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VI Preface

Our main application is to the so-called cooperative random and stochas-tic ordinary differential equations. These systems arise naturally from math-

ematical models in the eld of ecology, epidemiology, economics and bioche-mistry (see, e.g., the literature quoted in Smith [102]). Deterministic co-operative differential equations have been studied by many authors (see,e.g., Smith [102] and the references therein). The books by Krasnoselskii[68, 69] and the series of papers by Hirsch [52, 53, 54] (see also the referencesin Smith [102]) lay the groundwork for the qualitative theory of deterministiccooperative systems. Monotone methods and comparison arguments are of prime importance in the study of these systems.

The results presented in this book rely on ideas and methods developed

in collaboration with Ludwig Arnold (see Arnold/Chueshov [5], [6] and[7]). The author is extremely grateful to him for very stimulating and fruitfuldiscussions on the subject. Warmest thanks are also due to Gunter Ochs,James Robinson and Bj orn Schmalfuss for their comments and suggestions,all of which improved the book.

The book was written while the author was spending the 2000/2001 aca-demic year at the Institut f ur Dynamische Systeme, Universit at Bremen. Hewould like to thank the people at that institution for their very kind hospital-ity during this period. He also gratefully acknowledges the nancial support

of the Deutsche Forschungsgemeinschaft.September 2001 Igor Chueshov


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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. General Facts about Random Dynamical Systems . . . . . . . . 9

1.1 Metric Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Concept of RDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Random Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4 Dissipative, Compact and Asymptotically Compact RDS . . . . 241.5 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.6 Omega-limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1.7 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.8 Random Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.9 Dissipative Linear and Affine RDS . . . . . . . . . . . . . . . . . . . . . . . . 451.10 Connection Between Attractors and Invariant Measures . . . . . 49

2. Generation of Random Dynamical Systems . . . . . . . . . . . . . . . 552.1 RDS Generated by Random Differential Equations . . . . . . . . . . 552.2 Deterministic Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.3 The It o and Stratonovich Stochastic Integrals . . . . . . . . . . . . . . 65

2.4 RDS Generated by Stochastic Differential Equations . . . . . . . . 702.5 Relations Between RDE and SDE . . . . . . . . . . . . . . . . . . . . . . . . 76

3. Order-Preserving Random Dynamical Systems . . . . . . . . . . . 833.1 Partially Ordered Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 833.2 Random Sets in Partially Ordered Spaces . . . . . . . . . . . . . . . . . . 883.3 Denition of Order-Preserving RDS . . . . . . . . . . . . . . . . . . . . . . . 933.4 Sub-Equilibria and Super-Equilibria . . . . . . . . . . . . . . . . . . . . . . 953.5 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6 Properties of Invariant Sets of Order-Preserving RDS . . . . . . . 1053.7 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4. Sublinear Random Dynamical Systems . . . . . . . . . . . . . . . . . . .1134.1 Sublinear and Concave RDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2 Equilibria and Semi-Equilibria for Sublinear RDS. . . . . . . . . . . 1164.3 Almost Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122


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VIII Contents

4.4 Limit Set Trichotomy for Sublinear RDS . . . . . . . . . . . . . . . . . . 1254.5 Random Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.6 Positive Affine RDS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5. Cooperative Random Differential Equations . . . . . . . . . . . . . . 1435.1 Basic Assumptions and the Existence Theorem . . . . . . . . . . . . . 1435.2 Generation of RDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.3 Random Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.4 Equilibria, Semi-Equilibria and Attractors . . . . . . . . . . . . . . . . . 1565.5 Random Equations with Concavity Properties . . . . . . . . . . . . . . 1605.6 One-Dimensional Explicitly Solvable Random Equations . . . . . 166

5.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.7.1 Random Biochemical Control Circuit . . . . . . . . . . . . . . . 1715.7.2 Random Gonorrhea Model. . . . . . . . . . . . . . . . . . . . . . . . . 1755.7.3 Random Model of Symbiotic Interaction . . . . . . . . . . . . . 1765.7.4 Random Gross-Substitute System . . . . . . . . . . . . . . . . . . 178

5.8 Order-Preserving RDE with Non-Standard Cone . . . . . . . . . . . 180

6. Cooperative Stochastic Differential Equations . . . . . . . . . . . . 1856.1 Main Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6.2 Generation of Order-Preserving RDS . . . . . . . . . . . . . . . . . . . . . . 1866.3 Conjugacy with Random Differential Equations . . . . . . . . . . . . 1886.4 Stochastic Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . 1926.5 Equilibria and Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.6 One-Dimensional Stochastic Equations . . . . . . . . . . . . . . . . . . . . 199

6.6.1 Stochastic Equations on R + . . . . . . . . . . . . . . . . . . . . . . . 1996.6.2 Stochastic Equations on a Bounded Interval . . . . . . . . . 206

6.7 Stochastic Equations with Concavity Properties . . . . . . . . . . . . 2146.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.8.1 Stochastic Biochemical Control Circuit . . . . . . . . . . . . . . 2196.8.2 Stochastic Gonorrhea Model . . . . . . . . . . . . . . . . . . . . . . . 2216.8.3 Stochastic Model of Symbiotic Interaction . . . . . . . . . . . 2226.8.4 Lattice Models of Statistical Mechanics . . . . . . . . . . . . . 223

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .233


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Introduction

The state of many physical, chemical and biological systems can be described

by a single time-dependent variable x(t) which satises the ordinary differ-ential equation

x(t) = f (, x (t)) . (1)

This equation depends on parameters = ( 1 , . . . m ) which characterize theproperties of the environment and are usually called external parameters. Forexample, the equation

x = x x3 (2)can be used to describe the growth of a biological population. It contains the

parameter R which takes into account the properties of the environment.

If there is an existence and uniqueness theorem for (1), then we can de-ne an evolution operator S t in R by the formula S t x0 = x(t; x0 ), wherex(t; x0 ) is the solution to (1) with x(0; x0 ) = x0 . The uniqueness theorem for(1) and the fact that R is a totally ordered set imply that one-dimensionalequations generate monotone (or order-preserving) dynamical systems, i.e.S t x1 S t x2 provided x1 x2 . This property drastically simplies the dy-namics. For example, for equation (2) we have either one or three equilibriumpoints depending on the parameter and every solution is attracted by anequilibrium in a monotone way. Indeed, it is easy to see that any solution to(2) with initial data x0 has the form

x(t) =x0 et

(1 + x20 1 (e2 t 1))1 / 2 .

Therefore we have that (i) if < 0, then x(t) 0 as t + for any initialdata x0 ; (ii) if > 0 and x0 > 0, then x(t) as t + and (iii) if > 0 and x0 < 0, then x(t)

as t

+

. Thus for the case < 0

we have a unique globally asymptotically stable equilibrium and in the case > 0 we have two stable equilibria and one unstable. In the latter case theglobal attractor is the interval [ , ] (by denition, the global attractoris a strictly invariant set which uniformly attracts every bounded set). Thusequilibria and their stability properties completely determine the long-timedynamics of the system.

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2 Introduction

Similar behaviour is observed for one-dimensional systems with discretetime which are generated from a nondecreasing continuous mapping f : R R via the formula

xn +1 = f (xn ), n = 0 , 1, . . . .

The situation becomes more complicated in d-dimensional case. The phasespace R d is a partially ordered set with respect to the natural order relation(x = ( x1 , . . . , x d ) 0 if and only if x i 0 for all i) and there is no mono-tonicity in general. For example, it is easy to see that the linear system

x1 = a 11 x1 + a12 x2 ,x2 = a 21 x1 + a22 x2 ,

produces solutions which are monotone with respect to initial data if and onlyif a12 0 and a21 0. Nevertheless monotone multi-dimensional ordinarydifferential equations cover important classes of mathematical models arisingin modern natural science (see discussion in Smith [102]). The mathematicaltheory of deterministic monotone (order-preserving) systems is presently well-developed due to the efforts of many authors (see, e.g., Krasnoselskii [69],Hirsch [52, 53, 54] and also Smith [102] and the references therein). A well-posed autonomous system of ordinary differential equations

x i = f i (x1 , . . . , x d ), i = 1 , . . . , d ,

generates an order-preserving (with respect to the natural order relation)dynamical system in R d if and only if the mapping x (f 1 (x), . . . , f d (x))from R d into itself is cooperative (quasi-monotone), i.e.

f i (x1 , . . . , x d ) f i (y1 , . . . , y d )

for all (x1 , . . . , x d ) and ( y1 , . . . , y d ) fromR d

such that x i = yi and x j yj for j = i, where i = 1 , . . . , d . For example, this relation holds for the followingsystem of differential equations

x1 (t) = g(xd (t)) 1 x1 (t) ,x j (t) = x j 1 (t) j x j (t), j = 2 , . . . , d ,

where j > 0 for j = 1 , . . . , d and g(xd ) is a nondecreasing function. A systemof this type provides a simple model for positive feedback in biochemicalcontrol circuit (see, e.g., Selgrade [96] and Smith [102] and the referencestherein). The variables x j , j = 1 , . . . , d 1, could represent the concentrationsof a sequence of enzymes and xd , the concentrations of their substrate.

It was shown by Hirsch [52] that generic solutions to some classes of monotone systems converge to the set of equilibria. Thus, as in the one-dimensional case, we observe some simplication in the long-time dynam-ics. However an important construction due to Smale (see, e.g., Smith [102,


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Introduction 3

Chap.4]) shows that any (complicated) dynamics can occur on unstable in-variant sets for monotone systems of sufficiently large dimension.

If the system is coupled to a uctuating environment, then externalparameters can become stochastic quantities. In many cases these quanti-ties can be presented as stationary random processes. We refer to Hors-themke/Lefever [55, Chap.1] for a detailed discussion on the nature andsources of randomness in dynamical systems.

