invariant measures
for dynamical systems This is the reason why we introduce the name stationary for
the former
1

A MARKOV PROCESSES
has a limit point in the topology of weak convergence for T
for some
x E and every such limit point is a stationary probability
For discrete time T Z a standard state space E E a transition
probability Pt and a stationary we naturally extend see above to two
sided time T Z and obtain a metric DS
EZEZP ttZ
where the coordinate process n
n satis es the Markov property
on all of Z Further the sub algebra IP of mod P invariant sets
satis es IP
mod P i e for Markov processes invariant sets can
be seen in state space This is the basis of the equivalence of the concept
of invariance and ergodicity for introduced in Appendix A and the
following state space concept
Call a bounded measurable function g on E invariant if Pt g g
a s for all t Z A stationary is called ergodic if all invariant functions
are constant a s By the above there is a one to one correspondence
between invariant functions g and mod P invariant functions h via
g
h Hence is ergodic if and only if P is ergodic see Meyn and
Tweedie Chap or Kifer Sect
Every stationary on a standard space has an ergodic decomposition
Rd
where is a probability measure on the space of
all stationary measures concentrated on the ergodic stationary measures
see Kifer Appendix A for T Z and Skorokhod x for
TR
When does a Markov process with continuous time T R admit a
cadlag version Suppose E is locally compact with a countable base thus
Polish let C E be the Banach space of continuous real valued functions
vanishing at in nity Then a Markov process which is Feller in C E
see the above de nition with C E instead of Cb E admits a cadlag
version more precisely a version on D R E where E is the one point
compacti cation of E by the point
IfdisametriconEandlimt supxKtPtxfy dxy g
for every
and every compact set K E then can be realized on
CRE
If thus is stationary for the semigroup Pt which is Feller in C E we
obtain by natural extension a canonical metric DS on D R E
1

APPENDIX A MEASURABLE DYNAMICAL SYSTEMS

Appendix B
Smooth Dynamical
Systems
Throughout Appendix B time T is always equal to R
B Two Parameter Flows on a Manifold
B De nition Smooth manifold A d dimensional manifold M is
i a topological manifold in the sense that M is a Hausdor topological
space which is locally Euclidean of dimension d and has a countable base of
open sets It follows that M is locally connected locally compact normal
metrizable by a complete metric paracompact and separable hence Polish
with countably many connected components see e g Boothby pp
We assume once and for all that M is connected
ii It is smooth in the sense that the changes between local coordinates
are C functions The choice of a C structure is without loss of generality
for us since in most situations we need at least a C structure on M But
on a paracompact manifold M there is a C structure subordinate to a Ck
structure of M for k Whitney
B De nition A two parameter continuous local ow on a manifold
M is a continuous mapping
DMstx
stx D R R M anopenset
such that
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
iForeachs Rx M
DsxftRstxDgtsxtsx
is an open interval of R containing s
ii is a local ow in the following sense
ss idMforalls R
Foralls Rx Mandu Dsx wehavet Du sux if
and only if t D s x and in this case the two parameter local ow
property holds
utsux
stx
If the mappings
stDstMDstfxMstxDgx
stx
which are continuous are Ck k i e k times di erentiable with
respect to x and the derivatives are continuous with respect to s t x
then the local ow is called Ck
Thelocal owiscalledaglobal ow orjusta owifD R R M
or equivalentlyifDsx Rforallsx R MorifDst Mfor
allst R
B Theorem Let s t x be a local continuous Ck ow Then
ithefunction sx tsx
s
is lower semicontinuous
andthefunction s x t s x
s is upper semicontinuous
iiforallt Dsx
DsxtsDtstx
iiiforallst R Rthemapping
stDstDts x
stx
is a local homeomorphism Ck di eomorphism and
st ts Dts Dst
iv The mapping
stx
stx
tsx
is continuous Ck
A proof of these facts can be basically found in Amann p
or
Hartman
p
1

B SPACES OF FUNCTIONS IN RD
B Spaces of Functions in Rd
B Denition Letk Z and
Let Ck be the Frechet
space of functions f Rd Rd which are k times continuously di erentiable
and for
whose k th derivative is locally Holder continuous for
locally Lipschitz continuous with seminorms
kfkkKX
jjk
sup
x KjDfxj
kfkkKkfkkKX
jjk
sup
xyKx yjDfx Dfyj
jx yj
where K is a compact convex subset of Rd
d Zdisa
multi index j j
d and
Dfx Dxfx
jj
x
xd dfx
A complete metric is given by
fgX
n
nkfgkkKn
kf gkk Kn
where Kn is some increasing sequence of compact convex sets exhausting
Rd The topology of Ck is called the compact open topology While Ck
is separable hence Polish the space Ck for
is not separable
see Kufner John and Fucik Chap We have for any k Z resp
k
Z and
the continuous inclusions
CkCkCk
We will sometimes write Ck Ck and will say that f C if f Ck for all
k Composition f g f g is continuous in Ck for k f
g
and Ck is a Polish semigroup
B De nition Let
Di kRd ff Ck fbijectiveandf Ckg
put for k
Homeo Rd Di Rd
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
endowed with its relative compact open topology A complete metric gen
erating the same topology is given by
dfgfgfg
where is any complete metric of Ck making Di k Rd Homeo Rd a
Polish group with respect to composition the group of Ck di eomor
phisms homeomorphisms of Rd
B Denition Letk Z and
Let Ck
b be the Banach
space of functions f Rd Rd which are in Ck and for which the norm
kfkk
sup
x Rdjfxj
jxj X
jjk
sup
x RdjDfxj
kfkk kfkk X
jjk
sup
x yjDfx Dfyj
jx yj
is nite We have for any k Z respectively k Z and
the continuous inclusions
Ck
bCk
bCk
b
and sometimes write Ck
bCk
b
B Denition Letk Z and
LetLlocRCk bethe
set of measurable functions f R Rd Rd for which
ft
Ck for every t R for Lebesgue almost all t would
su ce
for every compact set K Rd and every bounded interval a b R
Zb
akftkk Kdt
B
With the seminorms given by B Lloc R Ck is a Frechet space We
have the continuous inclusions
Lloc R Ck
Lloc R Ck
Lloc R Ck
The Frechet spaces Lloc R Ck
b are de ned analogously and obey analo
gous inclusion properties
B DenitionLetk Z
fRRd Rdandlet
tx ftxbecontinuousWesaythatf Ck if

B DIFFERENTIAL EQUATIONS IN RD
foreacht Rft Ck
in case k the derivatives Dx f t x are continuous with respect
totxforall jjk
in case
for j j k the derivatives are locally Holder contin
uous for
locally Lipschitz continuous with respect to x
The spaces are Frechet with seminorms supa t b kf t kk K and we have
the continuous inclusions
CkCkCk
Further
Ck LlocRCk
The last inclusion is clear by the fact that local Holder continuity of f t x
with respect to x is equivalent to uniform Holder continuity on compact
setsab KseeegAmann p
Wesaythatf Ck
b iff Ck andforeacht Rft Ck
b
B Remark Observe that we have following Kunita
built into
the de nition of our spaces Ck
b that its elements are permitted to be un
bounded but with at most linear growth i e for each f Ck
b there are
constants and such that for all x Rd
jfxj jxj
It implies that for f Lloc R Ck
b
jftxj tjxj t
with locally integrable
This trick allows a more elegant for
mulation of the conditions for the existence of global ows
B Di erential Equations in Rd
In studying di erential equations xt f t xt and their solutions we will
adopt the dynamical systems or ow point of view Rather than studying
a particular solution for given initial values we consider the totality of
solutions starting at arbitrary times s R and at arbitrary points x Rd
We then look at the mappings de ned by initial value x at time s goes
to solution of xt f t xt at time t There is a basically one to one

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
correspondence between local continuous Ck two parameter ows s t
which are absolutely continuous with respect to t and di erential equations
Wesaythatt st x solvesorisasolutionofxt ft xt sometimes
called solution in the sense of Caratheodory see Aulbach and Wanner
or that the di erential equation generates the ow s t if it satis es
stx xZt
sfu suxdu
B
for all t D s x an open interval of R containing s this interval is all of
Rintheglobalcase NotethatB implies ss x xforallsandx
If the solution is in fact di erentiable with respect to t and satis es for
tDsx
d
dtstx ft stx
ssx x
B
it is called a classical solution of xt f t xt
B Theorem Local Flow from Di erential Equation Con
sider the di erential equation
xt ftxt
B
in Rdi If f Lloc R C then B uniquely generates through its max
imal solution a local continuous two parameter ow s t
ii If f Lloc R Ck for some k then B uniquely generates
alocalCk ow TheJacobianof statx
Dstx
stx i
xj
GldR
uniquely solves the variational equation vt Df t xt vt i e we have for
allt Dsx
Dstx IZt
sDfu suxDsuxdu
whereDftx
fitx
xj Rd distheJacobianofftx Finally the
determinant det D s t x satis es Liouville s equation
detD stx expZt
straceDfu suxdu tDsx
1

