An Introduction to Random Dynamical Systems for Climate
Franco Flandoli and Elisa Tonello
Septemebr 2019 2 Contents
0.1 Preface ...... 4
1 Non-Autonomous Dynamical Systems 1 1.1 Introduction ...... 1 1.2 Non autonomous dynamical systems: definition and examples ...... 2 1.3 Invariance. Criticalities of the concept for NADS ...... 5 1.4 Global attractor and omega-limit sets ...... 7 1.5 A criterium for existence of global attractor ...... 9 1.6 On the uniqueness of the global attractor ...... 12 1.7 A bimodal-climate example ...... 14 1.8 Invariant measures: attempts of qualifications to reduce non-uniqueness . . 16 1.9 Separation between non-autonomous input and autonomous dynamics: co- cycles ...... 17 1.10 Non-autonomous RDS (naRDS) ...... 19
2 Generalities on RDS 23 2.1 Random dynamical systems ...... 23 2.2 Definitions ...... 23 2.3 RDS from Stochastic Differential Equations ...... 27 2.3.1 Additive noise ...... 32 2.3.2 A finite dimensional model ...... 33 2.3.3 n-point motion ...... 36 2.4 RDS as products of i.i.d. random mappings ...... 38 2.5 An example in infinite dimensions: the stochastic Navier-Stokes equations ...... 38 2.6 Multiplicative ergodic theorem and Lyapunov exponents ...... 41
3 Random Attractors and Random Invariant Measures 43 3.1 Attractors for random dynamical systems ...... 43 3.1.1 Attractor for the finite dimensional approximation of SNSE . . . . . 46 3.2 Random Measures ...... 48
3 4 CONTENTS
3.3 Random measures and random dynamical systems ...... 49 3.3.1 Support of a random measure ...... 53 3.3.2 Markov measures ...... 54 3.4 From invariant measures to invariant random measures ...... 55
4 Synchronization 61
5 Physical measures 63 5.1 On the concept of physical measure ...... 63 5.2 Physical properties in deterministic settings ...... 64 5.3 What is known for stochastic systems ...... 68 5.4 Some results via two-point motion ...... 69 5.5 Statistical equilibrium and two-point motion ...... 69 5.6 Results on compact manifolds ...... 76 5.7 Conclusion and program ...... 79 5.8 On the study of the one and the two-point motion ...... 80 5.8.1 Existence of an invariant measure ...... 81 5.9 Ergodicity ...... 81 5.9.1 Strong Feller property ...... 82 5.9.2 Irreducibility ...... 85
6 Appendix 93 6.1 Attractors for deterministic systems ...... 93 6.2 Kernels ...... 96 6.3 Markov semigroups ...... 98 6.4 Existence and uniqueness of invariant measures ...... 99 6.5 Stationary Markov processes ...... 102
0.1 Preface
These notes are the result of a Master thesis and a series of lectures given at Institute Henri Poincaré on the occasion of the intensive research period "The Mathematics of Climate and the Environment". Chapter 1
Non-Autonomous Dynamical Systems
1.1 Introduction
In the following, to avoid misunderstandings, we have to always keep in mind the distinction between climate and climate models. When we say that climate is non-autonomous we do not criticize the several autonomous climate models, which may be perfectly justified by reasons of simplicity, idealization, interest in particular phenomena and so on. Climate is non-autonomous. Solar radiation is the most obvious non-autonomous ex- ternal input. It has a daily component of variability and an annual component. The annual component can be considered periodic; the daily one also if the random effect of clouds is not included in it, otherwise could be considered as a random rapidly varying input. Another fundamental non-autonomous input is produced by biological systems, in partic- ular the CO2 emission by humans in the last 150 years. The latter input is non-periodic, it has an increasing trend with some small variations. Another non-autonomous input, this time at the level of models instead of climate itself, is the randomization of sub-grid processes, more diffi cult to explain in a few words but fundamental in modern numerical codes; this randomization has usually a stationary character. We have thus exemplified non-autonomous inputs of periodic, increasing and random type. We thus have, in general, a multidimensional PDE system of the form
∂tu = Au + B (u) + C (u, q (t)) + D (u) ω· (t) (other generic descriptions are equally meaningful at this informal level). We collected in u (t) the state variables (wind velocity, temperature etc.) and in q (t) , ω· (t) the time-varying inputs, separting them, for conceptual convenience, in deterministic slowly varying terms q (t) (anthropogenic factors, annual variation of solar radiation) and stochastic fast varying terms ω· (t) assumed to be a - possibly state dependent - white noise in time (delay terms,
1 2 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS relevant, will be assumed of the form that allows to write the previous system enlarging the set of variables u (t)). It is then clear that we shall spend considerable effort to keep into account some degree of non-autonomy of the dynamics. We shall discuss different levels, concentrating on the one typical of autonomous Random Dynamical Systems (RDS) for the development of rigorous results, but with the hope that this more general presentation and insistence on the role of non-autonomous inputs will trigger more research on non-autonomous RDS.