Thus taking into account random uctuations of the environment for thesystem described by (1) leads to the equation

x(t) = f ( 0 + (t, ), x(t)) ,

where 0 corresponds to the mean state of the environment and the station-ary process (t, ) with zero expectation on some probability space ( , F , P )describes environmental uctuations around this mean state. For example,equation (2) turns into

x = ( + (t, )) x x3 . (3)As above we can show that the process x(t; , x0 ) which solves (3) with initial

data x0 has the form

x(t; , x0 ) =x0 exp{t + (t, )}

1 + 2 x20 t

0 exp{2s + 2 (s, )}ds1 / 2 ,

where (t, ) = t0 (, )d . It is clear that the solutions x(t; , x0 ) dependon x0 in a monotone way, i.e. the relation x0 x0 implies that x(t; , x0 ) x(t; , x0 ).Assume that the strong law of large numbers is valid for the process

(t, ), i.e. t 1 (t, ) 0 almost surely as t + . Then it is easy to seethat all solutions x(t; , x0 ) tend to 0 almost surely as t + in the case < 0. In the case > 0 the situation is a bit more complicated. How-ever it is possible to prove (see Arnold [3, Chap.9]) that there exists astationary process (t, ) > 0 which solves equation (3) and such that theinterval [ (t, ), (t, )] is a globally attracting set in some sense. Thus sta-tionary solutions to equation (3) play a role of equilibria and the interval[ (t, ), (t, )] should be treated as a global attractor. As we will see inChap.3, a similar picture is inherent in some classes of multi-dimensionalmonotone systems with both continuous and discrete time. However it iswell to bear in mind that random monotone systems may display the long-time behaviour which is impossible in deterministic (autonomous or periodic)order-preserving systems. As an example we can consider the following dif-ferential equation

x = (t, ) x(1 x)


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4 Introduction

in the interval [0 , 1]R . Under some conditions concerning the stationary

random process (t, ) the omega-limit set for any point from the open inter-

val (0, 1) is a non-trivial completely ordered set. We refer to Example 3.6.1and Sects.5.6 and 6.6 for details. This phenomenon does not take place indeterministic strongly order-preserving systems (see Smith [102]) and this isone of obstacles which prevent the direct expansion of the results availablefor deterministic monotone systems.

To make the analogy with the deterministic case more precise it is con-venient to involve the modern concept (see Arnold [3] and also Sect.1.2below) of a random dynamical system. This concept covers the most im-portant families of dynamical systems with randomness, including random

and stochastic ordinary and partial differential equations and random differ-ence equations, and makes it possible to study randomness in the frameworkof classical dynamical systems theory with all its powerful machinery. Ran-domness could describe environmental or parametric perturbations, internaluctuations, measurement errors, or just lack of knowledge. The theory of random dynamical systems has been developed intensively in recent yearsand contains a lot of interesting and deep results. From a probabilistic pointof view this theory offers a new approach to the study of qualitative proper-ties of stochastic differential equations. It became possible due to important

results on two-parameter ows generated by stochastic equations (see, e.g.,Belopolskaya/Dalecky [15], Elworthy [43], Kunita [74] and the liter-ature quoted there). For a detailed discussion of the theory and applicationsof random dynamical systems we refer to the monograph Arnold [3].

To present a clear explanation of the general concept of a random dynam-ical system (see Sect.1.2 for the formal denition) we consider the followingsimple discrete dynamical system.

Assume that f 0 and f 1 are continuous mappings of a metric space X intoitself. Let us consider X as the state space of some system that evolves as

follows: if x is the state of the system at time k then its state at time k + 1 iseither f 0 (x) or f 1 (x) with probability 1 / 2 and the choice of f 0 or f 1 does notdepend on time and the previous states. We can nd the state of the systemafter a number of steps in time if we ip a coin and write down the sequenceof events from right to left using 0 and 1. Assume, for example, that after 7ips we get a following set of outcomes: 1001101. Here 1 corresponds to thehead falling and 0 corresponds to the tail falling. Therewith the state of thesystem at time 7 will be written in the form

y = ( f 1 f 0 f 0 f 1 f 1 f 0 f 1 )(x).This construction can be formalized as follows. Let be the set of two-sidedsequences = {i | iZ }consisting of zeros and ones. On the set thereis a probability measure P such that

P (C i 1 ...i m ) = P 0 (C 1 ) . . . P 0 (C m )


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Introduction 5

for any cylindrical set

C i 1 ...i m = { |ik

C k , k = 1 , . . . , m },

where C k is one of the sets , {0}, {1}, {0, 1}and P 0 () = 0, P 0 ({0}) =P 0 ({1}) = 1 / 2, P 0 ({0, 1}) = 1. Here {i1 , . . . , i m }is an arbitrary m-tuple of integers. For every nZ we denote by n the left shift operator in , i.e.n {i | iZ }= {i + n | iZ }, nZ .

It is clear that the shift operator preserves probabilities of sets from .For each n

Z + and

we dene the mapping n (t, ) of

X into

itself by the formula

n = 1 n 1 , nN , 0 = id ,where 1 (, x) = ( 1 , f 0 (x)). This mapping n can be written in the form

n (, x) = ( n , (n, )x) , (4)

where (n, ) is dened by the formula

(n, ) = f n 1 f n 2 . . . f 1 f 0 , = {i | iZ }, nN ,and satises the cocycle property

(0, ) = id , (n + m, ) = (n, m ) (m, )for all n, m

Z + and . The pair ( n ,(n, )) is called a randomdynamical system with discrete time. The mapping n models the evolution of some random environment and (, n) describes the dynamics of the system.

If X = R and f 0 and f 1 are nondecreasing functions, then the mappings(n, ) are order preserving, i.e. the relation x1 x2 implies that (n, )x1 (n, )x2 for all n

Z + and .It is easy to see that satises the cocycle property if and only if n givenby (4) is a semigroup, i.e. n m = n + m for n, m Z + . Thus we obtain adynamical system in the classical sense (i.e. a semiow of mappings from somespace into itself). We note that semiows of a similar structure (see (4)) arisein the theory of nonautonomous (deterministic) differential equations andthey are known as skew-product ows (see, e.g., Chicone/Latushkin [19]

and the references therein). This observation is important in the study of thelong-time behaviour of random dynamical systems.

The aim of this book is to present a recently developed approach which issuitable for investigating a variety of qualitative aspects of order-preservingrandom dynamical systems and to give the backgrounds for further develop-ment of the theory. We try to demonstrate the effectiveness of this approach


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Introduction 7

asymptotically stable in each part of the cone. Our main result here is arandom limit set trichotomy, stating that in a given part either (i) all orbits

are unbounded, (ii) all orbits are bounded but their closure reaches out tothe boundary of the part, or (iii) there exists a unique, globally attractingequilibrium. Several examples, including Markov chains and affine systems,are given.

In Chapters 5 and 6 we apply the results of Chapters 3 and 4 to study thequalitative behaviour of random and stochastic perturbations of cooperativeordinary differential equations. These applications are the main motivationsfor the development of the general theory presented in Chapters 3 and 4and we believe that random and stochastic cooperative differential equations

merit a detailed study of its own.In Chapter 5 we consider random cooperative differential equations in

R d+ (real noise case). We rst give conditions under which these equationsgenerate order-preserving random dynamical systems in R d+ and then studymonotonicity properties of these systems. We prove several theorems on theexistence of equilibria and random attractors. Systems with concavity prop-erties are also considered. We apply general results from Chapters 3 and 4 tostudy the long-time behaviour of these systems and to obtain the limit settrichotomy theorem for random cooperative differential equations. We con-

clude Chapter 5 with a series of examples including a class of one-dimensionalexplicitly solvable equations to show possible scenarios of the long-time be-haviour in monotone systems.

Chapter 6 is devoted to stochastic cooperative differential equations(white noise case). The hypotheses that guarantee order-preserving prop-erties for this case lead to a special structure of the diffusion terms. In factwe consider here some class of stochastic perturbations of deterministic au-tonomous cooperative differential equations. We prove several assertions onthe long-time behaviour, investigate properties of systems that possess con-

cavity properties and establish a stochastic version of the limit set trichotomytheorem. We study the long-time dynamics in one-dimensional equations withdetails. We also discuss the stochastic versions of certain examples consid-ered in the previous chapter. Although the results for the stochastic case aresimilar to the random case, Chapter 6 is not at all a duplication of Chapter 5because the methods of proof are quite different.


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1. General Facts about Random Dynamical

Systems

In this chapter we recall some basic denitions and facts about random dy-

namical systems. For a more detailed discussion of the theory and applica-tions of random dynamical systems we refer to the monograph Arnold [3].We pay particular attention to dissipative systems and their random (pullback) attractors. These attractors were studied by many authors (see, e.g.,Arnold [3], Crauel/Debussche/Flandoli [35], Crauel/Flandoli [36],Schenk-Hopp e [89], Schmalfuss [92, 93] and the references therein). Theideas that lead to the concept of a random attractor have their rootsin the theory of deterministic dissipative systems which has been suc-cessfully developed in the last few decades ( see, e.g., the monographs

Babin/Vishik [13], Chueshov [20], Hale [50], Temam [104] and the liter-ature quoted therein). The proof of the existence of random attractors givenbelow follows almost step-by-step the corresponding deterministic argument(see, e.g., Chueshov [20], Temam [104]).

Throughout this book we will be concerned with a probability space bywhich we mean a triple ( , F , P ), where is a space, F is a -algebra of sets in , and P is a nonnegative -additive measure on F with P () = 1.We do not assume in general that the -algebra is complete. Below we willalso use the symbol T for either R or Z and we will denote by T + all non-negative elements of T . We will denote by B (X ) the Borel -algebra of setsin a topological space X . By denition B (X ) is the -algebra generated bythe collection of open subsets of X . If (X 1 , F 1) and ( X 2 , F 2) are measurablespaces, we denote by F 1 F 2 the product -algebra of subsets in X 1 X 2which is dened as the -algebra generated by the cylinder sets A = A1 A2 ,Ai F i . We refer to Cohn [30] for basic denitions and facts from themeasure theory.