B DIFFERENTIAL EQUATIONS IN RD
iii If f C then the solution of i is classical More precisely
s t x is Lipschitz continuous in the variable s t x and is C with respect
totivIff Ckforsomek
then the solution of ii is classi
cal The Jacobian and its determinant satisfy the classical versions of the
variational and Liouville s equation respectively
The proof of this theorem can be basically found in many textbooks see in
particular Amann Coddington and Levinson
or Walter
B Remark i Even if f t is C with respect to x the solution
s t x is in general only absolutely continuous with respect to t
ii Toassumethatft Ckforeach xedt Risingeneralnot
enough to guarantee the result
iii The integral equation B and the di erential equation B
are always with respect to the argument t for xed s Initial time s plays
the role of a parameter and is xed In s t s always denotes the initial
time no matter whether s t or s t
iv The maximal solution s t x has the property that if t s x
then
lim
tt sxjstxj
similarly for t s x
B Theorem Global Flow from Di erential Equation Con
sider the di erential equation B in Rd
i Supposef LlocRCb LlocRCk
b or more generally suppose
f LlocRC LlocRCk and
jftxj tjxj t
B
with
locally integrable Then B uniquely generates a
continuous Ck ow
iiSupposef Cb Ck
b
or more generally suppose f
C C k and B holds Then B uniquely generates a clas
sical continuous Ck ow
The fact that the linear growth condition B implies a global solution
follows from the Gronwall Bellman lemma which we state for convenience
B Lemma Gronwall Bellman Lemma Let a x t t R
be continuous and let b t t R be integrable If
xt atZt
tbsxsdsforallt tt
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
then
xt atZt
t asbs expZs
tbududsforallt tt
and the right hand side is nite
We now deal with the inverse problem of when for a given ow s t there
exists a di erential equation xt f t xt which generates it
B Theorem Di erential Equation from Flow i Let s t be
a local continuous ow for which t s t x is di erentiable at t s for
all x Put
fsx d
dtstxjts
Thenfismeasurable t stx isdierentiableforallt Ds x and
d
dtstx ft stx
ssx x
ii Let stbe alocalcontinuous owfor whicht stx is abso
lutely continuous with respect to t D s x for all x Then there exists a
measurable function f R Rd for which for all x
stx xZt
sfusuxdutDsx
The result is that we have a basically one to one correspondence between
two parameter ows and non autonomous di erential equations
B Autonomous Case Dynamical Systems
If the di erential equation is autonomous xt f xt then clearly f
Lloc R Ck
b ifandonlyiff Ck
b ifandonlyiff Ck
b similarly for
Ck This means that we are always in the case of classical solutions
Hence if e g f Cb the di erential equation uniquely generates a
continuous ow s t By substituting v u s we obtain
stx xZt
sfsuxdu xZtsfssvxdv
thus by uniqueness of solution
stx
tsx
B
1

B AUTONOMOUS CASE DYNAMICAL SYSTEMS
hence the one parameter family t
t satis es the one parameter
owproperty t sx t sx
A two parameter ow with the extra property B obtains the name
dynamical system
B De nition Dynamical system A local continuous dynamical
system on a manifold M is a continuous mapping
DMtx
tx
whereD R Misopen suchthatforeachx M
DxftRtxDgx
x
is an open interval of R containing and satis es the following conditions
idM
Forallx Mandalls Dx thelocal owpropertyholds We
havet D sx ifandonlyift s Dx andinthatcase
tsx
tsx
If the mapping t x
t x is k times continuously di erentiable with
respect to x where k the local continuous dynamical system is called
Ck
IfD R MequivalentlyifDx Rforallx MorifDt M
forallt R whereDt fx M tx Dgthenthelocaldynamical
system is called global dynamical system
ThesetsDx RandDt MfromDenitionB areopenas
sectionsoftheopensetD Wehavet Dx
x D t implying
Dx ft x DtgandDt fx t Dxg
B Theorem Let be a local continuous Ck DS on a manifold M
Theni x
x
is lower semicontinuous and x
x
is upper semicontinuous
iiforalltx D
DxtDtx
iiiforallt R
tDtDt
Dynamical system is abbreviated as DS
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
is a local homeomorphism Ck di eomorphism and
t
tDtDt
iv the mapping t x
tx
t x is continuous Ck
ForaproofseeegAmann pp
B Theorem Local DS from Vector Field i If f C
then the di erential equation xt f xt in Rd generates through its unique
maximal classical solution t x
t x a local continuous DS which is
Lipschitz with respect to t x and C with respect to t
iiIffCkk
thenthelocalDSisCk InthiscaseD tx
satis es locally the non autonomous variational equation
vt Dftxvt v idRd
and Liouville s equation
detD t x expZttraceDf s x ds
If for the maximal local solution
x
or
x
for some x
then
lim
t xjtxj orlim
t xjtxj
B Theorem Global DS from Vector Field Suppose f
Cb Ck
bforsomek
More generally suppose f C Ck and
for some constants jf x j jxj for all x Rd
Then the di erential equation xt f xt in Rd uniquely generates a
continuous Ck DS If f Ck
b then D isaCk DSonRd Rd
B Theorem Vector Field from DS Let be a local continuous
DSiIft
t x is di erentiable at t for all x then putting
fxd
dt txjt
is generated by xt f xt and t
txisdierentiableforallt Dx
and all x
ii Ift
t x is absolutely continuous for t D x and all x then
there exists a measurable function f Rd Rd for which for all x
tx xZtfsxds t Dx
1

B VECTOR FIELDS AND FLOWS ON MANIFOLDS
A consequence of the above is that the classes of DS and autonomous
di erential equations vector elds are basically the same symbolically
written as
dynamical systems exp vector eld
B Vector Fields and Flows on Manifolds
The de nitions B and B above are already formulated for the man
ifold case We claim that all local results stated above for Rd will remain
valid for the case where Rd is replaced by a manifold M As most things
are obvious we will be quite brief
Let now M be a d dimensional manifold For k
letXkM
be the separable Frechet space of Ck vector elds We will consider non
autonomous di erential equations xt f t xt on M with right hand side
fRXkMtftft
B
ThespacesC k C k R MTM cannowbedenedseeKunita
pp
as those functions B for which R M t x
f t x TxM is for any chart over which T M is locally trivialized in
C k the space introduced in De nition B
The spaces Lloc R Ck
LlocRCk MTM canalsobedened
as those functions B for which t kf t kk K is locally integrable
for any compact K M Here the seminorms kfkk K of a vector eld
f Xk M are de ned in an obvious manner by covering K by nitely
many charts over which T M is trivialized and then using De nition B
ThespaceCkMN ofCk mappingsf M N M Nmanifolds
k
with its Ck compact open topology can be similarly de ned
seeeg Baxendale Sect Ck M N isaPolishspace andCk M M
is a Polish semigroup
Further
DikM ff CkMM fbijectiveandf CkMMg
with
HomeoM Di M
given the relative compact open topology are Polish groups In fact if is
a complete metric of Ck M M then
dfgfgfg
is a complete metric of Di k M Homeo M still generating the compact
open topology
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS
LetfbeafunctionasinB Wecallafunction R R M M
stx
stx asolutionofxt ftxt ifforeachtestfunctionh
CK M R the space of real valued C functions with compact support
hstx hxZt
sfuhsuxdu
B
It will be called a classical solution if for each h CK M
d
dthstxfthstx hssx hx B
With those notions the existence and uniqueness part of Theorem B
holds verbatim
Let be a local C ow on M Denote by T the derivative of the
local continuous ow on the tangent bundle T M which covers and is on
bersdenedbythederivativeof statx Dst
Tstx TxM TstxM
where for a local di eomorphism for each x in the domain of the linear
isomorphism
TxTxMTxMvwTxv
isdenedbywh vh x h CKM
If solvesxt ftxt withf LlocRCk thenT isalocalCk
ow which uniquely solves
vt Tftvt
vsvTM
B
where Tf t is the natural lift of the vector eld f t on M to a vector
eld on TM in local coordinates Tf t x v
xvftxDftxv
If M is a Riemannian manifold we can use the Riemannian connec
tion to decompose TvT M into horizontal and vertical components Then
T f t v has horizontal component identi ed with f t x and vertical com
ponent rf t x v where r denotes the covariant derivative of the vector
eld f t at x in direction v TxM Equation B is then equiva
lent to the coupled system of the original equation xt f t xt and the
variational equation
vtrftstxvvsvTxM
B
Here the absolute or covariant derivative vt is taken along the integral
curve t s t x in the direction of the vector eld f t see Elworthy
Appendix B or
p
Klingenberg Sect
1