1.2 Non autonomous dynamical systems: definition and ex- amples
A relatively general mathematical scheme is made of a metric space (X, d) and a family of maps U (s, t): X X, s t → ≤ (to fix the ideas we assume for simplicity that time parameters s, t vary in R, but different time sets can be considered), with the two properties U (s, s) = Id and U (r, t) U (s, r) = U (s, t), r s t and various possible regularity properties, like U (s, t) continuous◦ in X, for every s≤ t.≤We shall always assume continuity. Thus let us formalize as a definition: ≤ Definition 1 A (forward in time) Non Autonomous Dynamical System (NADS) (also called an evolution operator) is a family of continuous maps indexed by two times U (s, t): X X s t → ≤ with the rules U (s, s) = Id for all s U (r, t) U (s, r) = U (s, t) for all s r t. ◦ ≤ ≤ Remark 2 Sometimes the definition of U (s, t) and the rules are required also for s > t; in that case we have a NADS with forward and backward properties. This is the case of finite dimensional systems and usually of hyperbolic problems. Since, however, we shall deal most often with parabolic problems, we restrict the exposition to the forward-in-time case.
Remark 3 The case, for T = [0, ), of a family depending on a single time parameter ∞ (t): X X t 0 U → ≥ with the rules (0) = Id U (t) (s) = (t + s) for all s, t 0 U ◦ U U ≥ 1.2. NON AUTONOMOUS DYNAMICAL SYSTEMS: DEFINITION AND EXAMPLES3 is a particular case, called simply a Dynamical System (DS). The correspondence with the more general concept is U (s, t) = (t s) U − (check that the properties of U (s, t) are satisfied).
Example 4 Let b : T Rd Rd continuous and satisfying × → d b (t, x) b (t, y) L x y for all t T and x, y R . | − | ≤ | − | ∈ ∈ For every s T , the Cauchy problem ∈
X0 (t) = b (t, X (t)) t s ≥ X (s) = x has a unique global solution of class C1 T ; Rd (classical Cauchy-Lipschitz theorem). De- note it by Xs,x (t). The map x Xs,x (t) 7→ is Lipschitz continuous (from Gronwall lemma). The family of maps
d d U (s, t): R R s t → ≤ defined as U (s, t)(x) = Xs,x (t) is a Lipschitz continuous NADS.
Example 5 Consider the stochastic equation in Rd
dXt = b (Xt) dt + dWt
d d d where b : R R is Lipschitz continuous and Wt is a Brownian motion (BM) in R . d → Let C0 R; R be the space of continuous functions null at t = 0. Let us first define the two-sided Wiener measure. Take, on some probability space, two independent copies of the (i) BM W , say Wt , i = 1, 2; define the two-sided BM:
(1) (2) Wt = Wt for t 0, Wt = W t for t 0; ≥ − ≤ d and call P its law, on Borel sets of C0 R; R ; this is the two-sided Wiener measure. d Now consider the canonical space Ω = C0 R; R with Borel σ-field and two-sided F Wiener measure P. On (Ω, , P) consider the canonical two-sided BM defined as F d Wt (ω) = ω (t) ω C0 R; R . ∈ 4 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Let us now interpret the stochastic equation in integral form on the canonical space: t Xt (ω) = x + b (Xr (ω)) dr + ω (t) ω (s) . − Zs d Given ω C0 R; R , we may solve uniquely this equation by the contraction principle, first locally∈ but then with global prolungation step by step, and construct a solution Xs,x (ω) t defined for all t, continuously dependent (in a Lipschitz way) on x. Again we set s,x U (s, t)(x) = Xt (ω) . In this example the dynamical system is parametrized by ω, feature that we shall discuss more extensively in later sections.