1.1 Metric Dynamical Systems

The random dynamical system is an object consisting of a metric dynamicalsystem and a cocycle over this system. We need a metric dynamical systemfor modeling of random perturbations.

I. Chueshov: LNM 1779, pp. 953, 2002.c Springer-Verlag Berlin Heidelberg 2002


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10 1. General Facts about Random Dynamical Systems

Denition 1.1.1. A metric dynamical system (MDS) (, F , P , {t , t T }) with (two-sided) time T is a probability space (, F , P ) with a family of

transformations {t : , t T } such that 1. it is one-parameter group, i.e.

0 = id , t s = t + s for all t, s T ;

2. (t, ) t is measurable;3. t P = P for all t T , i.e. P (t B ) = P (B ) for all B F and all t T .

A set B F is called -invariant if t B = B for all t T . A metric dynamical

system is said to be ergodic under P

if for any -invariant set B F

wehave either P (B ) = 0 or P (B ) = 1 .

We refer to Cornfeld/Fomin/Sinai [29], Ma ne [79], Rudolph [88], Si-nai [100] and Walters [106] for the references and presentation of MDSand ergodic theory.

From an applied point of view the use of metric dynamical systems tomodel external perturbations assumes implicitly that the external inuenceis stationary in some sense (see examples below). This means that we donot consider possible transient (random) process in the environment, i.e.we assume that all these processes are nished before we start to observethe dynamics of our system. This is also the reason why we consider MDSwith two-sided time. We note that any one-sided MDS (with time T + ) pos-sesses a natural two-sided extension (see, e.g., Cornfeld/Fomin/Sinai [29,Sect.10.4] or Arnold [3, Appendix A]).

Now we give several important examples of metric dynamical systems.They show what kind of time dependence we can allow in the equationsconsidered in Chaps.5 and 6.

Example 1.1.1 (Periodic Case). Consider the probability space ( , F , P ),where is a circle of unit circumference, F is its -algebra of Borel sets and Pis the Lebesgue measure on . Let {t , t R } be the group of rotations of thecircle. It is easy to see that we obtain an ergodic MDS ( , F , P , {t , t R })with continuous time.

Example 1.1.2 (Quasi-Periodic Case). Let be d-dimensional torus, =Tor d . Assume that its points are written as x = ( x1 , . . . , x d ) with x i [0, 1).Let F be the -algebra of Borel sets of Tor d and P be the Lebesgue measure

on Tord

. We dene transformations {t , t T } by the formulat x = ( x1 + t a1(mod 1), . . . , x d + t ad (mod 1)) , t T ,

for a given a = ( a1 , . . . , a d ). Thus we obtain an MDS. If the numbersa1 , . . . , a d , 1 are rationally independent, then this MDS is ergodic (see, e.g.,Rudolph [88]).


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1.1 Metric Dynamical Systems 11

Example 1.1.3 (Almost Periodic Case). Let f (x) be a Bohr almost periodicfunction on R . We dene the hull H (f ) of the function f as the closure of

the set {f (x + t), t R } in the norm f = sup xR |f (x)|. The hull H (f ) isa compact metric space, and it has a natural commutative group structure.Therefore it possesses a Haar measure which, if normalized to unity, makesH (f ) into probability space. If we dene transformations {t , t T } as shifts:(t g)(x) = g(x + t), g H (f ), we obtain an ergodic MDS with continuoustime. For details we refer to Ellis [42] and Levitan/Zhikov [77].

Example 1.1.4 (Ordinary Differential Equations). MDS can be also gener-ated by ordinary differential equations (ODE). Let us consider a system of

ODEs inR n

: dx idt

= f i (x1 , . . . x n ), i = 1 , . . . , n . (1.1)

Assume that the Cauchy problem for this system is well-posed. We dene{t , t R } by the formula t x = x(t), where x(t) is the solution of (1.1) withx(0) = x. Assume that a nonnegative smooth function (x1 , . . . , x n ) satisesthe stationary Liouville equation

n

i =1

x i((x1 , . . . x n ) f i (x1 , . . . x n )) = 0 (1.2)

and possesses the property R n (x) dx = 1. Then (x) is a density of aprobability measure on R n . By Liouvilles theorem R n f (t x)(x) dx = R n f (x)(x) dx

for any bounded continuous function f (x) on R n and therefore in this situ-ation an MDS arises with = R n , F = B (R n ) and P (dx) = (x)dx. HereB (R n ) is the Borel -algebra of sets in R n . Sometimes it is also possibleto construct an MDS connected with the system (1.1), when the solution to (1.2) is not integrable but the problem (1.1) possesses a rst integral(e.g., if (1.1) is a Hamiltonian system) with appropriate properties (see, e.g.,Ma ne [79] or Sinai [100] for details).

Example 1.1.5 (Bernoulli Shifts). Let (0 , F 0 , P 0) be a probability spaceand ( , F , P ) be the probability space of innite sequences = {i }, wherei 0 , i Z . Here F is the -algebra generated by nite-dimensionalcylinders

C i 1 ...i m = { | i k C k , k = 1 , . . . , m } ,

where C k F 0 and {i1 , . . . , i m } is an arbitrary m-tuple of integers. Theprobability measure P is dened such that P (C i 1 ...i m ) = P 0(C 1) . . . P 0(C m ).


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We dene transformations {t , t Z } by the formula t = , where = {i } and = {i + t }. Since

t C i 1 ...i m = { | i k t C k , k = 1 , . . . , m } ,

the probability measure P is invariant under t . Thus we obtain an MDS.In the particular case when 0 = {0, 1} is a two-point set and P 0({0}) =P 0({1}) = 1 / 2, we have the standard Bernoulli shift. In the general case wecan interpret this MDS as one generated by an innite sequence of indepen-dent identically distributed random variables.

Example 1.1.6 (Stationary Random Process). Let = {(t), t T } be astationary random process on a probability space ( , F , P ), where F is the-algebra generated by . Assume that in the continuous case ( T = R ) theprocess possesses the cadlag property: all trajectories are right-continuousand have limits from the left. Then the shifts (t) ( )( t) = (t + )generate an MDS. See Arnold [3] and the references therein for details.

In the framework of stochastic equations the following example of an MDSis of importance.

Example 1.1.7 (Wiener Process). Let W t = ( W 1t , . . . , W dt ) be a Wiener pro-cess with values in R d and two-sided time R . Let (, F , P ) be the corre-sponding canonical Wiener space. More precisely, let C 0(R , R d ) be the spaceof continuous functions from R into R d such that (0) = 0 endowed withthe compact-open topology, i.e. with the topology generated by the metric

(, ) :=

n =1

12n

n (, )1 + n (, )

, n (, ) = maxt[ n,n ]

|(t) (t)| .

Let F be the corresponding Borel -algebra of C 0(R , R d ), and let P be theWiener measure on F . We suppose is the subset in C 0(R , R d ) consisting of

the functions that have a growth rate less than linear for t andF

isthe restriction of F to . In this realization W t () = (t), where () ,i.e. the elements of are identied with the paths of the Wiener process.We dene a metric dynamical system by t () := (t + ) (t). Thesetransformations preserve the Wiener measure and are ergodic. Thus we havean ergodic MDS. The ow {t } is called the Wiener shift. We note that the -algebra F is not complete with respect to P and we cannot use its completionF P to construct MDS because ( t, ) t is not a measurable mappingfrom (R , B (R ) F P ) into ( , F P ). This is one of the reasons why the

completeness of F

is not assumed in the basic denitions. See Arnold [3]for details. We also note that this realization of a Wiener process makes itpossible to introduce the white noise process as the derivative W t of W t withrespect to t in the sense of generalized functions. From an applied point of view white noise processes correspond to an extremely short memory of theenvironment in comparison with the memory of the system (see the discussionin Horsthemke/Lefever [55], for instance).


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1.2 Concept of RDS 13

1.2 Concept of RDS

Let X be a Polish space, i.e. a separable complete metric space. We equipX with the Borel -algebra B = B (X ) generated by open sets of X . Weneed the following concept of a (continuous) random dynamical system (cf.Arnold [3]).

Denition 1.2.1 (Random Dynamical System). A random dynamicalsystem (RDS) with (one-sided) time T + and state (phase) space X is a pair (,) consisting of a metric dynamical system (, F , P , {t , t T }) and a cocycle over of continuous mappings of X with time T + , i.e. a measurable

mapping : T + X X, (t ,,x ) (t ,,x ) ,

such that

(i) the mapping x (t ,,x ) (t, )x is continuous for every t 0 and ,

(ii) the mappings (t, ) := (t,, ) satisfy the cocycle property :

(0, ) = id , (t + s, ) = (t, s ) (s, )

for all t, s T + and . Here means composition of mappings.

We emphasize the following peculiarities of this denition.

Remark 1.2.1. (i) While the metric dynamical system (modeling the randomperturbations) is assumed to have two-sided time T = R or Z , the cocycle isonly required to have one-sided time T + = R + or Z + . This reects the factthat evolution operators are often non-invertible. However this set-up allowsus to consider (t, s ) for t T + , but starting at an arbitrary (possibly

negative) time s T

which will be crucial for the construction of equilibriaand attractors. In the case of continuous time ( T = R ) the standard denitionof a continuous RDS requires the continuity of the mappings ( t, x ) (t, )xfor all (see Arnold [3, Sect.1.1]). This property is usually true forthe RDS generated by nite-dimensional random and stochastic equations.However, as we will see, many general results on the long-time behaviourcan be proved under a weaker assumption of the continuity of the mappingx (t, )x for each t 0 and . We also note that the cocycleproperty reduces to the classical semiow property if is independent of .