B VECTOR FIELDS AND FLOWS ON MANIFOLDS
The expression for the variational equation B in a chart around
x Mis
vk
tvk
t
k
ijfjvi rifk vi rifk fk
xi
k
ijfj
with summation of repeated indices
Moreover
divftx tracerftx
dX
ihrftxviviix
where vi is an orthonormal basis of TxM h ix and we have Liouville s
equation Fort Dsx
detT stx expZt
sdivfu sux du expZt
stracerfu sux du
AlocalCk owonMgeneratedbyxt ftxt withf LlocRCk
isglobalifMiscompact orifbyembeddingMintoanRNft isthe
restriction of a vector eld f t on RN for which f Lloc R Ck
b
Dynamical Systems on Manifolds
If f Xk M then xt f xt generates a local Ck DS The derivative T
satis es
vt Tfvt
v vTM
B
and T is a Ck DS as well as a continuous linear bundle DS on T M over
the DS If M is a Riemannian manifold then B is equivalent to
xt f xt and the variational equation vt rf t x vt and
detT tx expZtdivf s x ds expZttracerf s x ds
1

APPENDIX B SMOOTH DYNAMICAL SYSTEMS

Bibliography
H Amann Gewohnliche Di erentialgleichungen Walter de Gruyter
Berlin
D V Anosov and V I Arnold Dynamical systems I Springer Berlin
Heidelberg New York
R Arens Topologies for homeomorphism groups Amer J Math
S T Ariaratnam Some illustrative examples of stochastic bifurca
tion In J M T Thompson and S R Bishop editors Nonlinearity
and chaos in engineering dynamics IUTAM Symposium
pages
Wiley New York
S T Ariaratnam and N Sri Namachchivaya Stochastically per
turbed Hopf bifurcation In F M A Salam and G I L Levi editors
Dynamical Systems Approaches to Nonlinear Problems in Systems
and Circuits pages
SIAM
S T Ariaratnam and W C Xie Stochastic bifurcations in engineer
ing mechanics In Proceedings of the th Session of the International
Statistical Institute Paris France Volume L III pages
S T Ariaratnam and W C Xie Lyapunov exponents in stochastic
structural dynamics In Arnold et al pages
L Arnold The unfolding of dynamics in stochastic analysis Comput
Appl Math
L Arnold and P Boxler Stochastische Bifurkationstheorie Report
Institut fur Dynamische Systeme Universitat Bremen
1

BIBLIOGRAPHY
L Arnold and P Boxler Additive noise turns a hyperbolic xed point
into a stationary solution In Arnold et al pages
L Arnold and P Boxler Stochastic bifurcation instructive ex
amples in dimension one In M Pinsky and V Wihstutz edi
tors Di usion processes and related problems in analysis volume II
Stochastic ows volume of Progress in Probability pages
Birkhauser Boston Basel Stuttgart
L Arnold and H Crauel Iterated function systems and multiplicative
ergodic theory In M Pinsky and V Wihstutz editors Di usion pro
cesses and related problems in analysis volume II Stochastic ows
volume
of Progress in Probability pages
Birkhauser
Boston Basel Stuttgart
L Arnold H Crauel and J P Eckmann editors Lyapunov Expo
nents Proceedings Oberwolfach
volume
of Springer Lec
ture Notes in Mathematics Springer Berlin Heidelberg New York
L Arnold L Demetrius and M V Gundlach Evolutionary for
malism for products of positive random matrices Annals of Applied
Probability
L Arnold M Doyle and N Sri Namachchivaya Small noise expan
sion of moment Lyapunov exponents for two dimesnional systems
Dynamics and Stability of Systems
To appear
L Arnold A Eizenberg and V Wihstutz Large noise asymp
totics of invariant measures with applications to Lyapunov expo
nents Stochastics and Stochastics Reports
L Arnold and P Imkeller Furstenberg Khasminskii formulas for Lya
punov exponents via anticipative calculus Stochastics and Stochastics
Reports
L Arnold and P Imkeller Normal forms for stochastic di erential
equations Report Institut fur Dynamische Systeme Universitat
Bremen
L Arnold and P Imkeller Stratonovich calculus with spatial pa
rameters and anticipative problems in multiplicative ergodic theory
Stochastic Processes and their Applications
1

BIBLIOGRAPHY
L Arnold and P Imkeller On the integrability condition of the multi
plicative ergodic theorem for stochastic di erential equations Report
Institut fur Dynamische Systeme Universitat Bremen
L Arnold and P Imkeller Rotation numbers for linear stochastic
di erential equations Report Institut fur Dynamische Systeme Uni
versitat Bremen
L Arnold and R Khasminskii Stability index for nonlinear stochastic
di erential equations In M C Cranston and M A Pinsky editors
Stochastic Analysis Proceedings of Symposia in Pure Mathematics
Volume pages
American Mathematical Society Provi
dence Rhode Island
L Arnold and W Kliemann Qualitative theory of stochastic systems
In A T Bharucha Reid editor volume of Probabilistic Analysis
and Related Topics pages
Academic Press New York
L Arnold and W Kliemann Large deviations of linear stochastic
di erential equations In H J Engelbert and W Schmidt editors
Stochastic Di erential Systems pages
Springer Lec
ture Notes in Control and Information Sciences Volume
L Arnold and W Kliemann On unique ergodicity for degenerate
di usions Stochastics
L Arnold W Kliemann and E Oeljeklaus Lyapunov exponents of
linear stochastic systems In L Arnold and V Wihstutz editors Lya
punov Exponents Proceedings Bremen
pages
Springer
Berlin Heidelberg New York
L Arnold and P Kloeden Discretization of a random dynamical
system near a hyperbolic point Mathematische Nachrichten
L Arnold and Nguyen Dinh Cong Generic properties of Lyapunov
exponents Random Computational Dynamics
L Arnold and Nguyen Dinh Cong On the simplicity of the Lyaponov
spectrum of products of random matrices Ergodic Theory and Dy
namical Systems
L Arnold Nguyen Dinh Cong and V I Oseledets Jordan normal
form for linear cocycles In preparation
1

BIBLIOGRAPHY
L Arnold E Oeljeklaus and E Pardoux Almost sure and moment
stability for linear Ito equations In L Arnold and V Wihstutz
editors Lyapunov Exponents Proceedings Bremen
pages
Springer Berlin Heidelberg New York
L Arnold and L San Martin A control problem related to the lya
punov spectrum of stochastic ows Matematica Aplicada e Com
putacional
L Arnold and L San Martin A multiplicative ergodic theorem for
rotation numbers Journal of Dynamics and Di erential Equations
L Arnold and M Scheutzow Perfect cocycles through stochastic
di erential equations Probab Theory Relat Fields
L Arnold and B Schmalfu Fixed points and attractors for random
dynamical systems In A Naess and S Krenk editors IUTAM Sym
posium on Advances in Nonlinear Stochastic Mechanics pages
Kluwer Dordrecht
L Arnold N Sri Namachchivaya and K R Schenk Hoppe To
ward an understanding of stochastic Hopf bifurcation a case study
International Journal of Bifurcation and Chaos
L Arnold and V Wihstutz Stationary solutions of linear systems
with additive and multiplicative noise Stochastics
L Arnold and V Wihstutz editors Lyapunov Exponents Proceed
ings Bremen
volume
of Springer Lecture Notes in Math
ematics Springer Berlin Heidelberg New York
L Arnold and Xu Kedai Normal forms for random di eomorphisms
J Dynamics and Di erential Equations
L Arnold and Xu Kedai Simultaneous normal form and center
manifold reduction for random di erential equations In C Perello
C Simo and J Sola Morales editors EQUADIFF volume
pages
World Scienti c Singapore
L Arnold and Xu Kedai Invariant measures for random dynamical
systems and a necessary condition for stochastic bifurcation from a
xed point Random Computational Dynamics
L Arnold and Xu Kedai Normal forms for random di erential equa
tions Journal of Di erential Equations
1

BIBLIOGRAPHY
V I Arnold Geometrical methods in the theory of ordinary di eren
tial equations Springer Berlin Heidelberg New York
V I Arnold Dynamical systems V Springer Berlin Heidelberg New
York
B Aulbach and T Wanner Invariant ber bundles in dynamical
processes Preliminary version of a book
B Aulbach and T Wanner Integral manifolds for Caratheodory type
di erential equations in Banach spaces In B Aulbach and F Colo
nius editors Six lectures on dynamical systems pages
World
Scienti c Singapore
M Barlow and M Yor Semi martingale inequalities via the Garsia
Rodemich Rumsey lemma and applications to local times Journal
of Functional Analysis
M F Barnsley Fractals everywhere Academic Press New York
M F Barnsley and J Elton A new class of Markov processes for
image encoding Adv Appl Probab
H Bauer Ma und Integrationstheorie Walter de Gruyter Berlin
H Bauer Wahrscheinlichkeitstheorie Walter de Gruyter Berlin
P Baxendale Wiener processes on manifolds of maps Proc Royal
Soc Edinburgh A
P Baxendale Brownian motion in the di eomorphism group I Com
positio Mathematica
P Baxendale Asymptotic behavior of stochastic ows of di eomor
phisms two case studies Probab Theory Relat Fields
P Baxendale Asymptotic behaviour of stochastic ows of di eomor
phisms In K Ito and T Hida editors Stochastic processes and their
applications volume
of Springer Lecture Notes in Mathematics
pages Springer Berlin Heidelberg New York
1