Example 6 On the torus T2 = R2/Z2 (just for simplicity) consider the Navier-Stokes equations 2 ∂tu (t, x) + u (t, x) xu (t, x) + xp (t, x) = ν∆xu (t, x) + f (t, x)(t, x) T T · ∇ ∇ ∈ × 2 divx u (t, x) = 0 (t, x) T T ∈ × with periodic boundary conditions, zero average, and the "initial" condition u (s, x) = ϕ (x) . 2 Let H be the Hilbert space obtained as the closure in the L -topology of C∞ periodic zero average divergence free vector fields on T2. If f L2 (T ; H) ϕ H ∈ loc ∈ then there exists a unique weak solution, denoted by us,ϕ (t, x), with the property us,ϕ (t, ) H and several others. The concept of weak solution of the 2D Navier-Stokes equations· will∈ be explained later. The family of maps U (s, t): H H s t → ≤ defined as U (s, t)(ϕ) = us,ϕ (t, ) · is a Lipschitz continuous NADS.
Example 7 On a smooth bounded open domain D R3 consider the diffusion equation ∈ ∂tθ (t, x) + u (t, x) xθ (t, x) = κ∆xθ (t, x) + f (θ (t, x)) + Q (t, x)(t, x) T D · ∇ ∈ × where u is a velocity field, for instance solution of equations of Navier-Stokes type, f is a nonlinear function, for instance f (θ) = Cθ4 − and Q (t, x) is a time-dependent heat exchange, e.g. corresponding to solar radiation. Also this equation generates a NADS in suitable function spaces. More details a due time. 1.3. INVARIANCE. CRITICALITIES OF THE CONCEPT FOR NADS 5
Can we introduce interesting dynamical concepts for such general objects? We discuss (more or less everywhere in these lectures) mainly two concepts: attractors and measures with suitable invariance properties. Behind these concepts there is invariance.
1.3 Invariance. Criticalities of the concept for NADS
In the autonomous case we say that a set A X is invariant if ⊂
ϕt (A) = A for all t
(also the simpler property of positive invariance, ϕt (A) A, sometimes is already quite ⊂ useful). To avoid trivialities (like ϕt (X) = X), one usually add the requirement of bound- edness or compactness of invariance sets. This identifies a notion, for instance compact invariant set, which is already quite useful and interesting. A probability measure µ is invariant if
(ϕt)] µ = µ for all t where ] stands for the push-forward ((ϕt)] µ is the probability measure defined by (ϕt)] µ (B) = 1 µ ϕt− (B) for all Borel sets B). Being a probability measure implies it is almost sup- ported on compact sets (in the sense of tightness), hence it incorporates a requirement similar to the boundedness or compacteness of invariant sets. In the non-autonomous case, invariance can be defined (presumably) only in the fol- lowing way (we add compactness to go faster to the main points):
Definition 8 A family of sets At X, t R is called a compact invariant set for U (s, t) { ⊂ ∈ } if At is compact for every t and
U (s, t) As = At for all s t. ≤
A family of Borel probability measures µt, t R is invariant for U (s, t) if { ∈ }
U (s, t) µs = µt for all s t. ] ≤ Apparently they look similar to the autonomous case but they are almost empty con- cepts, satisfied by trivial object of no long-time interest: assume the dynamics can be run both forward and backward globally in time - as it is for usual ordinary differential equations) and let x (t) , t R ∈ be a trajectory. It satisfies U (s, t) x (s) = x (t) 6 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS and thus the singleton x (t) is a compact invariant set; the same for any suitable bunches { } of trajectories. Let µt, t R be simply defined by { ∈ }
µt = δx(t).
It is an invariant measure. This genericity is very far from the specificity of invariant sets and measures of the autonomous case!
Example 9 Consider the equation
x0 (t) = x (t) + f (t) , t s and t < s − ≥ 0 x (s) = xs
0 with a given initial condition xs and a continuous function f. Its solution is
t (t s) 0 (t r) x (t) = e− − xs + e− − f (r) dr Zs (t s) 0 =: e− − xs + xs (t) .
This is an example of full-time trajectory, hence x (t) is an invariant set, supporting a { } 0 delta Dirac invariant measure µt = δx(t); and this independently of the choice of s and xs.