Hence deterministic dynamical systems are particular cases of RDS.(ii) If in Denition 1.2.1 the cocycle is dened on a -invariant set of full measure, then we can extend it to the whole by the formula

(t, ) :=(t, ) if ,id if /.

(1.3)


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Thus we obtain the cocycle (t, ) which is indistinguishable from (t, ).We recall that by denition the indistinguishability of (t, ) and (t, )

means that there exists a set N F such that P ( N ) = 0 and{ : (t, ) = (t, ) for some t R + } N .

In our case the cocycles coincide on the -invariant set and we can set N = \ . In further considerations we do not distinguish cocycles whichcoincide on -invariant sets of full measure.

(iii) In the denition of an RDS we require some properties to be valid forall . However the stochastic analysis deals usually with almost all ele-

mentary events . Solutions to stochastic differential equations are dened al-most surely, for example. Therefore to construct RDS connected with stochas-tic equations we need extend the corresponding evolution operator to all and prove the cocycle property for this extension. This can be done formany cases which are important from the point of view of applications. Thisprocedure is usually referred to as perfection . Roughly speaking the perfectionof cocycles (or other objects) can be done in the following way. First we provea property for some -invariant set of full measure. After that we dene thecocycle on \ in an appropriate way (cf. (1.3)). Perfection theorems have

been shown in various different cases, see, e.g., Arnold/Scheutzow [10],Scheutzow [90], Kager/Scheutzow [61], Sharpe [98] and also the dis-cussion in Arnold [3].

We also recall the following denitions Arnold [3].

Denition 1.2.2 (Smooth RDS). Let X be an open subset of a Banach space. A random dynamical system (,) is said to be a smooth RDS of classC k or a C k RDS, where 1 k , if it satises the following property: for each (t, ) T + the mapping x (t, )x from X into itself is k times

Frechet differentiable with respect to x and the derivatives are continuous with respect to x.

Denition 1.2.3 (Affine RDS). Let X be a linear Polish space. The RDS (,) is said to be affine if the cocycle is of the form

(t, )x = (t, )x + (t, ) , (1.4)

where (t, ) is a cocycle over consisting of bounded linear operators of X ,and : T + X is a measurable function. If (t, ) 0 then the affineRDS is said to be linear .

If (, ) is a linear RDS, then the cocycle property for the mapping denedby (1.4) is equivalent to the relation

(t + s, ) = (t, s )(s, ) + (t, s ), t, s 0 . (1.5)


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A thorough treatment of affine RDS in R d can be found in Sect. 5.6 of Arnold [3].

Any RDS ( ,) generates a skew-product semiow { t , t T + } on X bythe formula

t (, x) = ( t , (t, )x), t T + . (1.6)

Since (, x) t (, x) is an (F B )-measurable mapping from X intoitself, we obtain a measurable dynamical system on ( X, F B ). Here Bis the -algebra of Borel sets in X . The cocycle property for is equivalentto the semigroup property for . We note that the standard theory of skew-product ows (see, e.g., Shen/Yi [99], Chicone/Latushkin [19] and the

references therein) usually requires that both and X are topological spacesand {t } are continuous mappings. In the RDS case we have no topology on in general.

The simplest examples of RDS are described below.

Example 1.2.1 (Markov Chain). This is a generalization of the example con-sidered in the Introduction. Let ( 0 , F 0 , P 0) be a probability space and X be aPolish space. Assume that f (, x ) is a measurable mapping from 0 X intoX which is continuous with respect to x for every xed 0 . Let (, F , P )

be the probability space of innite sequences = {i }, where i 0 , i Z

,and = ( , F , P , {t , t Z }) be the metric dynamical system constructed inExample 1.1.5. For every = {i : i Z } we introduce the functionf : X X by the formula f (x) = f (0 , x) and for each n Z + and we dene the mapping (n, ) by the formula

(n, ) = f n 1 f n 2 . . . f 1 f , = , n N . (1.7)

We also suppose (0, ) = id. It is easy to see that the sequence (n, )xsolves the difference equation

xn +1 = f n (xn ), n Z + , x0 = x ,

and the mappings (n, ) possess the cocycle property. Thus we obtain adiscrete RDS. It is a C k -RDS, if X R d and f (, ) C k (X, X ). If X is alinear Polish space and f (, ) are affine mappings, i.e. f (, x ) = K x + h ,where K are continuous linear operators in X and h are elements from X ,then the RDS constructed above is affine. It is a linear RDS when h = 0 for 0 .

Since all random mappings f n , n Z , are independent and identicallydistributed (i.i.d.), the RDS constructed above generates (see Arnold [3,p.53]) the homogeneous Markov chain

{xn := (n, )x : n Z + , x X }

with state space X and transition probability


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P (x, B ) := P {n +1 B | n = x}

= P { : f (x) B } P 0{ : f (, x ) B }, B B (X ) .

For a detailed presentation of the theory of Markov chains we refer to Gih-man/Skorohod [48, Chap.2], for example. We note that the inverse prob-lem of constructing an RDS of i.i.d. mappings with a prescribed transitionprobability is not unique in general and so far largely unsolved. We refer toArnold [3] and Kifer [66] for discussions of this problem.

Example 1.2.2 (Kick Model). Let {k : k Z } be a stationary random

process (chain) in X on a probability space ( ,F

,P

) and be the corre-sponding metric dynamical system such that k () = 0(k ) for all k Z(cf. Example 1.1.6). Suppose that mappings f : X X have the form

f (x) = g(x, 1()) , ,

where g is a continuous function from X X into X . In this case the cocycle dened by (1.7) generates the sequence xn = (n, )x which solves thedifference equation

xn +1 = g(xn , n +1 ()) , n Z + , x0 = x .

If X is a Banach space and g(x, ) = g(x) + , then this equation has theform

xn +1 = g(xn ) + n +1 (), n Z + , x0 = x . (1.8)

A kick force model corresponds to the case when the mapping g : X X hasthe form g(x) = y(T ; x), where T > 0 is a xed number and y(t) := y(t; x)solves the equation

y(t) = h(y(t)) , t > 0, y(0) = x . (1.9)

Here h is a mapping from X into itself such that equation (1.9) generates a(deterministic) continuous dynamical system. In this case

(n, )x = z(n T + 0 , ; x), n Z + .

Here z(t) := z(t, ; x) is a generalized solution to the problem

z(t) = h(z(t)) +kZ

k () (t k T ), z(+0) = x ,

where (t) is a Dirac -function of time. Thus the kick model describes thesituation when the deterministic system (1.9) gets random kicks with someperiod T and evolves freely between kicks. We note that kick models aresufficiently popular in the study of turbulence phenomena.


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The next examples present the simplest versions of RDS considered in Chaps.2, 5 and 6 with details.

Example 1.2.3 (1D Random Equation). Let = ( , F , P , {t , t R }) be ametric dynamical system. Consider the pathwise ordinary differential equa-tion

x(t) = f (t , x(t)) . (1.10)

Under some natural conditions (see Sect. 2.1 below) on the function f : R R this equation generates an RDS with state space R and with thecocycle given by the formula (t, )x = x(t), where x(t) is the solution to(1.10) with x(0) = x. This RDS is affine if f (, x) = a() x + b() for somerandom variables a() and b().

Example 1.2.4 (Binary Biochemical Model). Consider the system of ordinarydifferential equations

x1 = g(x2) 1(t )x1 ,x2 = x1 2(t )x2 ,

(1.11)

over a metric dynamical system . This is a two-dimensional version of the

deterministic model considered in the Introduction. If we assume that g(x)is a globally Lipschitz function and i () is a random variable such that i (t ) L1loc (R ) for i = 1 , 2 and , then equations (1.11) generate anRDS in R 2 with (t, )x = x(t), where x(t) = ( x1(t), x2(t)) is the solutionto (1.11) with x(0) = x.

Example 1.2.5 (1D Stochastic Equation). Let {W t } be the one-dimensionalWiener process (see Example 1.1.7). Then the It o stochastic differential equa-tion in R

dx(t) = b(x(t))dt + (x(t))dW t , (1.12)

where the scalar functions b(x) and (x) possess some regularity properties(see Sect. 2.4 below), also generates an RDS. Of course, the same conclu-sion remains true, if we understand the stochastic equation (1.12) in theStratonovich sense. We note that formally equation (1.12) can be written inthe form

x(t) = b(x(t)) + (x(t))W t

and the corresponding RDS can be interpreted as a system in a white noiseenvironment.

More detailed presentation of the last three examples and their generaliza-tions can be found in Chaps.5 and 6. We also refer to Sects. 2.1 and 2.4 in


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Chap.2 for a description of the basic properties of random and stochasticdifferential equations.

As in the deterministic case the following concept of topological equivalence(or conjugacy) of two random dynamical systems is of importance in ourstudy. In particular below we will use equivalence between some classes of random and stochastic differential equations.

Denition 1.2.4 (Equivalence of RDS). Let (,1) and (,2) be twoRDS over the same MDS with phase spaces X 1 and X 2 resp. These RDS (,1) and (, 2) are said to be (topologically) equivalent (or conjugate) if there exists a mapping T : X 1 X 2 with the properties:

(i) the mapping x T (, x) is a homeomorphism from X 1 onto X 2 for every ;

(ii) the mappings T (, x1) and T 1(, x2) are measurable for every x1 X 1 and x2 X 2 ;

(iii) the cocycles 1 and 2 are cohomologous, i.e.

2(t ,,T (, x)) = T (t , 1(t ,,x )) for any x X 1 . (1.13)

We refer to Arnold [3], Keller/Schmalfuss [63] and also to the recentpapers Imkeller/Lederer [58] and Imkeller/Schmalfuss [59] for moredetails concerning equivalence of RDS.