BIBLIOGRAPHY
P Baxendale The Lyapunov spectrum of a stochastic ow of dif
feomorphisms In L Arnold and V Wihstutz editors Lyapunov ex
ponents Proceedings Bremen
pages
Springer Berlin
Heidelberg New York
P Baxendale Moment stability and large deviations for linear
stochastic di erential equations In N Ikeda editor Proc Taniguchi
Symposium on probabilistic methods in mathematical physics pages
Kinokuniya Tokyo
P Baxendale Lyapunov exponents and relative entropy for a stochas
tic ow of di eomorphisms Probab Th Rel Fields
P Baxendale Invariant measures for nonlinear stochastic di erential
equations In Arnold et al pages
P Baxendale Statistical equilibrium and two point motion for
a stochastic ow of di eomorphisms In K S Alexander and
J C Watkins editors Spatial stochastic processes pages
Birkhauser Boston
P Baxendale Stability and equilibrium properties of stochastic ows
of di eomorphisms In M Pinsky and V Wihstutz editors Di u
sion processes and related problems in analysis volume II Stochastic
ows volume of Progress in Probability pages
Birkhauser
Boston Basel Stuttgart
P Baxendale A stochastic Hopf bifurcation Probab Theory Relat
Fields
P Baxendale and D Stroock Large deviations and stochastic ows
of di eomorphisms Probab Theory Relat Fields
Y I Belopolskaya and Y L Dalecky Stochastic equations and dif
ferential geometry Kluwer Dordrecht
E Beno t editor Dynamic bifurcations volume
of Springer
Lecture Notes in Mathematics Springer Berlin Heidelberg New York
M Berger Introduction to probability and stochastic processes
Springer Berlin Heidelberg New York
N Bhatia and G Szego Stability theory of dynamical systems
Springer Berlin Heidelberg New York
1

BIBLIOGRAPHY
P Billingsley Convergence of probability measures Wiley New York
J M Bismut Flots stochastiques et formule de Ito Stratonovitch
generalisee C R Acad Sci Paris Ser I
J M Bismut Mecanique aleatoire volume
of Springer Lecture
Notes in Mathematics Springer Berlin Heidelberg New York
J P Bismut A generalized formula of Ito and some other proper
ties of stochastic ows Z Wahrscheinlichkeitstheorie verw Gebiete
T Bogenschutz Equilibrium states for random dynamical systems
PhD thesis Institut fur Dynamische Systeme Universitat Bremen
W M Boothby An introduction to di erentiable manifolds and Rie
mannian geometry Academic Press New York
P Bougerol Comparaison des exposants de Lyapounov des processus
markoviens multiplicatifs Ann Inst Poincare Probab Stat
P Bougerol Theoremes limite pour les systemes lineaires a coe
cients markovien Probab Th Rel Fields
P Bougerol and J Lacroix Products of random matrices with appli
cations to Schrodinger operators Birkhauser Boston Basel Stuttgart
N Bourbaki Elements de mathematique Fascicule XXV Livre Inte
gration Hermann Paris
P Boxler A stochastic version of center manifold theory Probab
Th Rel Fields
P Boxler How to construct stochastic center manifolds on the level
of vector elds In Arnold et al pages
L Breiman Probability SIAM Philadelphia
M Brin and Y Kifer Dynamics of Markov chains and stable man
ifolds for random di eomorphisms Ergodic Theory and Dynamical
Systems
1

BIBLIOGRAPHY
I Y Bronshtein Non autonomous dynamical systems Akademia
Nauk Moldavskoi SSR Kishinev
H Bunke Gewohnliche Di erentialgleichungen mit zufalligen Param
etern Akademie Verlag Berlin
B F Bylov R E Vinograd D M Grobman and V V Nemytskii
Theory of Lyapunov exponents Nauka Moscow in Russian
R Carmona and J Lacroix Spectral theory of random Schrodinger
operators Birkhauser Boston Basel Stuttgart
R Carmona and D Nualart Nonlinear stochastic integrators equa
tions and ows Gordon and Breach New York
A Carverhill A pair of stochastic dynamical systems which have
the same in nitesimal generator but of which one is strongly com
plete and the other is not Technical report Mathematics Institute
University of Warwick
A Carverhill Flows of stochastic dynamical systems ergodic theory
Stochastics
A Carverhill A formula for the Lyapunov numbers of a stochastic
ow Application to a perturbation theorem Stochastics
A Carverhill A nonrandom Lyapunov spectrum for nonlinear
stochastic dynamical systems Stochastics
A Carverhill Survey Lyapunov exponents for stochastic ows on
manifolds In L Arnold and V Wihstutz editors Lyapunov Expo
nents Proceedings Bremen
volume
of Springer Lecture
Notes in Mathematics pages
Springer Berlin Heidelberg
New York
A Carverhill M Chappell and D K Elworthy Characteristic ex
ponents for stochastic ows In S Albeverio P Blanchard and
L Streit editors Stochastic processes
mathematics and physics
volume
of Springer Lecture Notes in Mathematics pages
Springer Berlin Heidelberg New York
A Carverhill and K D Elworthy Flows of stochastic dynamical
systems the functional analytic approach Probab Th Rel Fields
1

BIBLIOGRAPHY
A Carverhill and K D Elworthy Lyapunov exponents for a stochas
tic analogue of the geodesic ow Trans Amer Math Soc
C Castaing and M Valadier Convex analysis and measurable mul
tifunctions volume
of Springer Lecture Notes in Mathematics
Springer Berlin Heidelberg New York
L Cesari Asymptotic behavior and stability problems in ordinary
di erential equations Springer Berlin Heidelberg New York
M Chappell Bounds for average Lyapunov exponents of gradient
stochastic systems In L Arnold and V Wihstutz editors Lyapunov
Exponents Proceedings Bremen
volume
of Springer Lec
ture Notes in Mathematics pages
Springer Berlin Heidel
berg New York
S Chow and Y Yi Center manifold and stability for skew product
ows J Dynamics and Di erential Equations
S N Chow and J K Hale Methods of bifurcation theory Springer
Berlin Heidelberg New York
S N Chow K Lu and Y Q Shen Normal forms for quasiperiodic
evolutionary equations Discrete and continuous dynamical systems
E Coddington and N Levinson Theory of ordinary di erential equa
tions McGraw Hill New York
J E Cohen H Kesten and C M Newman Oseledec s multiplicative
ergodic theorem a proof In Random matrices and their applications
pages
J E Cohen H Kesten and C M Newman editors Random Matri
ces and Their Applications volume of Contemporary Mathematics
American Mathematical Society Providence Rhode Island
F Colonius and W Kliemann Dynamics of control systems
Birkhauser Boston Basel Stuttgart
I P Cornfeld S V Fomin and Y G Sinai Ergodic theory Springer
Berlin Heidelberg New York
P H Coullet C Elphick and E Tirapegui Normal form of a Hopf
bifurcation with noise Physics Letters A
1

BIBLIOGRAPHY
H Crauel Ergodentheorie linearer stochastischer Systeme Re
port Institut fur Dynamische Systeme Universitat Bremen
H Crauel Random dynamical systems positivity of Lyapunov ex
ponents and Markov systems PhD thesis Institut fur Dynamische
Systeme Universitat Bremen
H Crauel Extremal exponents of random dynamical systems do not
vanish Journal of Dynamics and Di erential Equations
H Crauel Lyapunov exponents of random dynamical systems on
Grassmannians In Arnold et al pages
H Crauel Markov measures for random dynamical systems Stochas
tics and Stochastics Reports
H Crauel Non Markovian invariant measures are hyperbolic Sto
chastic Processes and their Applications
H Crauel Global random attractors are uniquely determined by
attracting deterministic compact sets Submitted
H Crauel Random probability measures on Polish spaces Habilita
tionsschrift Universitat Bremen
H Crauel A Debussche and F Flandoli Random attractors Jour
nal of Dynamics and Di erential Equations
H Crauel and F Flandoli Attractors for random dynamical systems
Probab Theory Relat Fields
H Crauel and F Flandoli Additive noise destroys a pitchfork bifur
cation Journal of Dynamics and Di erential Equations
to
appear
H Crauel and F Flandoli Hausdor dimension of invariant sets for
random dynamical systems Journal of Dynamics and Di erential
Equations to appear
S Dahlke Invariante Mannigfaltigkeiten fur Produkte zufalliger Dif
feomorphismen PhD thesis Institut fur Dynamische Systeme Uni
versitat Bremen
S Dahlke Invariant manifolds for products of random di eomor
phisms J Dynamics and Di erential Equations
1