Example 10 There is however a special case of the following example: the function
t (t r) x (t) := e− − f (r) dr −∞ Z−∞ when defined, for instance when f has at most polynomial growth at . This function satisfies the differential equation, the singleton x (t) is an invariant−∞ set but it has something special, intuitively speaking. For instance:{ −∞ assume} f = 1 for simplicity; then the function x (t) = 1, it is also bounded, but the other solutions above are all unbounded −∞ 0 as t except when xs = 1 (which is the case of x (t)). → −∞ −∞ The way to focus on interesting objects is to require, as in the previous example, that they come form bounded elements at "time ". One can formulate several conditions; let us focus first on the concept of global attractors.−∞
Remark 11 Also in the autonomous case it happens for many examples that invariant sets and invariant measures are non unique but this corresponds to interesting different long time objects of the dynamics. In the example above the non-uniqueness is simply related to different initial conditions, in spite of the fact that all trajectories tend to approach each other whn time increases. Thus it is an artificial non-uniqueness due to a drawback of the concept. 1.4. GLOBAL ATTRACTOR AND OMEGA-LIMIT SETS 7
1.4 Global attractor and omega-limit sets
What does it mean that a set attracts others in the non-autonomous case?
Definition 12 We say that A (t) attracts B at time t if for every > 0 there exists s0 < 0 such that for all s < s0 we have
U (s, t)(B) (A (t)) . ⊂ U We say tha a family of sets A (t) , t R attracts B if the set A (t) attracts B at time t, { ∈ } for every t R. ∈ This definition can be formulated by means of the non-symmetric distance between sets. Given A, B X define ⊂ d (B,A) = sup d(x, A) x B ∈ where d(x, A) = inf d(x, y). y A ∈ Then A (t) attracts B if lim d (U (s, t)(B) ,A (t)) = 0. s →−∞ Definition 13 We call pull-back omega-limit set of B at time t the set
Ω(B, t) = U (s, t)(B)
s0 0s s0 \≥ [≤ = y X : xn B, sn ,U (sn, t)(xn) y . { ∈ ∃ ⊂ → −∞ → } Notice that obviously it can be an empty set.
Proposition 14 A (t) attracts B if an only if Ω(B, t) A (t). ⊂ Proof. Let us prove that if A (t) attracts B then Ω(B, t) A (t). Take y Ω(B, t) and ⊂ ∈ xn B, sn such that ∈ → −∞ U (sn, t)(xn) y. → We have U (sn, t)(xn) (A (t)) eventually, hence y is in the closer of (A (t)). By arbitrarity of , y A (t∈). U U Let us prove the∈ converse statement by contradiction. Assuming that A (t) does not attract B means that there exists > 0 such that, for every s0 < 0 there exists s < s0 and x B such that U (s, t)(x) / (A (t)). We can thus construct a sequence with this ∈ ∈ U property so that a point y Ω(B, t) does not belong to (A (t)). This contradicts the assumption. ∈ U Below we give the definition of compact absorbing family; we anticipate here for com- pactness of exposition a criterium partially based on such concept. 8 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Proposition 15 In general,
U (s, t)Ω(B, s) Ω(B, t) . ⊂ If there is a compact absorbing family, then
U (s, t)Ω(B, s) = Ω (B, t) .