1.3 Random Sets

One of the goals in this book is to describe the long-time behaviour of RDSand the limit regimes of these systems. These limit regimes typically dependon an event and therefore to characterize their attractivity properties weshould at least be able to calculate the distance between (random) trajecto-ries and (random) limit objects and treat this distance as a random variable.It is also crucial to decide whether the limit regimes contain a random vari-able representing the different states of the system. These circumstances leadto a notion of a random set which is stronger than simply a collection of setsdepending on . We introduce this notion of a random set following to Cas-taing/Valadier [18] and Hu/Papageorgiou [56] (see also Crauel [32]and Arnold [3]).

Below any mapping from into the collection of all subsets of X is saidto be a multifunction (or a set valued mapping) from into X .

Denition 1.3.1 (Random Set). Let X be a metric space with a metric. The multifunction D () = is said to be a random set if the mapping

dist X (x, D ()) is measurable for any x X , where dist X (x, B ) is thedistance in X between the element x and the set B X . If D () is closed for each then D is called a random closed set . If D () are compact sets


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1.3 Random Sets 19

for all then D is called a random compact set . A random set {D ()}is said to be bounded if there exist x0 X and a random variable r () > 0

such that

D () {x X : (x, x 0) r ()} for all .

For ease of notation we denote the random set D () by D or {D ()}.

Remark 1.3.1. (i) The property of D being a random closed set is slightlystronger than

graph( D ) = {(, x) X : x D ()}

being F B (X )-measurable and D () being closed; the two properties areequivalent if F is P -complete, i.e. if for any set A F with zero probabilityall subsets of A also belong to F (see Castaing/Valadier [18]).

(ii) For any x X and bounded sets A and B from X we have the relation

|dist X (x, A) dist X (x, B )| h(A|B ) ,

where h(A|B ) is the Hausdorff distance dened by the formula

h(A|B ) = supaA

dist X (a, B ) + supbB

dist X (b, A) .

Therefore, if for a multifunction D () there exists a sequence {Dn } of random bounded sets such that

limn

h(Dn ()|D ()) = 0 for all ,

then D () = n 0k n Dk () for every and D () is a randombounded set ( D denotes the closure of D in X ).

Example 1.3.1 (Random Ball). Let X = R d . Suppose that r () 0 is arandom variable and a() is a random vector from R d . Then the multifunction

B () = {x : |x a()| r ()}

is a random compact set. Here | | is the Euclidean distance in R d . This factfollows from the formula

dist X (y, B ()) =0 if y B () ,|y a()| r () if y /B () ,

which implies that dist X (y, B ()) = max {0, |y a()| r ()}. It is alsoclear that int B () = {x : |x a()| < r ()} is a random (open) set.


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More general examples are described in Proposition 1.3.1(vi) and in Propo-sition 1.3.6.

We need the following properties of random sets (for the proofs we referto Hu/Papageorgiou [56, Chap.2], see also Castaing/Valadier [18],Crauel [32] and Arnold [3]).Proposition 1.3.1. Let X be a Polish space. The following assertions hold:

(i) D is a random set in X if and only if the set { : D () U = }ismeasurable for any open set U X ;

(ii) D is a random set in X if and only if {D ()} is a random closed set ( D () denotes the closure of D () in X );

(iii) D is a random compact set in X if and only if D () is compact for every and the set { : D () C = }is measurable for any closed set C X ;

(iv) if {Dn , n N } is a sequence of random closed sets with non-void inter-section and there exists n0 N such that Dn 0 is a random compact set,then nN D n is a random compact set in X ;

(v) if {Dn , n N } is a sequence of random sets, then D = nN Dn is also a random set in X ;

(vi) if f : X X is a mapping such that f (, ) is continuous for all and f (, x) is measurable for all x, then f (, D()) is a random set in X provided D is a random set in X ; similarly, f (, D()) is a random compact set in X provided D is a random compact set.

The following representation theorem (see Ioffe [60]) provides us with aconvenient description of random closed sets.Theorem 1.3.1. Let D be a random closed set in a Polish space X . Then there exist a Polish space Y and a mapping g(, y) : Y X such that (i) g(, ) is continuous for all and g(, y) is measurable for all y Y ;

(ii) for all and y1 , y2 Y one has(g(, y1), g(, y2)) (1 + (g(, y1), g(, y2))) r (y1 , y2) ,

where (, ) and r (, ) are distances in X and Y ;(iii) for all one has D () = g(, Y ), the range of g(, ).This theorem immediately implies the following assertion.Proposition 1.3.2 (Measurable Selection Theorem). Let a multifunc-tion D () take values in the subspace of closed non-void subsets of a

Polish space X . Then {D ()} is a random closed set if and only if thereexists a sequence {vn : n N } of measurable maps vn : X such that

vn () D () and D () = {vn (), n N } for all .

In particular if {D ()} is a random closed set, then there exists a measurableselection, i.e. a measurable map v : X such that v() D () for all .


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1.3 Random Sets 21

Below we also need the following assertion on the measurability of projections(see, e.g., Castaing/Valadier [18, p.75]). It deals with the -algebra F u

of universally measurable sets associated with the measurable space ( , F )which is dened by the formula

F u =

F ,

where the intersection taken over all probability measures on (, F ) and F denotes the completion of the -algebra F with respect to the measure . Wecall F u the universal -algebra and F the -completion of F for shortness.

Recall that theP

-completionF P

is the -algebra consisting of all subsets Aof for which there are sets U and V in F such that U A V andP (U ) = P (V ). The probability measure P can be extended from F to F P suchthat F P is a complete -algebra with respect to the extended probabilitymeasure. For details we refer to Cohn [30], for instance. We also note thatt F P = F P for any xed t R . This property follows from the relationP (t U ) = P (U ) for any U F and t R .

Proposition 1.3.3 (Projection Theorem). Let X be a Polish space and M X be a set which is measurable with respect to the product -algebra F B (X ). Then the set

proj M = { : (, x) M for some x X }

is universally measurable, i.e. belongs to F u . In particular it is measurablewith respect to the P -completion F P of F .

Now we introduce the following set valued analog of a separable process (cf.Gihman/Skorohod [48, p.165]).

Denition 1.3.2. Let I be a set in R . A collection {C t : t I } of random sets is said to be separable if there exists an everywhere dense countable set Q in I such that

C t () nN

{C () : [t n 1 , t + n 1] Q} (1.14)

for all t I and . The set Q is called the separability set of thecollection {C t }. A process {v(t, ) : t I } is said to be separable if the

collection of random sets C t () = {v(t, )} is separable.It is easy to see that {C t : t I } is a separable collection with a separabilityset Q if and only if for any t I and x C t () there exist sequences {tn } Qand {xn } X such that tn t and xn x as n and xn C t n ().

The following proposition gives examples of separable collections of ran-dom closed sets.


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Proposition 1.3.4. Let D be a random closed set and I = ( , ) R .Assume that the function h(t ,,x ) : I X satises

(i) for each t I the function h(t,, ) is continuous for all and h(t, , x) is measurable for all x X ;

(ii) h(, ,x) is a right continuous function for all and x X .

Then h(t ,,D ()) is a separable collection of random closed sets whoseseparability set Q is an arbitrary everywhere dense countable set from (, ).The same conclusion holds if h(, ,x) is a left continuous function.

Proof. Proposition 1.3.1(vi) implies that h(t ,,D ()) is a random

closed set for every t. From Theorem 1.3.1 we have thath(t ,,D ()) = {h(t ,,g(, y)) : y Y }

Thus by (ii) for any t I there exists a sequence {tk } Q such that tk > tand

h(t ,,g(, y)) = limt k t

h(tk , ,g(, y))

for every y Y and . This property easily implies

h(t ,,g(, y)) nN

{C () : [t n 1 , t + n 1] Q}

for all y Y and , where C t () = h(t ,,D ()). This relation givesthe separability of {h(t ,,D ())}. 2

The main property of separable collections of random closed sets which isimportant in the considerations below is given in the following proposition.

Proposition 1.3.5. Let {C t : t I } be a separable collection of random sets. Then the multifunction

C () =tI

C t ()

is a random closed set.

Proof. It follows from (1.14) that tI C t () = tI Q C t (). Therefore wecan apply Proposition 1.3.1(v). 2

Below we also need the following assertion.

Proposition 1.3.6. Let V : X R be a continuous function on a Polish space X and R() be a random variable. If the set V R () := {x : V (x) R()} is non-empty for any , then it is a random closed set.


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1.3 Random Sets 23

Proof. The idea of the proof is borrowed from Schenk-Hopp e [89]. It is clearthat V R () is closed for any . Due to Proposition 1.3.1(i) it is sufficient

to prove that { : V R () U = }is measurable for every open set U X .This is equivalent to measurability of the set

{ : V R () U = } { : U X \ V R ()} .

This measurability follows from the relation

{ : U X \ V R ()} = { : R() < s for any s V (U )} (1.15)

which we now prove. Since

X \ V R () = V 1(R ) \ V 1(( , R ()]) = V 1((R(), + )) ,

we have that U X \ V R () if and only if V (U ) (R(), + ). This implies(1.15) and therefore

{ : U X \ V R ()} =nN

{ : R() < s n },

where sn V (U ) and sn inf V (U ) as n .2

The following notions of random tempered sets and variables play an impor-tant role in applications of the general theory of RDS connected with ran-dom and stochastic equations (cf. Chaps. 4 and 5). Roughly speaking, thata random variable which describes an inuence of the random environmentis tempered means that this environment evolves in non-explosive way.