BIBLIOGRAPHY
J de Sam Lazaro and P A Meyer Methodes de martingales et theorie
des ots Z Wahrscheinlichkeitstheorie verw Gebiete
J de Sam Lazaro and P A Meyer Questions de theorie des ots
Seminaire de Probabilites IX
Springer Lecture Notes in
Mathematics Volume
A Debussche On the nite dimensionality of random attractors
Stochastic Analysis and Applications to appear
F Delyon and P Foulon Complex entropy for dynamical systems
Preprint
J D Deuschel and D Stroock Large Deviations Academic Press
New York
J Dieudonne Grundzuge der modernen Analysis Volume VEB
Deutscher Verlag der Wissenschaften Berlin
J Doob Stochastic processes Wiley New York
J Doob Classical potential theory and its probabilistic counterpart
Springer Berlin Heidelberg New York
D Dubischar Representation of Markov processes by random dy
namical systems Report Institut fur Dynamische Systeme Uni
versitat Bremen
R M Dudley Real Analysis and Probability Wadsworth
Brooks Cole Paci c Grove California
N Dunford and J T Schwartz Linear Operators Part I General
theory Interscience New York
R Durrett Probability theory and examples Wadsworth
Brooks Cole Paci c Grove California
W Ebeling H Herzel W Richert and L Schimansky Geier In u
ence of noise on Du ng Van der Pol oscillators Zeitschrift f Angew
Math u Mechanik
C Elphick E Tirapegui M E Brachet P H Coullet and G Iooss
A simple global characterization for normal forms of singular vector
elds Physica D
1

BIBLIOGRAPHY
J Elton A multiplicative ergodic theorem for Lipschitz maps
Stochastic Processes and their Applications
K D Elworthy Stochastic dynamical systems and their ows In
A Friedman and M Pinsky editors Stochastic Analysis pages
Academic Press New York
K D Elworthy Stochastic di erential equations on manifolds Cam
bridge University Press Cambridge
K D Elworthy Geometric aspects of di usions on manifolds Ecole
d Ete de Probabilites de Saint Flour pages
Springer
Lecture Notes in Mathematics Volume
K D Elworthy Stochastic ows on Riemannian manifolds In M Pin
sky and V Wihstutz editors Di usion processes and related prob
lems in analysis volume II Stochastic ows volume of Progress in
Probability pages
Birkhauser Boston Basel Stuttgart
K D Elworthy Stochastic di erential geometry Bull Sc math e
serie
K D Elworthy Y Le Jan and X M Li Concerning the geometry
of stochastic di erential equations and stochastic ow Preprint
Warwick University
K D Elworthy and X M Li Derivative ows of stochastic di eren
tial equations moment exponents and geometric properties In M C
Cranston and M A Pinsky editors Stochastic Analysis Proceedings
of Symposia in Pure Mathematics Volume pages
Amer
ican Mathematical Society Providence Rhode Island
K D Elworthy and S Rosenberg Homotopy and homology vanishing
theorems and the stability of stochastic ows Preprint Warwick
University
K D Elworthy and M Yor Conditional expectations for derivatives
of certain stochastic ows Seminaire de Probabilites XXVII
Springer Lecture Notes in Mathematics Volume
M Emery Stochastic calculus in manifolds Springer Berlin Heidel
berg New York
S Ethier and T Kurtz Markov processes Characterization and con
vergence Wiley New York
1

BIBLIOGRAPHY
M Farkas Periodic motion Springer Berlin Heidelberg New York
A Fathi M R Herman and J C Yoccoz A proof of Pesin s stable
manifold theorem In J Palis editor Geometric Dynamics pages
Springer Lecture Notes in Mathematics Volume
F Flandoli and K U Schaumlo el A multiplicative ergodic theorem
with applications to a rst order stochastic hyperbolic equation in a
bounded domain Stochastics and Stochastics Reports
F Flandoli and B Schmalfu Random attractors for the D stochas
tic Navier Stokes equation with multiplicative white noise Stochas
tics and Stochastics Reports
M I Freidlin and A Wentzell Random perturbations of dynamical
systems Springer Berlin Heidelberg New York
H Furstenberg Noncommuting random products Trans Amer
Math Soc
H Furstenberg and H Kesten Products of random matrices Ann
Math
H Furstenberg and Y Kifer Random matrix products and measures
on projective spaces Israel Journal of Mathematics
P Ganssler and W Stute Wahrscheinlichkeitstheorie Springer
Berlin Heidelberg New York
F R Gantmacher The theory of matrices Chelsea New York
T C Gard Introduction to stochastic di erential equations Marcel
Dekker New York
I Gihman and A Skorohod Stochastic di erential equations
Springer Berlin Heidelberg New York
I Gihman and A Skorohod The theory of stochastic processes
I II III Springer Berlin Heidelberg New York
I Y Goldsheid and G A Margulis Lyapunov indices of a product
of random matrices Russian Mathematical Surveys
1

BIBLIOGRAPHY
J Guckenheimer and P Holmes Nonlinear oscillations dynamical
systems and bifurcation of vector elds Springer Berlin Heidelberg
New York
Y Guivarc h and A Raugi Frontiere de Furstenberg proprietes de
contraction et theoreme de convergence Z Wahrscheinlichkeitstheo
rie Verw Gebiete
V M Gundlach Random homoclinic orbits Random Computa
tional Dynamics
W Hackenbroch and A Thalmaier Stochastische Analysis Teubner
Stuttgart
W Hahn Stability of motion Springer Berlin Heidelberg New York
J Hale and H Kocak Dynamics and bifurcations Springer Berlin
Heidelberg New York
P Hartman Ordinary di erential equations Birkhauser Boston
Basel Stuttgart
H Hennion Loi de grands nombres et perturbations pour des pro
duits reductibles de matrices aleatoires independantes Z Wahr
scheinlichkeitstheorie verw Gebiete
B Hess M Kiwi M Markus and J Rossler Periodically and ran
domly modulated non linear processes In E Tirapegui and D Villar
roel editors Instabilities and Nonequilibrium Structures pages
Kluwer Dordrecht
S Hilger Smoothness of invariant manifolds J Functional Analysis
G Hognas and A Mukherjea Probability measures on semigroups
convolution products random walks and random matrices Plenum
Press
P Holmes and D Rand Phase portraits and bifurcations of the
nonlinear oscillator x
xx
x
x
International
Journal of Non linear Mechanics
W Horsthemke and D K Kondepudi editors Fluctuations and sen
sitivity in nonequilibrium systems volume of Springer Proceedings
in Physics Springer Berlin Heidelberg New York
1

BIBLIOGRAPHY
W Horsthemke and R Lefever Noise induced transitions Springer
Berlin Heidelberg New York
D Husemoller Fibre bundles Springer Berlin Heidelberg New York
third edition edition
J E Hutchinson Fractals and self similarity Indiana Univ Math
J
N Ikeda and S Watanabe Stochastic di erential equations and dif
fusion processes North Holland Kodansha Tokyo
P Imkeller The smoothness of laws of random ags and Oseledets
spaces of linear stochastic di erential equations Preprint Humboldt
Universitat Berlin
P Imkeller The smoothness of laws of random ags and Oseledets
spaces of linear stochastic di erential equations Potential Analysis
To appear
G Iooss Global characterization of the normal form for a vector eld
near a closed orbit Journal of Di erential Equations
M C Irwin Smooth dynamical systems Academic Press New York
K Ito Di erential equations determining a Markov process Journal
of the Pan Japanese Mathematical Colloquium
K Ito On stochastic di erential equations Memoirs of the American
Mathematical Society Volume
K Ito Stochastic integral Proc Imp Acad Tokyo
J Jacod Calcul stochastique et problemes de martingales Springer
Berlin Heidelberg New York
Lecture Notes in Mathematics
Volume
J Jacod and A N Shiryaev Limit theorems for stochastic processes
Springer Berlin Heidelberg New York
R Johnson On a Floquet theory for almost periodic two dimensional
linear systems Journal of Di erential Equations
R Johnson m functions and Floquet exponents for linear di erential
systems Ann Mat Pura Appl
1

BIBLIOGRAPHY
R Johnson The Oseledec and Sacker Sell spectra for almost periodic
linear systems an example Proceedings of the Amer Math Soc
R Johnson Exponential dichotomy and rotation number for linear
Hamiltonian systems J of Di erential Equations
R Johnson and J Moser The rotation number for almost periodic
potentials Commun Math Phys
R Johnson K Palmer and G R Sell Ergodic properties of linear
dynamical systems SIAM J Math Anal
G Kager Zur Perfektionierung nicht invertierbarer grober Kozykel
PhD thesis Fachbereich Mathematik Technische Universitat Berlin
G Kager and M Scheutzow Generation of one sided random dy
namical systems by stochastic di erential equations Preprint
V A Kaimanovich Lyapunov exponents symmetric spaces and a
multiplicative ergodic theorem for semisimple Lie groups Zapiski
Nauchn Sem Leningr Otdel Matemat Inst Steklova
J Kao and V Wihstutz Stabilization of companion form systems by
mean zero noise Stochastics and Stochastics Reports
I Karatzas and S E Shreve Brownian motion and stochastic calcu
lus Springer Berlin Heidelberg New York
S Karlin and H Taylor A second course in stochastic processes
Academic Press New York
T Kato Perturbation theory for linear operators Springer Berlin
Heidelberg New York second edition edition
A Katok and B Hasselblatt Introduction to the modern theory of
dynamical systems Cambridge University Press
A Katok and J M Strelcyn Invariant manifolds entropy and bil
liards smooth maps with singularities volume
of Springer Lec
ture Notes in Mathematics Springer Berlin Heidelberg New York
H Keller Attraktoren und Bifurkationen des stochastischen Lorenz
Systems Diplomarbeit Universitat Bremen
1