Proof. Take y Ω(B, t) and xn B, sn such that ∈ ∈ → −∞
U (sn, s)(xn) y. → Then U (sn, t)(xn) = U (s, t) U (sn, s)(xn) U (s, t) y → namely U (s, t) y Ω(B, t), i.e. U (s, t)Ω(B, s) Ω(B, t). Conversely, take z Ω(B, t) ∈ ⊂ ∈ and xn B, sn such that ∈ → −∞
U (sn, t)(xn) z. → Then U (s, t) U (sn, s)(xn) z. → The existence of a compact absorbing set implies that U (sn, s)(xn) is included, eventually, in a compact set, hence there is a converging subsequence U (snk , s)(xnk ) y, hence y Ω(B, s) and z = U (s, t) y, therefore Ω(B, t) U (s, t)Ω(B, s). → ∈ ⊂ Definition 16 Given the NADS
U (s, t): X X s t, s, t R → ≤ ∈ we say that a family if sets A (t) t R ∈ is a pull-back global compact attractor (PBCGA) if: i) A (t) is compact for every t R ∈ ii) A ( ) is invariant: U (s, t)(A (s)) = A (t) for every s t, s, t R iii) A (· ) pull-back attracts bounded sets: ≤ ∈ · Ω(B, t) A (t) ⊂ for all bounded set B X. ⊂ Remark 17 The climate we observe today is the result of a long tme-evolution of some (unknown) initial condition. We thus observe configurations in the pull-back attractor. This is why the notion is at the core of conceptual climatology. 1.5. A CRITERIUM FOR EXISTENCE OF GLOBAL ATTRACTOR 9
Example 18 Consider again Example 9, with f bounded (one can consider a more general case). The singleton x (t) is a compact global attractor, while each other invariant set of the form x (t) (or{ −∞ compact} unions of such sets not containing x (t) , see Section 1.6) is not.{ What} distinguishes the two cases is only property (iii).{ Take−∞ a bounded} set B. Then (t s) 0 0 U (s, t)(B) = e− − x + xs (t); x B s s ∈ where n o t (t r) xs (t) = e− − f (r) dr Zs We have
(t s) 0 d (U (s, t)(B), x (t) ) = sup d e− − xs + xs (t) , x (t) { −∞ } x0 B −∞ s∈ (t s) 0 e− − sup xs + xs (t) x (t) 0 −∞ ≤ xs B | − | ∈ 0 which is easily seen to go to zero as s . On the contrary, taken x0 = x (0), we have → −∞ 6 −∞
t 0 (t s) 0 t 0 d U (s, t)(B), e− x0 + x0 (t) = sup d e− − xs + xs (t) , e− x0 + x0 (t) x0 B s∈ (t s) 0 t 0 = sup e− − xs + xs (t) e− x0 x0 (t) x0 B − − s∈ (t s) 0 t 0 = sup e− − xs + (xs (t) x (t)) e− x0 x (0) x0 B − −∞ − − −∞ s∈ where we have used t x (t) = e− x (0) + x0 (t) . −∞ −∞ (t s) 0 The quantities supx0 B e− − xs and xs (t) x (t) converge to zero as s while s∈ | − −∞ | → −∞ the quantity e t x0 x (0) is not zero and independent of s. Hence the distance above − 0 does not tend to zero− as −∞s . → −∞ 1.5 A criterium for existence of global attractor
Definition 19 A family of sets D (t), t R is called a bounded (resp. compact) pull-back absorbing family if: ∈ i) D (t) is bounded (resp. compact) for every t R ∈ ii) for every t R and every bounded set B X there exists tB < t such that ∈ ⊂
U (s, t)(B) D (t) for every s < tB. ⊂ 10 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Theorem 20 Let U (s, t) be a continuous NADS. Assume that there exists a compact pull- back absorbing family. Then a PBCGA exists.
In infinite dimensional examples of parabolic type the existence of a compact absorbing family is usually proved by means of the following lemma; hence what we usually apply in those examples is the next corollary.
Lemma 21 Let U (s, t) be a continuous NADS. Assume that: i) (compact NADS) for every t and bounded set B X, the set U (t, t + 1) (B) is compact ⊂ ii) there exists a pull-back absorbing family. Then there exists a compact pull-back absorbing family.
Proof. Call D (t) the (bounded) absorbing family. Set D0 (t) = U (t 1, t)(D (t 1)). It is easy to check that this is a compact absorbing family. − −
Remark 22 In the compactness assumption for U (t, t + 1) the time lag 1 is obviously ar- bitrary. Over this remark and the peculiarities of certain examples, the literature developed a more general criterium based on asymptotic compactness.
Corollary 23 Let U (s, t) be a continuous NADS. Assume that: i) (compact NADS) for every t and bounded set B X, the set U (t, t + 1) (B) is compact ⊂ ii) there exists a pull-back absorbing family. Then a PBCGA exists.