Denition 1.3.3 (Tempered Random Set). A random set {D ()} issaid to be tempered with respect to MDS = ( , F , P , {t , t T }) if there

exist a random variable r () and an element y X such that

D () {x | dist X (x, y ) r ()} for all

and r () is a tempered random variable with respect to , i.e

suptT

e | t | |r (t )| < for all and > 0 . (1.16)

A random variable v() with values in X is said to be tempered if the one-point random set {v()} is tempered.It is clear that every deterministic set is tempered. We note that non-tempered random variables exist on any standard probability space withergodic and aperiodic (see Arnold/Cong/Oseledets [9]). Sometimes(see, e.g., Arnold [3, p.164]) the denition of a tempered random variableis based on the relation


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lim| t |

1|t |

log {1 + |r (t )|} = 0 for all .

which is weaker than (1.16). However we prefer to use (1.16) because it allowsus to simplify some calculations in the applications below. We also note thatif is ergodic, the only alternative to property (1.16) is that

lim| t |

1|t |

log {1 + |r (t )|} = + for almost all ,

see Arnold [3, p.165].As in the deterministic case we need a notion of an invariant set for the

description of qualitative properties of RDS. It is convenient to introduce thisnotion for multifunctions to cover all types of random sets.

Denition 1.3.4 (Invariance Property). Let (,) be a random dynam-ical system. A multifunction D () is said to be

(i) forward invariant with respect to (,) if (t, )D () D (t ) for all t > 0 and , i.e. if x D () implies (t, )x D (t ) for all t 0and ;

(ii) backward invariant with respect to (, ) if (t, )D () D (t ) for all t > 0 and , i.e. for every t > 0, and y D (t ) there existsx D () such that (t, )x = y;

(iii) invariant with respect to (,) if (t, )D () = D (t ) for all t > 0 and , i.e. if it is both forward and backward invariant.

We note that the forward invariance of the multifunction D () meansthat

graph( D ) = {(, x) X : x D ()}

is a forward invariant set in X with respect to the semiow { t } denedby (1.6), i.e. t graph( D ) graph( D ) for all t > 0. The same is true for theproperty of invariance.

1.4 Dissipative, Compact and Asymptotically CompactRDS

In this section we start to develop methods for studying the qualitative be-

haviour of random dynamical systems. Our main goal is to investigate thebehaviour of expressions of the form x(t) = (t, t )x when t + . Atrst sight this object looks a bit strange. However there are at least threereasons to study the limiting structure of (t, t )x.

The rst one is connected with the question of what limiting dynamicswe want to observe. The point is that in many applications RDS are generated


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1.4 Dissipative, Compact and Asymptotically Compact RDS 25

by equations whose coefficients depend on t . These coefficients describethe internal evolution of the environment and t represents the state of

the environment at time t which transforms into the real state ( ) atthe time of observation (time 0, after a time t has elapsed). Furthermore thetwo-parameter mapping U (, s ) := ( s, s ) describes the evolution of the system from moment s to time , > s . Therefore the limiting structureof U (0, t)x = (t, t )x when t + can be interpreted as the state of our system which we observe now ( t = 0) provided it was in the state x in theinnitely distant past ( t = ). Thus the union of all these limits providesus with the real picture of the present state of the system.

The second reason is that the asymptotic behaviour of (t, t )x pro-

vides us with some information about the long-time future. Indeed, since {t }are measure preserving, we have that

P { : (t, )x D} = P { : (t, t )x D}

for any x X and D B (X ). Therefore

limt +

P { : (t, )x D} = limt +

P { : (t, t )x D } ,

if the limit on the right hand side exists. Thus the limiting behaviour of

(t, t )x for all determines the long-time behaviour of (t, )x withrespect to convergence in probability.The third reason is purely mathematical. If on the set of random variables

a() with values in X we dene the operators T t by the formula

(T t a)() = (t, t )a( t ), t R + ,

then the family {T t , t R + } is a one-parameter semigroup. Indeed, usingthe cocycle property we have

(T s [T t a])() = (s, s )(T t a)( s ) = (s, s )(t, t s )a( t s )

= (t + s, t s )a( t s ) = ( T t + s a)() .

Thus it becomes possible to use ideas from the theory of deterministic (au-tonomous) dynamical systems for which the semigroup structure of the evo-lution operator is crucial. Below we introduce several important dynamicalnotions and study the qualitative behaviour of RDS relying on this observa-tion.

Let D be a family of random closed sets which is closed with respect toinclusions (i.e. if D1 D and a random closed set {D 2()} possesses theproperty D2() D1() for all , then D 2 D ). Sometimes thecollection D is called a universe of sets (see, e.g., Schenk-Hopp e [89]) oran IC-system (see Flandoli/Schmalfuss [44]). The simplest example of auniverse is the collection of all one-point subsets of X . However the concept


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of a universe allows us to include the consideration of local regimes of thesystem into the theory in a natural way. We refer to Schenk-Hopp e [89]) for

a further discussion of this concept. In the applications presented in Chaps.5and 6 we deal with the universe of all tempered subsets of the phase space.

Denition 1.4.1 (Absorbing Set). A random closed set {B ()} is said to be absorbing for the RDS (, ) in the universe D , if for any D D and for any there exists t0() such that

(t, t )D ( t ) B () for all t t0() and .

Denition 1.4.2 (Dissipative RDS). An RDS (,) is said to be dissi-

pative in the universe D , if there exists an absorbing set B for the RDS (,)in the universe D such that

B () B r ( ) (x0) { x : dist X (x, x 0) r ()}, (1.17)

for some x0 X and random variable r () and for all . If X is a linear space and x0 = 0 , then the variable r () is said to be a radius of dissipativityof the RDS (,) in the universe D .

The simplest examples of dissipative RDS are the following ones.

Example 1.4.1 (Discrete Dissipative RDS). Let us consider the RDS con-structed in Example 1.2.1. Let X = R and 0 = {0, 1} be a two-point set.Assume that the continuous functions f 0 and f 1 possess the property

|f i (x)| a |x| + b with some 0 a < 1, b 0 .

In this case is the set of two-sided sequences = {i | i Z } consisting of zeros and ones and

(n, ) = f n 1 f n 2 . . . f 1 f 0 , = {i | i Z }, n N .

Using the cocycle property it is easy to see that

|(n + 1 , )x| a |(n, )x | + b, n Z + . (1.18)

Therefore after n iterations we obtain

|(n, )x | an |x | + b (1 a) 1 , n Z + . (1.19)

Let D be the family of all tempered (with respect to ) random closed sets

inR

. Let D D

and D () {x : |x | r ()}, where r () possesses theproperty (1.16) (i.e. is a tempered random variable). Then (1.19) implies that

|(n, n )x( n )| an r ( n ) + b (1 a) 1 , for all x() D () .

Since 0 a < 1, it follows from (1.16) that an r ( n ) 0 as n + .Therefore for every there exists n0() such that an r ( n ) 1 forn n0(). Consequently we have


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(n, n )D ( n ) B := [ 1 b (1 a) 1 , 1 + b (1 a) 1]

for n n0(). Thus the RDS considered is dissipative in the universe D of all tempered random closed sets from R . Using (1.18) with n = 0 one caneasily see that B is a forward invariant set from D .

Example 1.4.2 (Kick Model). Let X be a Banach space and g : X X bea continuous mapping such that

g(x) a x + b, 0 a < 1, b 0 . (1.20)

Consider the RDS ( ,) generated by the difference equation

xn +1 = g(xn ) + (n +1 ), n Z + , (1.21)

over a metric dynamical system ( , F , P , {n , n Z }), where () is a tem-pered random variable in X . Using (1.20) and (1.21) we have

(n, )x an x + R(n ), n Z + ,

where

R() = b(1 a) 1 +

k =0ak ( k )

is a tempered random variable. It is easy to see that for every > 0 the ballB () = {x : x (1 + )R()} is a forward invariant absorbing set for(,) in the universe D of all tempered random closed sets from X .

Example 1.4.3 (Continuous Dissipative RDS). Let ( ,) be the RDS consid-ered in Example 1.2.3 from the random ODE x = f (t , x). Assume addi-tionally that the function f (, x) possesses the property

xf (, x) |x |2 + , for all ,

where > 0 and 0 are nonrandom constants. Then it is easy to see that

12

ddt

|x(t)|2 |x(t)|2 + , t > 0 ,

for any solution to (1.10). Therefore, since (t, )x = x(t), we have

|(t, )x |2 e 2t |x |2 + 1 e 2t , t > 0 .

As in Example 1.4.1 this property implies that ( ,) is dissipative in theuniverse D of all tempered (with respect to ) random closed sets from R .Moreover the absorbing set B = {x : |x| 1 + / } is a forward invariantset from D .


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The situation described in Example 1.4.3 admits the following generalizationwhich can be also considered as an extension of well-known deterministic

results (see, e.g., Babin/Vishik [13], Chueshov [20] or Hale [50]) to therandom case.

Proposition 1.4.1. Assume that the phase space X of RDS (,) is a sepa-rable Banach space with the norm and there exists a continuous function V : X R with the properties:

(i) V ((t, )x) is absolutely continuous with respect to t for any (, x) X ;

(ii) there exists a constant > 0 and a tempered random variable () 0such that for every (, x) X we have the inequality

ddt

V ((t, )x) + ( + (t )) V ((t, )x) (t ) (1.22)

for almost all t > 0, where () is a random variable such that (t )lies in L1loc (R ) for every and

limt +

1t

t

0

( ) d = limt +

1t

0

t

( ) d = 0 (1.23)

for all ;(iii) there exist positive constants b1 , b2 , 1 , 2 and nonnegative numbers c1 and

c2 such that

b1 x 1 c1 V (x) b2 x 2 + c2 , x X . (1.24)

Then the RDS (,) is dissipative in the universe D of all tempered random closed sets in X . Moreover there exists a tempered random variable R() 0

such that for any positive the set B () = {x : V (x) (1 + )R()} (1.25)

is a forward invariant absorbing tempered random closed set.

Proof. Let D D and x() D () for all . From (1.22) we have that

V ((t, )x()) V (x()) exp t

t

0( ) d

+ t

0 (s ) exp (t s)

t

s( ) d ds .


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Therefore

V ((t, t )x( t )) V (x( t )) exp t 0

t( ) d

+ 0

t (s ) exp s

0

s( ) d ds .