BIBLIOGRAPHY
H Keller Attractors and bifurcations of the stochastic Lorenz system
Report Institut fur Dynamische Systeme Universitat Bremen
E S Key Lyapunov exponents for matrices with invariant subspaces
Ann Probab
R Z Khasminskii A limit theorem for solutions of di erential equa
tions with random right hand side Theory of Probability and its
Applications
R Z Khasminskii Necessary and su cient conditions for the asymp
totic stability of linear stochastic systems Theory of Probability and
its Applications
R Z Khasminskii Stochastic stability of di erential equations Si
jtho and Noordho Alphen Translation of the Russian edi
tion Nauka Moscow
Y Kifer Ergodic theory of random transformations Birkhauser
Boston Basel Stuttgart
Y Kifer A note on integrability of Cr norms of stochastic ows
and applications In A Truman and I M Davies editors Stochastic
Mechanics and Stochastic Processes pages
Springer
Lecture Notes in Mathematics Volume
J F C Kingman The ergodic theory of subadditive stochastic pro
cesses J Royal Statist Soc Ser B
J F C Kingman Subadditive ergodic theory Ann Probab
J F C Kingman Subadditive processes Ecole d Ete de Proba
bilites de Saint Flour pages
Springer Lecture Notes
in Mathematics Volume
W Kliemann Recurrence and invariant measures for degenerate dif
fusions Ann Probab
W Klingenberg Riemannian geometry Walter de Gruyter Berlin
P E Kloeden and E Platen Numerical solution of stochastic di er
ential equations Springer Berlin Heidelberg New York
1

BIBLIOGRAPHY
O Knill The upper Lyapunov exponent of Sl R cocycles Dis
continuity and the problem of positivity In Arnold et al pages
O Knill Positive Lyapunov exponents for a dense set of bounded
measurable Sl R cocycles Ergodic Theory and Dynamical Systems
S Kobayashi and K Nomizu Foundations of di erential geometry
volume I Wiley New York
A N Kolmogorov The Wiener helix and other interesting curves in
Hilbert space Dokl Akad Nauk
H J Kowalsky Lineare Algebra Walter de Gruyter Berlin th
edition edition
M Krebs Bifurkationsverhalten des stochastischen Brusselators
Diplomarbeit Universitat Bremen
U Krengel Ergodic theorems Walter de Gruyter Berlin
A Kufner O John and S Fucik Function spaces Academia Pub
lishing House of the Czechoslovak Academy of Sciences Prague
H Kunita On the decomposition of solutions of stochastic di erential
equations In D Williams editor Stochastic Integrals pages
Springer Lecture Notes in Mathematics Volume
H Kunita Stochastic di erential equations and stochastic ows of
di eomorphisms Ecole d Ete de Probabilites de Saint Flour pages
Springer Lecture Notes in Mathematics Volume
H Kunita Stochastic ows and stochastic di erential equations
Cambridge University Press Cambridge
H Kushner Stochastic stability and control Academic Press New
York
A Lasota and M Mackey Chaos fractals and noise Springer
Berlin Heidelberg New York
Y Le Jan Equilibre statistique pour les produits de di eomorphismes
aleatoires independants Ann Inst H Poincare Probab Statist
1

BIBLIOGRAPHY
Y Le Jan and S Watanabe Stochastic ows of di eomorphisms
In K Ito editor Stochastic Analysis Proceedings of the Taniguchi
Symposium
pages
North Holland Amsterdam
R Leandre Un exemple en theorie des ots stochastiques Seminaire
de Probabilites XVII
Lecture Notes in Mathematics
Volume
F Ledrappier Quelques proprietes des exposants caracteristiques
Ecole d Ete de Probabilites de Saint Flour pages
Springer Lecture Notes in Mathematics Volume
F Ledrappier and L S Young Stability of Lyapunov exponents
Ergod Th Dynam Sys
R Lefever and J Turner Sensitivity of a Hopf bifurcation to external
multiplicative noise In W Horsthemke and D K Kondepudi editors
Fluctuations and sensitivity in nonequilibrium systems pages
Springer Berlin Heidelberg New York
R Lefever and J Turner Sensitivity of a Hopf bifurcation to multi
plicative colored noise Physical Review Letters
Li Xue Mei Strong p completeness of stochastic di erential equa
tions and the existence of smooth ows on noncompact manifolds
Probab Theory Relat Fields
Li Xue Mei Stochastic ows on noncompact manifolds PhD thesis
University of Warwick
M Liao Stochastic ows on the boundaries of Lie groups Stochastics
and Stochastics Reports
M Liao Stochastic ows on the boundaries of SL n R Probab
Theory Relat Fields
M Liao The Brownian motion and the canonical stochastic ow on
a symmetric space Trans Amer Math Soc
M Liao Lyapunov exponents of stochastic ows Preprint
M A Liapouno Probleme general de la stabilite du mouvement
Annales Fac Sciences Toulouse
Translation of the Rus
sian edition Kharkov
Reprinted by Princeton University Press
Princeton N J
and
1

BIBLIOGRAPHY
I Lindemann Stability of hyperbolic PDE s with random loads
SIAM J Appl Math
I Lindemann Lyapunov exponents of random block diagonal systems
in in nite dimensions Stochastics and Stochastics Reports
R Liptser A strong law of large numbers for local martingales
Stochastics
P D Liu and M Qian Smooth ergodic theory of random dynami
cal systems volume
of Springer Lecture Notes in Mathematics
Springer Berlin Heidelberg New York
M Lucke Bifurcation behaviour under modulated control param
eters In F Moss and P McClintock editors Noise in nonlinear
dynamical systems Cambridge University Press
M Lucke and F Schank Response to parametric modulation near
an instability Phys Rev Letters
R Mane Lyapunov exponents and stable manifolds for compact
transformations In J Palis editor Geometric Dynamics pages
Springer Lecture Notes in Mathematics Volume
P Malliavin Stochastic analysis Springer Berlin Heidelberg New
York
R Mane Ergodic theory and di erentiable dynamics Springer Berlin
Heidelberg New York
M S Marasimhan and S Ramanan Existence of universal connec
tions American J Math
G A Margulis Discrete subgroups of semisimple Lie groups
Springer Berlin Heidelberg New York
P A Meyer Flot d une equation differentielle stochastique Semi
naire de Probabilites pages
Springer Lecture Notes in
Mathematics Volume
S Meyn and R Tweedie Markov chains and stochastic stability
Springer Berlin Heidelberg New York
V M Millionshchikov Metric theory of linear systems of di erential
equations Math USSR Sbornik
1

BIBLIOGRAPHY
S E Mohammed The Lyapunov spectrum and stable manifolds for
stochastic linear delay equations Stochastics and Stochastics Reports
S E Mohammed Lyapunov exponents and stochastic ows of linear
and a ne hereditary systems In M Pinsky and V Wihstutz edi
tors Di usion processes and related problems in analysis volume II
Stochastic ows volume of Progress in Probability pages
Birkhauser Boston Basel Stuttgart
S E Mohammed and M Scheutzow Lyapunov exponents of linear
stochastic functional di erential equations driven by semimartingales
Part I the multiplicative ergodic theory Ann Inst Henri Poincare
Prob et Stat
Nguyen Dinh Cong Structural stability and topological classi cation
of continuous time linear hyperbolic cocycles Random Computa
tional Dynamics
Nguyen Dinh Cong Topological dynamics of random dynamical sys
tems Oxford University Press
C Nicolis and G Nicolis Normal form analysis of stochastically
forced dynamical systems Dynamics and Stability of Systems
G Nicolis and I Prigogine Self organization in non equilibrium sys
tems Wiley New York
G L O Brien The occurence of large values in stationary sequences
Z Wahrscheinlichkeitstheorie verw Gebiete
G Ochs On the stability of Oseledets spaces under random pertur
bations Report Institut fur Dynamische Systeme Universitat
Bremen
T Ohno Asymptotic behaviors of dynamical systems with random
parameters Publ RIMS Kyoto Univ
S Orey Stationary solutions for linear systems with additive noise
Stochastics
V I Oseledets A multiplicative ergodic theorem Lyapunov charac
teristic numbers for dynamical systems Trans Moscow Math Soc
1