Example 24 We show by a simple example that the property of pull-back absorbing set can be verified in examples. Consider, over all t R, the equation ∈ 3 X0 (t) = f (t) X (t) X (t) − where f (t) is a given bounded function. In the following arguments we can think that we work on a "pull-back" interval [s, t] or equivalently on a standard forward interval [0,T ]: the bounds are the same, depending only on f . We prove now that a pull-back absorbingk familyk∞ exists, made of a single bounded set (hence backward frequently bounded); moreover, the NADS is obviously compact (it is al- wasy true in finite dimensions). Hence the unique PBCGA exists. As done also before,
d 2 2 4 2 4 X = 2XX0 = 2f (t) X 2X 2 f X 2X . dt − − ≤ k k∞ − Let y (t) (for us = X2 (t)) be a non negative differentiable functions which satisfies, for every t 0, ≥ 2 y0 (t) 2 f y (t) 2y (t) . ≤ k k∞ − 1.5. A CRITERIUM FOR EXISTENCE OF GLOBAL ATTRACTOR 11
Since the function y 2 f y 2y2, for y 0, is positive in the interval (0, f ), 7→ k k∞ − ≥ k k∞ negative for y > f , the same happens to y0 (t). Therefore, if y (0) [0, f ], we can show that y (tk) k∞[0, f ] for every t 0. If y (0) > f , y (t)∈decreasesk k∞ until y (t) > f . More∈ precisely,k k∞ if y (0) [0, f≥ + 1], then y (0)k k∞[0, f + 1] for every t 0;k if yk∞(0) > f + 1, in a finite∈ timek k depending∞ on y (0) ∈the functionk k∞ y (t) enters [0≥, f + 1] andk thenk∞ (for what already said) it never leaves it. Translated to X (t): the ballkBk∞0, f + 1 is an absorbing set. k k∞ Proof. (Proofp of Theorem 20) Set
A (t) = Ω(B, t). B bounded [ Let us prove it fulfills all properties of a PBGCA. By definition, Ω(B, t) A (t), hence we have pull-back attraction. ⊂ From property (ii) of absorbing set,
Ω(B, t) D (t) ⊂ for every t and every bounded set B. In particular,
Ω(B, t) D (t) = D (t) . ⊂ B bounded [ Hence A (t) D (t) namely it is compact (being closed subset of a compact set). Let us prove⊂ forward invariance. It will be an easy consequence of the forward invariance of omega-limt sets: U (s, t)Ω(B, s) Ω(B, t). It implies ⊂ U (s, t)Ω(B, s) Ω(B, t) = A (t) . ⊂ B bounded B bounded [ [ We always have the set-theoretical property
U (s, t) Ω(B, s) U (s, t)Ω(B, s) ⊂ B bounded ! B bounded [ [ hence we have proved that
U (s, t) Ω(B, s) A (t) . ⊂ B bounded ! [ But, by continuity of U (s, t) and definition of A (s) we have
U (s, t) A ((s)) = U (s, t) Ω(B, s) U (s, t) Ω(B, s) ⊂ B bounded ! B bounded ! [ [ 12 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS which implies U (s, t) A ((s)) A (t) . ⊂ The opposite inclusion A (t) U (s, t) A ((s)) ⊂ is the most tricky step of the proof. Let z A (t). There exists a sequence of bounded sets ∈ Bn and points zn Ω(Bn, t) such that ∈ z = lim zn. n →∞ Each zn is equal to n n zn = lim U (sk , t)(xk ) k →∞ where lim sn = and (xn) B . Hence k k k k N n →∞ −∞ ∈ ⊂ n n n zn = lim U (s, t)(U (sk , s)(xk )) = lim U (s, t)(yk ) k k →∞ →∞ n n n yk := U (sk , s)(xk ) . The existence of a compact absorbing set D (s) implies that there exists a subsequence n ykn with a limit m m N ∈ n n y = lim y n m km →∞ yn D (s) . ∈ Then, by continuity of U (s, t), n zn = U (s, t) y . Again by compactness of D (s) there is a subsequence (ynj ) with a limit y D (s). By j N continuity of U (s, t), ∈ ∈ z = U (s, t) y. It remains to check that y A (s). Due to the closure in the definition of A (s), it is nj ∈ n suffi cient to prove that y A (s). Looking at the definition of ykn , we see that its limit n ∈ m (in m) y is in Ω(Bn, s), hence in A (s).
1.6 On the uniqueness of the global attractor
The attraction property has a strong power of identification, with respect to the poor property of invariance. However, given a global attractor A (t) we may always add to it trajectories x (t) and still have all properties, because compactness and invariance are satisfied and{ attraction} continue to hold when we enlarge the attracting sets. One way to escap this artificial non-uniqueness is by asking a property of minimality. We call it so because of the use in the literature of attractors, although minimal in set theory is a different notion. 1.6. ON THE UNIQUENESS OF THE GLOBAL ATTRACTOR 13
Definition 25 We say that a global compact attractor A (t) is minimal if any other global compact attractor A0 (t) satisfies A (t) A0 (t) ⊂ for all t R. ∈ When it exists, the minimal global compact attractor is obviously unique. Proposition 26 Under the assumptions of Theorem 20, the attractor
A (t) = Ω(B, t) B bounded [ is minimal.