(1.26)

It follows from (1.23) that for any > 0 and there exists c() > 0such that

t

0 ( ) d |t | + c(), t R

, . (1.27)Therefore, since () is tempered, for all the integral

R() = 0

(s ) exp s

0

s( ) d ds (1.28)

exists. It follows from (1.27) that

R(t ) C ()e | t |

0

es

e( + ) | t + s |

ds sup e | |

( )

1

e( +2 ) | t | sup

e | | ( )

for all > 0 and > 0 such that + < . This implies that R() is atempered random variable. Proposition 1.3.6 and relation (1.24) imply thatB () given by (1.25) is a tempered random closed set. Let

e(t, ) = exp t 0

t( ) d .

Then from (1.26) for any x() B () we have that

V ((t, t )x( t )) (1 + )R( t ) e( t, ) + 0

t (s ) e(s, )ds .

Since e( t, ) e(s, t ) = e(s t, ), it follows from (1.28) that

R( t ) e( t, ) = t

(s ) e(s, )ds .

ThereforeV ((t, t )x( t )) (1 + )R() .


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Thus B () is forward invariant. It follows from (1.24) and (1.26) that

V ((t, t )x( t )) b2 x( t )2

+ c2 e t

+ R() .This relation implies that B () is absorbing in the universe D . 2

Remark 1.4.1. If is an ergodic metric dynamical system, assumption (ii) inProposition 1.4.1 can be replaced by the inequality

ddt

V ((t, )x) + (t ) V ((t, )x) (t ) , (1.29)

there () 0 is a tempered random variable and () L1

(, F , P ) isa random variable such that E > 0. Indeed, it follows from the Birkhoff-Khintchin ergodic theorem (see, e.g., Arnold [3, Appendix]) that

lim| t |

1t

t

0 ( )d = E , ,

where is a -invariant set of full measure. Without loss of generalitywe can suppose that = (see Remark 1.2.1(ii)). Therefore we can applyProposition 1.4.1 with = E and () = () E .

Example 1.4.4 (Binary Biochemical Model). Consider the RDS ( ,) gen-erated in R 2 by equations (1.11) over an ergodic metric dynamical system. Let the hypotheses concerning g and i listed in Example 1.2.4 hold. Weassume in addition that

min () = min { 1(), 2()} L1(, F , P ) and 0 = E min > 0 .

If

x1 (x2 + g(x2)) 02 (x

21 + x

22) + 0 , (x1 , x2) R

2+ ,

where 0 0 is a constant, then (1.29) holds with V (x) = x21 + x22 , =2min () 0 and () 2 0 . Thus the RDS ( ,) is dissipative in theuniverse of all tempered random closed sets from R 2 .

The following concepts are useful when the phase space X is innite-dimensional.

Denition 1.4.3 (Compact RDS). An RDS (,) is said to be compactin the universe D , if it is dissipative in D and the absorbing set B is a random compact set.

If the phase space X of an RDS (, ) is compact, then ( ,) is a compactRDS. If X is a nite-dimensional space, then any dissipative RDS is compact.


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Example 1.4.5 (Kick Model). Let ( ,) be the RDS considered in Exam-ple 1.4.2. Assume additionally that g is a compact mapping, i.e. g(B ) is a

compact set for every bounded set B from X . The setC () = (1, 1)B ( 1) = g(B ( 1)) + ()

is an absorbing forward invariant random compact set for ( ,) in the uni-verse D of all tempered random closed sets from X .

Denition 1.4.4 (Asymptotically Compact RDS). An RDS (,) issaid to be asymptotically compact in the universe D , if there exists an at-tracting random compact set {B0()}, i.e. for any D D and for any

we have limt +

dX {(t, t )D ( t ) | B0()} = 0 , (1.30)

where dX {A|B } = sup xA dist X (x, B ).

It is clear that any compact RDS is asymptotically compact. Deterministicexamples of asymptotically compact systems which are not compact can befound in Babin/Vishik [13], Chueshov [20], Hale [50] and Temam [104].

The following assertion shows that every asymptotically compact RDS isdissipative.

Proposition 1.4.2. Let (,) be an asymptotically compact RDS in D with an attracting random compact set {B0()}. Then it is dissipative in D .

Proof. For any x0 X we can nd a random variable r () (0, + ) suchthat

B0() {x : dist X (x, x 0) r ()} for all . (1.31)To prove this we note that by Theorem 1.3.1

B0() = {g(, y) : y Y } for all ,

where Y is a Polish space and the mapping g(, y) : Y X is suchthat g(, ) is continuous for all and g(, y) is measurable for all y Y .Since B0() is a compact set and Y is separable, r () dened by

r () := supyY

dist X (x0 , g(, y)) (0, + ), ,

is a random variable and (1.31) holds.It follows from (1.30) that for any D D and for any there exists a

t0() such that(t, t )D ( t ) B() := {x : distX (x, x 0) 1+ r ()} for t t0() .

Thus ( , ) is dissipative. 2

The notions of dissipative, compact and asymptotically compact random sys-tems differ only in innite-dimensional phase spaces.


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1.5 Trajectories

In this section we describe some measurable properties of the trajectories of RDS.

Denition 1.5.1. Let D : D () be a multifunction. We call the mul-tifunction

tD () := t

(, )D ( )

the tail (from the moment t) of the pull back trajectories emanating from D .If D () = {v()} is a single valued function, then v () 0D () issaid to be the (pull back) trajectory (or orbit) emanating from v.In the deterministic case is a one-point set and (t, ) = (t) is a semigroupof continuous mappings. Therefore in this case the tail tD has the form

tD = t

( )D = 0

( )((t)D ) = 0 ( t )D ,

i.e. tD is a collection of the normal trajectories emanating from (t)D .We note that any tail is a forward invariant multifunction. It also follows

from Proposition 1.3.1(v) that in the case of discrete time ( T = Z ) the closure tD () of any tail

tD () is a random closed set. For continuous time we have

the following proposition.

Proposition 1.5.1. For any random closed set {D ()} the closure tD ()of any tail tD () of the pull back trajectories emanating from D is a random closed set with respect to the -algebra F u of universally measurable sets.

Proof. The idea of the proof is borrowed from Crauel/Flandoli [36]. The

Representation Theorem 1.3.1 gives that D () = g(, Y ), where Y is a Polishspace, g(, ) is continuous for all and g(, y) is measurable for all y Y .Therefore for every x X we have

d(t, ) := dist X (x,(t, t )D ( t )) = inf k

dist X (x,(t, t )g( t , yk )) ,

where {yk } is a dense sequence in Y . Since (t, ) (t, t ) is a measurablemapping and ( t, ) dk (t, ) := dist X (x,(t, )g(, yk )) is a measurablefunction, the function ( t, ) dk (t, t ) is also measurable. Consequently

the function ( t, ) d(t, ) is B (R + ) F -measurable. It is also clear that

dist( x, tD ()) = dist( x, tD ()) = inf t d(, ) .


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1.5 Trajectories 33

For any a R + we have

: inf t

d(, ) < a = proj {(, ) : d(, ) < a, t} ,

where proj is the canonical projection of R + on dened by

proj M = { : (t, ) M for some t R + }.

Hence Proposition 1.3.3 implies that { : inf t d(, ) < a } is a universallymeasurable set and therefore tD () is a random closed set with respectto F u . 2

As a direct consequence of Proposition 1.3.5 we also have the following as-sertions.

Proposition 1.5.2. Let a() be a random variable in X . Assume that t (t, t )a( t ) is a separable process, t R + . Then ta () is a forward invariant random closed set with respect to F . In particular, if for some x X the mapping t (t, t )x is a right continuous function for all t > 0 and , then tx () is a forward invariant random closed

set with respect toF

.Proof. It is clear that

tD () = {(, )a( ) : t, Q} ,

where Q is a separability set of the process t (t, t )a( t ). Thereforewe can apply Proposition 1.3.1(v). 2

Proposition 1.5.3. Let (,) be an RDS such that the function

(t, x ) (t, t )x is a continuous mapping (1.32)

from R + X into X . Assume that D is a random closed set such that {D (t ) : t 0} is a separable collection. Then the closure tD of the tail tDis a forward invariant random closed set with respect to F for every t 0. In particular, tD possesses this property for every deterministic D .

Proof. Since {D (t ) : t 0} is a separable collection, we can nd an

everywhere dense countable set Q such that for any t 0 and x D ( t )there exist tn Q and xn D ( t n ) such that xn x and tn tas n . Property (1.32) implies that (tn , t n )xn (t, t )x asn . Therefore {(t, t )D ( t ) : t t0} is a separable collection forany t0 0. Thus we can apply Propositions 1.3.5 and 1.3.1(v). 2


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Remark 1.5.1. Assume that the mappings (t, ) are restrictions to R + of mappings (t, ) which satisfy the conditions listed in Denition 1.2.1 for

all t, s R and such that ( t, x ) (t, )x is a continuous mapping fromR X into X for every . This situation is typical for RDS generated bynite-dimensional random and stochastic differential equations (for instance,this is true for the RDS considered in Examples 1.2.4 and 1.4.4). The cocycleproperty for implies that

(t, t ) ( t, ) = ( t, ) (t, t ) = id , t R , .

Hence (t, x ) (t, ( t, )) is a bijective mapping from R X into itself and

(t, t ) = (t, t ) = [ ( t, )] 1 , t 0 .

Therefore by Proposition 1.1.6 ( Arnold [3]) (t, x ) (t, t ) is a continu-ous mapping from R X into X for every provided that X is either acompact Hausdorff space or a nite-dimensional topological manifold. There-fore in this case by Proposition 1.5.2 { ta ()} is a forward invariant randomclosed set with respect to F for every a() such that the mapping t a(t )is continuous for all . By Proposition 1.5.3 the same is true for tD ,where D is a deterministic subset in X .