BIBLIOGRAPHY
K J Palmer A proof of Oseledec s multiplicative ergodic theorem
Preprint
E Pardoux and V Wihstutz Lyapunov exponent and rotation num
ber of two dimensional linear stochastic systems with small di usion
SIAM J Appl Math
E Pardoux and V Wihstutz Lyapunov exponent of linear stochastic
systems with large di usion term Stochastic Processes and their
Applications
K P Parthasarathy Probability measures on metric spaces Aca
demic Press New York
L Pastur and A Figotin Spectra of random and almost periodic
operators Springer Berlin Heidelberg New York
Y B Pesin Families of invariant manifolds corresponding to nonzero
characteristic exponents Math USSR Izvestija
Y B Pesin Characteristic Lyapunov exponents and smooth ergodic
theory Russian Mathematical Surveys
K Petersen Ergodic theory Cambridge University Press Cambridge
R Phelps Lectures on Choquet s theorem Van Nostrand Princeton
NJ
M Pinsky Extremal character of the Lyapunov exponent of the
stochastic harmonic oscillator The Annals of Applied Probability
M Pinsky and W Wihstutz Lyapunov exponents of nilpotent Ito
systems Stochastics
P Protter Semimartingales and measure preserving ows Ann Inst
Poincare Probab Stat
P Protter Stochastic integration and di erential equations Springer
Berlin Heidelberg New York
C Pugh and M Shub Ergodic attractors Trans Amer Math Soc
A Quas On representations of Markov chains by random smooth
maps Bull London Math Soc
1

BIBLIOGRAPHY
A Quas Representation of Markov chains on tori Random Com
putational Dynamics
M S Ragunathan A proof of Oseledec s multiplicative ergodic the
orem Israel J Math
D Revuz and M Yor Continuous martingales and Brownian motion
Springer Berlin Heidelberg New York
L C G Rogers and D Williams Di usions Markov processes and
martingales Volume Ito calculus Wiley New York
V A Rokhlin On the fundamental ideas of measure theory Mat
Sb NS
English translation AMS Transl
V A Rokhlin Selected topics from the metric theory of dynamical
systems Usp Mat Nauk
English translation AMS
Transl
V A Rokhlin Exact endomorphisms of a Lebesgue space Izv Acad
Sci USSR Ser Mat
English translation AMS
Transl
D J Rudolph Fundamentals of measurable dynamics Oxford Uni
versity Press Oxford
D Ruelle Ergodic theory of di erentiable dynamical systems Publ
Math Inst Hautes Etud Sci
D Ruelle Characteristic exponents and invariant manifolds in
Hilbert space Annals of Mathematics
D Ruelle Rotation numbers for di eomorphisms and ows Ann
Inst Henri Poincare
D Ruelle Elements of di erentiable dynamics and bifurcation theory
Academic Press New York
P Ru no Rotation numbers for stochastic dynamical systems PhD
thesis University of Warwick
P Ru no Rotation numbers for stochastic dynamical systems
Stochastics and Stochastics Reports
To appear
1

BIBLIOGRAPHY
R Sacker and G Sell Lifting properties in skew product ows with
applications to di erential equations Memoirs of the American Math
ematical Society Volume
R Sacker and G Sell A spectral theory for linear di erential systems
Journal of Di erential Equations
L San Martin Rotation numbers in higher dimensions Report
Institut fur Dynamische Systeme Universitat Bremen
K U Schaumlo el Multiplicative ergodic theorems in in nite di
mensions In Arnold et al pages
K R Schenk Hoppe Bifurcation scenarios of the noisy Du ng van
der Pol oscillator Nonlinear Dynamics
K R Schenk Hoppe Deterministic and stochastic Du ng van der
Pol oscillators are non explosive ZAMP
K R Schenk Hoppe The stochastic Du ng van der Pol equation
PhD thesis Institut fur Dynamische Systeme Universitat Bremen
K R Schenk Hoppe Stochastic Hopf bifurcation an example Int
J Non Linear Mechanics
K R Schenk Hoppe Attractors and invariant measures of the
stochastic Du ng van der Pol equation Report Institut fur
Dynamische Systeme Universitat Bremen
K R Schenk Hoppe Random attractors I Report Department
of Economics University of Bielefeld
K R Schenk Hoppe Random attractors II Report Department
of Economics University of Bielefeld
B Schmalfu Backward cocycles and attractors of stochastic di er
ential equations In V Reitmann T Riedrich and N Koksch editors
International Seminar on Applied Mathematics Nonlinear Dynam
ics Attractor Approximation and Global Behaviour pages
Teubner Leipzig
B Schmalfu Measure attractors and stochastic attractors Report
Institut fur Dynamische Systeme Universitat Bremen
B Schmalfu Attractors for the non autonomous Navier Stokes
equation Submitted
1

BIBLIOGRAPHY
B Schmalfu The random attractor of the stochastic Lorenz system
ZAMP
To appear
B Schmalfu A random xed point theorem and the random graph
transform Report Institut fur Dynamische Systeme Universitat
Bremen
K Schmidt Cocycles on ergodic transformation groups MacMillan
Delhi
K Schmidt Algebraic ideas in ergodic theory American Mathemat
ical Society Providence R I
Regional Conference Series in
Mathematics Number
G Schoner and H Haken The slaving principle for Stratonovich
stochastic di erential equations Z Phys B
G Schoner and H Haken A systematic elimination procedure for Ito
stochastic di erential equations and the adiabatic approximation Z
Phys B
G Sell The structure of a ow in the vicinity of an almost periodic
motion Journal of Di erential Equations
S Siegmund Zur Di erenzierbarkeit von Integralmannigfaltigkeiten
Diplomarbeit Universitat Augsburg
Y G Sinai editor Dynamical systems II Encyclopaedia of Math
ematical Sciences Volume Springer Berlin Heidelberg New York
A V Skorokhod Random linear operators Reidel Dordrecht
Translation of the Russian edition Naukova Dumka Kiev
A V Skorokhod Asymptotic methods in the theory of stochastic dif
ferential equations Volume Transl Amer Math Soc Providence
R I Translation of the Russian edition Nauka Moscow
C Sparrow The Lorenz equations bifurcations chaos and strange
attractors Springer Berlin Heidelberg New York
N Sri Namachchivaya Stochastic bifurcation Applied Mathematics
and Computation
N Sri Namachchivaya and Y K Lin Methods of stochastic normal
forms Int J Nonlin Mech
1

BIBLIOGRAPHY
J M Steele Kingman s subadditive ergodic theorem Ann Inst
Poincare Probab Stat
D Stroock On the rate at which a homogeneous di usion approaches
a limit an application of the large deviation theory of certain stochas
tic integrals Annals of Probability
D W Stroock and S R S Varadhan Multidimensional di usion
processes Springer Berlin Heidelberg New York
D Tanny A zero one law for stationary sequences Z Wahrschein
lichkeitstheorie verw Gebiete
R Temam In nite dimensional dynamical systems in mechanics and
physics Springer Berlin Heidelberg New York
P Thieullen Fibres dynamiques asymptotiquement compacts
exposants de Lyapunov Entropie Dimension Ann Inst Henri
Poincare Anal Non Lineaire
C W S To and D M Li Largest Lyapunov exponents and bifurca
tions of stochastic nonlinear systems Shock and Vibration
C W S To D M Li and K L Huang Supercritical and subcritical
Hopf bifurcations in a stochastically excited system Journal of Sound
and Vibration
A Vanderbauwhede Center manifolds normal forms and elementary
bifurcations In U Kirchgraber and H O Walter editors Dynamics
Reported volume pages
Teubner and Wiley New York
A Vanderbauwhede and S Van Gils Center manifolds and contrac
tions on a scale of banach spaces Journal of Functional Analysis
J B Walsh The perfection of multiplicative functionals Semi
naire de Probabilites pages
Springer Lecture Notes in
Mathematics Volume
W Walter Gewohnliche Di erentialgleichungen Springer Berlin
Heidelberg New York th edition
P Walters An introduction to ergodic theory Springer Berlin Hei
delberg New York
1

BIBLIOGRAPHY
P Walters A dynamical proof of the multiplicative ergodic theorem
Trans Amer Math Soc
T Wanner Invariante Faserbundel und topologische Aquivalenz bei
dynamischen Prozessen Diplom Arbeit Universitat Augsburg
T Wanner Zur Linearisierung zufalliger dynamischer Systeme PhD
thesis Universitat Augsburg
T Wanner Linearization of random dynamical systems In C Jones
U Kirchgraber and H O Walther editors Dynamics Reported vol
ume pages
Springer Berlin Heidelberg New York
T Wanner Qualitative behavior of random di erential equations In
D Bainov and A Dishliev editors Proceedings of the Fifth Inter
national Colloquium on Di erential Equations pages
SCT
Publishing
W Wedig Vom Chaos zur Ordnung GAMM Mitteilungen
W Wedig Lyapunov exponents and invariant measures of equilibria
and limit cycles In Arnold et al pages
W Wedig Simulation and analysis of mechanical systems with pa
rameter uctuations International Series of Numerical Mathematics
A D Wentzell Theorie zufalliger Prozesse Akademie Verlag Berlin
S Wiggins Introduction to applied nonlinear dynamical systems and
chaos Springer Berlin Heidelberg New York
Xu Kedai Bifurcations of random di erential equations in dimension
one Random Computational Dynamics
Y Yi A generalized integral manifold theorem Journal of Di eren
tial Equations
Y Yi Stability of integral manifold and orbital attraction of quasi
periodic motion Journal of Di erential Equations
M I Zakharevich Characteristic exponents and a vector ergodic
theorem Vestn Leningr Gos Univ
1