Proof. If A0 (t) is a global attractor, it includes all omega-limit sets, hence it includes (being closed) A (t). The previous solution of the uniqueness problem is very simple and effi cient under the assumptions of Theorem 20, which are those we verify in examples. However, at a more conceptual level one could ask whether there are cses when we can establish uniqueness from the definition itself. Indeed, in the autonomous case (see for instance the first chapter of [49]) it is known that the analogous concept is unique as an immediate consequence of the definition, because the attractor is itself a bounded set and the attraction property of bounded sets easily implies uniqueness. We thus mimic the autonomous case by a definition and a simple criterium.
Definition 27 A family of compact sets A (t), t R, is called backward frequently bounded ∈ if there is a bounded set B and a sequence sn such that A (sn) B for every n N. → −∞ ⊂ ∈ Proposition 28 In the category of backward frequently bounded families, the PBCGA is unique, when it exists. Proof. Assume A ( ) and A ( ) are two PBCGA, both backward frequently bounded and · · let B be a bounded set and sn be a sequence such that A (sn) B for every n N. → −∞ ⊂ ∈ Take z A (t) and, thankse to property U (sn, t) A (sn) = A (t), let xn A (sn) be such ∈ ∈ that U (sn, t) xn = z. We have xn B. Hence we have ∈ z y X : xn B, sn ,U (sn, t)(xn) y . ∈ { ∈ ∃ ⊂ → −∞ → } This proves A (t) Ω(B, t) . ⊂ But Ω(B, t) A (t) ⊂ hence A (t) A (et) . ⊂ The converse is also true, hence the two families coincide. e 14 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Example 29 In Example 9 with f bounded, both x (t) and any compact invariant set containing x (t) is a compact global attractor.{ −∞ But} only x (t) is backward frequently bounded.{ −∞ } { −∞ }
1.7 A bimodal-climate example
The climate iterpretation of the following example is just immaginary; it is left to educated students to find a more realistic model of climatology that has similar features, if it exists. Consider the equation 3 X0 (t) = f (t) X (t) X (t) − where f (t) is a given function. Let us start considering a few special cases.
Example 30 Assume f (t) = 1. Then all trajectories tend exponentially to zero for t : − → ∞ d 2 2 4 2 X = 2XX0 = 2X 2X 2X dt − − ≤ − 2 2t 2 X (t) e− X (0) ≤ (from Gronwall inequality). The classical compact global attractor, and thus the PBCGA, is equal to 0 . { }
Example 31 Assume f (t) = 1. Then X+ (t) = 1, X0 (t) = 0 and X (t) = 1 are three solutions. With elementary qualitative study one deduces that all trajectories− starting− from x > 0 tend to X+, and all trajectories starting from x < 0 tend to X . "Pointwise" − (a simple rigorous definition can be given) all trajectories tend to X+,X0,X . A simple argument then gives that the global attractor is the full interval [ 1, 1]. Thus− also the PBCGA is [ 1, 1]. − − Example 32 Assume
1 for t [2n, 2n + 1) f (t) = −1 for t ∈[2n + 1, 2n + 2) ∈ for every n Z. The intuition is that over all intervals any bounded set B shrinks at least to [ 1, 1]∈ (in infinite time); on the intervals [2n, 2n + 1) it further shrinks to a small − interval [ x0, x0] around zero; on the intervals [2n + 1, 2n + 2) the set [ x0, x0] enlarges − − again to a set [ x1, x1] close to [ 1, 1]. The precise computation of x0 and x1 is made as − − 3 follows. Call y (t), t [0, 1], the trajectory satisfying y0 = y y starting from x1 and ∈ 3 − − call z (t), t [0, 1], the trajectory satisfying z = z z starting from x0. There is a unique ∈ 0 − 1.7. A BIMODAL-CLIMATE EXAMPLE 15 pair (x0, x1) such that y (1) = x0, z (1) = x1. The final result is that, computed this pair, the NACGA is
[ y (t 2n) , y (t 2n)] for t [2n, 2n + 1) A (t) = . [ z (−t 2n− 1) , z (t − 2n 1)] for t ∈[2n + 1, 2n + 2) − − − − − ∈ We omit the proof.