We note that if X is a separable Banach space, then the set of randomvariables v() such that t v(t ) is a C -function for every is densein the set of all random variables with respect to convergence in probability(see the argument given in the proof of Proposition 8.3.8 Arnold [3]). Wealso note that in the case considered the function t (t, t )a( t ) isa stochastically continuous process (i.e. it is continuous with respect to con-vergence in probability) for any random variable a(). This property followsfrom the stochastic continuity of the process t a(t ) (see Arnold [3,Appendix A.1]).

1.6 Omega-limit Sets

To describe the asymptotic behaviour of RDS as in the deterministic case (cf.Hartman [51] and also Hale [50], Temam [104], Chueshov [20], for exam-ple) we use the concept of an omega-limit set. As in Crauel/Flandoli [36]our denition concerns pull back trajectories.

Denition 1.6.1. Let D : D () be a multifunction. We call the mul-tifunction

D () :=t> 0

tD () =t> 0 t

(, )D ( )

the (pull back) omega-limit set of the trajectories emanating from D .


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1.6 Omega-limit Sets 35

The following assertion gives another description of omega-limit sets.

Proposition 1.6.1. Let D

() be the omega-limit set of the trajectories em-anating from a multifunction D . Then x D () if and only if there exist sequences tn + and yn D ( t n ) such that

x = limn +

(tn , t n )yn . (1.33)

Proof. Let x D (). Then we have

x

n

(, )D ( ) for all n = 1 , 2, . . . .

Therefore there exists an element bn such that

bn n

(, )D ( ) (1.34)

and dist( x, bn ) 1/n , n = 1 , 2, . . . . It follows from (1.34) that there existtn n and yn D (t n ) such that bn = (tn , t n )yn . It is clear that wehave (1.33) for these tn and yn .

Vice versa, assume that an element x possesses property (1.33). It isobvious that for any t > 0 there exists tn such that

(tn , t n )yn t

(, )D ( ) t

(, )D ( ) .

Thereforex

t

(, )D ( ) for all t > 0.

This implies that x D (). 2

We note that Proposition 1.6.1 provides us with a description of omega-limit sets. But it does not guarantee that they are nonempty. The followingassertion gives us conditions under which D () is nonempty.

Proposition 1.6.2. Assume that the RDS (, ) is asymptotically compact in a universe D with the attracting random compact set {B0()}. Then for any D D and for all the omega-limit set D () is a nonempty

compact set and D () B0(). The multifunction D () is invariant and it is a random compact set with respect to the -algebra F u of universally measurable sets (with respect to F , in the case of discrete time).


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Proof. Let tn and yn D ( t n ) be arbitrary sequences. From (1.30)we have that

(tn , t n )yn B0() when n + ,

i.e. there exists a sequence {bn } B0() such that

dist X ((tn , t n )yn , bn ) 0 when n + .

The compactness of B0() implies that for some subsequence {n k } and somebB0() we have that bn k b. This implies that

(tn k , t n k )yn k bB0() when k + .

Thus D () is nonempty. It is clear from (1.30) that any element of the form(1.33) belongs to B0(). Therefore we have D () B0() and, since D ()is closed, D () is a compact set.

Let us prove that D () is invariant. Using the cocycle property wehave

(t, )x = limn

(t, ) (tn , t n )yn = limn

(t + tn , t t n t )yn

for any x D () of the form (1.33). Due to Proposition 1.6.1 this impliesthat (t, )x D (t ). Thus (t, ) D () D (t ) for all t > 0 and .

Assume that x D (t ) for some t > 0 and . Proposition 1.6.1implies that

x = limn

(tn , t n t )yn , (1.35)

where yn D ( t n t ) and tn . The cocycle property gives that

x = limn

(t, )zn with zn = (tn t, t n + t )yn . (1.36)

From (1.30) we have that zn B0() as n . Since B0() is compact,there exist {n k } and b B0() such that zn k b as k . MoreoverProposition 1.6.1 implies that b D (). From (1.36) we obtain that x =(t, )b. Therefore D (t ) (t, ) D () for all t > 0 and . Thus{ D ()} is invariant.

To prove that { D ()} is a random compact set with respect to F u we

use Proposition 1.5.1 and the obvious formula D () = nZ + nD () whichimplies in our case that

dist( x, D ()) = limn

dist( x, nD ()) , . (1.37)


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1.6 Omega-limit Sets 37

Indeed, since D () n +1D () nD (), we have that

dist( x, nD ()) dist( x,

n +1D ()) dist( x, D ())

for any x X . Therefore the limit in (1.37) exists and

dist( x, D ()) limn

dist( x, nD ()) , .

Let xn nD () be such that

dist( x, x n ) dist( x, nD ()) +1n

, n = 1 , 2, . . .

Since nD () B0() as n for all , there exist a subsequencen k = nk () and b B0() such that xn k b. By Proposition 1.6.1 b D (). Therefore

dist( x, D ()) dist( x, b) = limk

dist( x, x n k ) limn

dist( x, nD ()) .

Thus we obtain (1.37). By Proposition 1.5.1 dist( x, nD ()) is F u -measurable. Therefore dist( x, D ()) is also F u -measurable. Hence Dis a random set with respect to the universal -algebra F u . 2

Remark 1.6.1. The existence and measurability of omega-limit sets with re-spect to the universal -algebra can be proved under a weaker property thanthe asymptotic compactness of RDS ( ,). Assume that {D ()} is a randomclosed set and for every there exists a compact set BD () X suchthat

limt +

dX {(t, t )D ( t ) | BD ()} = 0 ,

where dX {A|B } = sup xA dist X (x, B ). Then, as in the proof of Proposi-tion 1.6.2, it follows from Proposition 1.5.1 that D exists and D ()is an invariant random compact set with respect to the universal -algebraF u . If we additionally assume that the closure tD () of the tail

tD () is a

random closed set for every t 0 (cf. Proposition 1.5.3 and Remark 1.5.1),then D is a random compact set with respect to F . We refer to Crauel [33]for other results concerning the measurability of omega-limit sets.

The following two assertions provide us with conditions which guarantee that{ D ()} is a random compact set with respect to the -algebra F .

Proposition 1.6.3. If {D ()} is a forward invariant random compact set for the RDS (,), then the multifunction D () is an invariant random compact set with respect to F and D () D ().


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Proof. Since {D ()} is a forward invariant set, we have

D () =t> 0(t, t )D ( t ) =

nZ +(n, n )D ( n ) . (1.38)

Proposition 1.3.1(vi) implies that Dn () := (n, )D () is a randomcompact set. Therefore Dn ( n ) is also a random compact set. Conse-quently it follows from Proposition 1.3.1(iv) that D () is a random compactset. It is clear from (1.38) that D () is a forward invariant set. Let us proveits backward invariance. Let x D (t ) for some t > 0 and . Then asabove by Proposition 1.6.1 we have (1.35) and (1.36) with zn D (). Since

D () is compact, we can choose a convergent subsequence {zn k } and applythe same argument as in the proof of Proposition 1.6.2. 2

Proposition 1.6.4. Let a() be a random variable in X . Assume that theprocess t (t, t )a( t ) is separable for t R + and for each there exists t = t() such that t a () is a compact set. Then the omega-limit set a () is a random compact set with respect to F .

Proof. The compactness of t a () implies that a () is a nonempty com-pact set for all . Therefore we can use Proposition 1.5.2, the for-mula D () = nZ + nD () and the argument given in the proof of Proposi-tion 1.6.2. 2

1.7 Equilibria

A special case of omega-limit sets are random equilibria. They are the randomanalog of deterministic xed points and generate stationary stochastic orbits(cf. Arnold [3], Arnold/Schmalfuss [11] and Schmalfuss [94]).

Denition 1.7.1. A random variable u : X is said to be an equilibrium(or xed point, or stationary solution) of the RDS (,) if it is invariant under , i.e. if

(t, )u() = u(t ) for all t 0 and all .

It is clear that if u = u() is an equilibrium, then u () = u().

Example 1.7.1 (Kick Model). If in Example 1.4.2 we additionally assumethat g is a linear mapping such that g a < 1, then it is easy to see that

u() =

k =0

gk (( k ))

is an equilibrium for the RDS generated by (1.21).


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1.7 Equilibria 39

Remark 1.7.1. The problem of the construction of equilibria for general RDSis rather complicated. The following example demonstrates the difficulties in

the construction of equilibria. Let us consider the RDS on R + constructedin the Introduction (cf. also Example 1.4.1) with f 0(x) = 12 x and f 1(x) =12 + f 0(x) =

12 (1 + x). Both functions f 0(x) and f 1(x) have a xed point:

f 0(0) = 0 and f 1(1) = 1. To obtain an equilibrium we should look for asolution to the equation f 0 (u()) = u(1), where = {i | i Z } is a two-sided sequence consisting of zeros and ones and 1 is the left one-symbol shiftoperator. It is clear that an equilibrium u() is not simply a random variablewhich takes as its values the xed points 0 and 1 of the mappings f 0(x) andf 1(x). The variable u() can really depend on the sequence = {i | i Z } in

a very complicated way. However we prove in Chap.3 that this RDS possessesa unique globally asymptotically stable equilibrium in R + with its valuesinside the interval (0 , 1).

We also note that the results by Ochs/Oseledets [87] and Ochs [85]show that it is impossible to generalize topological xed point theorems tothe case of random dynamical systems. However, as we will see in Chaps.36,there are more simple approaches which allow us to construct equilibria formonotone RDS.

The following simple assertion makes it possible to prove the uniqueness of equilibria, if they exist, in several important cases (see, e.g., Sect. 4.2 below).

Proposition 1.7.1. Let D () be a forward invariant multifunction for the RDS (, ). Assume that on the set

G = {(,u,v) : u, v D (), } X X

there exists a function V : G R satisfying

(i) V (, u(), v()) is measurable for any random variables u() and v() from D ();

(ii) for any u and v from D () we hav

monotone random systems

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