BIBLIOGRAPHY
E C Zeeman On the classi cation of dynamical systems Bull
London Math Soc
E C Zeeman Stability of dynamical systems Nonlinearity
R J Zimmer Ergodic theory and semisimple groups Birkhauser
Boston Basel Stuttgart
1

Index
absolute derivative
absorbing set
abstract bifurcation point
local
action of group on space
additive noise
a ne cocycle
a neRDE
a neRDS
invariant measure
a neSDE
almost periodic function
anglebetween subspaces
between vectors
attractor
average Lyapunov exponent
backward cocycle
backward Markov measure
backward semimartingale
bifurcation
Hopf
pitchfork
saddle node
transcritical
bifurcation point
abstract
dynamical
phenomenological
Borel space
Brownian motion
Brusselator
bundle RDS
continuous
linear
smooth
Ck RDS
cadlag
canonical realization
center manifold
characteristic exponent
coboundary
cocycle
ane
backward
crude
group valued
induced by action
local
on homogeneous space
on principal bundle
perfect
semimartingale
very crude
cocycle property
cohomological equation
RDE case
SDE case
cohomological operator
RDE case
1

INDEX
SDE case
cohomology
Lyapunov
compact open topology
complete RDS
continuous DS
global
local
continuous RDS
contraction theorem
countably generated
covariant derivative
D bifurcation
local
su cient condition
determinant
diagonalization
of linear RDE
of linear SDE
dichotomy spectrum
di erential equation
classical solution
solution in the sense of Cara
theodory
disintegration see factorization
domain of attraction
DS see dynamical system
Du ng van der Pol oscillator
averaged
normal form
dynamical bifurcation
dynamical characterization
dynamical system
Ck
continuous
ergodic
ltered
metric
on manifold
slowly varying
equivalence of RDS
local topological
smooth
topological
equivariant measure
ergodic decomposition
ergodic DS
ergodic theorem
extended version
exhaustive algebra
explosion time
extension of DS
exterior power
invariant measure
factor of DS
factorization
of image measure
of measure
Feller Markov process
ltered DS
ltration
measurability of
ag of stable manifolds
of unstable manifolds
ag manifold
Floquet exponent
ow global
local
formal linearization
of RDE
of RDS
of SDE
forward invariant set
forward Markov measure
forward semimartingale
Furstenberg Kesten theorem

INDEX
for backward cocycles
for one sided time
for two sided time
Furstenberg Khasminskii formu
las
for discrete time
for linear RDE
for linear SDE
for top exponent
for SDE on manifold
for sum of exponents
future of RDS
gap conditions
generation of RDS
by Ito SDE
by RDE
by Stratonovich SDE
one sided discrete time
two sided discrete time
global ow
global RDS
from Ito SDE
from RDE
from scalar SDE
from Stratonovich SDE
globalization of local manifold
gradient Brownian ow
Gronwall Bellman lemma
continuous time
discrete time
group valued cocycle
Hartman Grobman theorem
hyperbolic case
local
non hyperbolic case
helix
semimartingale
Hopf bifurcation
independent increments of RDS
indistinguishable
invariant foliations
invariant function
invariant manifolds
Ck smoothness
center
classical vs generalized
dynamical characterization
of
foliation by
for RDE
global
global center
global stable
global unstable
globalization of local
local
Oseledets
stable
unstable
invariant measure
factorization of
fora neRDS
for continuous RDS
for local RDS
for RDS induced on Gk d
for RDS induced on Sd
for state space R
lift of
on random set
on Stiefel bundle
on unit sphere bundle
invariant mod P
invariant set
forward
isomorphism

INDEX
metric
isomorphism of RDS
linear
metric
iterated function system
Kingman s theorem see subaddi
tive ergodic theorem
Kolmogorov s fundamental theo
rem
Kronecker product
Krylov Bogolyubov procedure
Langevin equation
Lebesgue space
lift of measure
linear bundle RDS
linear isomorphism of RDS
linear RDE
diagonalization
linear RDS
on invariant bundle
on quotient space
linear SDE
diagonalization
linearization of RDS
Liouville s equation
for RDE
for RDE on manifold
for SDE
for SDE on manifold
on manifold
local characteristics
local cocycle property
local ow
local RDS
from RDE
from RDE on manifold
from SDE
from SDE on manifold
local topological equivalence of
RDS
Lorenz equation
Lyapunov cohomology
for di erent bases
for random norms
for random Riemannian met
ric
Lyapunov exponent
average
backward
forward
multiplicity of
of a system
small noise expansion
top
Lyapunov index
and scale of Banach spaces
of angle between subspaces
of angle between vectors
of determinant
of stationary process
of volume
Lyapunov metric norm
Lyapunov spectrum
of exterior power
of inverse and adjoint
of RDS induced on Sd
of tensor product
simplicity of
manifold
Markov measure
and stationary measure
backward
forward
on Sd
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INDEX
Markov process
Markov property
Markovian RDS
measurability of ltration
measurable bundle
measurable isomorphism of RDS
measurable RDS
measurable selection theorem
measurable space
memoryless RDE
MET see multiplicative ergodic
theorem
metric DS
metric isomorphism
metric isomorphism of RDS
multiplicative ergodic theorem
for exterior power
for inverse and adjoint
for linear RDE
for linear SDE
for local RDS
for one sided time
for RDE
for RDS on manifolds
for rotation numbers
for SDE
for subbundle and quotient
for tensor product
for two sided time
independent of norm
Lyapunov spectrum
on Euclidean bundle
strong version of
total measure
multiplicative noise
n point motion
backward
forward
natural extension
of canonical DS
Nemytskii operator
regularity of
never exploding initial values
noisy parameters
nonresonant of order n
normal form
for discrete time
for RDE
for SDE
normal form and center manifold
for RDE
with parameters
normed linear bundle
orbit
Oseledets manifolds
Oseledets spaces
Oseledets splitting
for exterior power
for inverse and adjoint
for RDS induced on Sd
for tensor product
Oseledets s theorem see multi
plicative ergodic theo
rem
P bifurcation
su cient condition
past of RDS
perfection of a cocycle
for continuous time
for discrete time
periodic linear system
phenomenological bifurcation
pitchfork bifurcation
1

INDEX
prepared RDS
principal ber bundle
probability space
complete
product of random mappings
product of random matrices
random attractor
random averaging
random coordinate transforma
tion
random di erential equation
ane
classical solution
linear
solution in the sense of Cara
theodory
random Dirac measure
random dynamical system
ane
Ck
complete
continuous
future of
generation of
by RDE
by SDE
for discrete time
independent increments
linear
local
Markov chain for
Markovian
measurable
on bundle
on subset
past of
prepared
smooth
stationary increments
strictly complete
topological
random eigenvalue
random eigenvectors
basis of
random homeomorphism
random norm
random Riemannian metric
random scalar product
random set
RDE see random di erential
equation
RDS see random dynamical sys
tem
reduction principle
regular linear system
resonant of order n
right invariant
RDE
SDE
rotation number
for canonical plane
for invariant measure
for linear di erential equa
tion
for linear SDE
for nonlinear differential
equation
for nonlinear RDE
MET
Sacker Sell spectrum
saddle node bifurcation
sample measure see factorization
scale of Banach spaces
contraction theorem
SDE see stochastic di erential
equation
semimartingale
backward
forward
semimartingale cocycle
semimartingale helix
1

INDEX
Ck
bCk
local characteristics of
with spatial parameters
shift
simple Lyapunov spectrum
singular value
skew product
slowly varying
smooth equivalence of RDS
smooth RDS
spectral theory of linear RDS
stable manifolds
stable reference solution
asymptotically
exponentially
standard measurable space
stationary increments of RDS
stationary measure
and Markov measure
for extreme exponents
stationary process
Stiefel bundle
Stiefel manifold
stochastic di erential equation
Ito
Stratonovich
stochastic integral
backward Ito
backward Stratonovich
forward Ito
forward Stratonovich
stochastic process
equivalent
indistinguishable
measurable
stationary
with stationary increments
strictly forward invariant set
strictly complete RDS
backward
forward
subadditive ergodic theorem
subadditive process
support of a measure
tail algebra
tangent bundle
tempered
ball
from above
from below
set
tensor product
time T
time reversible
topological decoupling
topological equivalence of RDS
local
topological linearization
topology of weak convergence
transcritical bifurcation
two parameter ow
universally measurable set
universe of sets
unstable manifolds
graph of
variational equation
for RDE
for RDE on manifold
for SDE
for SDE on manifold
on manifold
weak convergence
white noise
Wiener process
1