Consider now the case of external forcing
3 X0 (t) = X (t) X (t) + f (t) . − Without forcing f (t) we know that all trajectories, except for X = 0, are attracted by X+ = 1 and X = 1; the global attractor is [ 1, 1]. Assume f has, sometimes, large fluctuations. To− make− an example, assume −
f (t) = 0 for t [t2n 1, t2n) ∈ − and
f (t) = Fn for t [t2n, t2n+1) ∈ where the values of the sequences (tn) and (Fn) will be chosen. Let us start with heuristic arguments. If a period [t2n, t2n+1) is long enough, we just see a subset I (t) of the form [ 1 , 1 + ]. But then assume a large fluctuation occurs. The set I (t2n+1) is rapidly − − shifted very far from [ 1, 1], where contraction takes place. If Fn and t2n+2 t2n+1 − | | − are large enough, we observe a very small set I (t2n+2). During the subsequent interval [t , t ) this small interval I (t ) converge to a very small interval around Fn . 2n+2 2n+3 2n+2 Fn Unless something strange and special occurs, subsequently we shall only observe very small| | intervals around +1 or -1, depeding on subsequent signs of the fluctuations. The attractor will be one point, in the long run; but moving, changing in time.
Theorem 33 Given T, F > 0, let (t ) be such that t t = T and (F ) be such n n Z n+1 n n n Z n ∈ − ∈ that Fn = ( 1) F . If T and F are suffi ciently large, then A (t) = a (t) for a suitable − { } function a : R R. → We omit the proof. The fenomenon of this example, that we may call synchronization, occurs for many systems. But presumably not for fluid mechanics problems, where we do not expect the attractor to be one point but still describing the complexity of motion (think to an analog of Lorenz example). In such a case the statistical behaviour becomes very important and this requires a notion of invariant measure for non-autonomous dynamical system. 16 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
1.8 Invariant measures: attempts of qualifications to reduce non-uniqueness
For invariant measures it is more diffi cult to remedy the criticalities in the definition of invariance, already outlined above where we have shown, as an example, that the delta Dirac supported by any trajectory is invariant (opposite to the autonomous case). In the properties of attraction we just ask that limit points are included; in the case of measures there is no concept like inclusion. We are forced to require something stronger, maybe much more diffi cult to obtain. The idea, supported by physical meaning, is however always that we are interested in objects, at time t, which come from visible objects at time . Call (X) the set of Borel probability measures on X. −∞ P
Definition 34 We say that a time-dependent probability measure µt, invariant for U (s, t), is the time-development of ρ (X) if ∈ P lim U (s, t) ρ = µt for all t . s ] R →−∞ ∈ We say it is the average-time-development if 1 t lim U (s, t)] ρds = µt for all t R. T + T t T ∈ → ∞ Z − When one or the other of these concepts hold only up to a subsequence, we say that µt is a (average-) time-development up to a subsequence.
In the previous definitions the limits are understood in the weak sense: for every φ ∈ Cb (X) we have
lim φ (U (s, t) x) ρ (dx) = φ (y) µt (dy) s →−∞ ZX ZX in the first case (time-development) and
1 t lim φ (U (s, t) x) ρ (dx) ds = φ (y) µt (dy) T + T t T X X → ∞ Z − Z Z in the second case (average-time-development); we have also used the theorem on trans- formation of integrals on the left-hand-side of these reformulations. By comparison with autonomous dynamical systems, it must be a priori clear that uniqueness is out of question for most examples, without this fact being a pathology. In the autonomous case any system with multiple equilibria has a dirac Delta invariant measure for each equilibrium, plus infinitely many others (convex combnations and maybe others). Similar examples can be done in the non-autonomous case. However, the source of non uniqueness due to the complexity of the dynamics is another fact with respect to the source of non-uniqueness due to the artefact of the definition of 1.9. SEPARATION BETWEEN NON-AUTONOMOUS INPUT AND AUTONOMOUS DYNAMICS: COCYCLES17 invariance. The second source is presumably strongly eliminated by these two concepts of relevance. In a later chapter we shall also discuss the very important concept of physical measure, which relates with the concept of time-development of a measure ρ, but with a further specification about ρ. Time-developments and physical measures are the most important ones, those which really capture essential features of the dynamics.
1.9 Separation between non-autonomous input and autonomous dynamics: cocycles
In all examples of non-autonomous sources described above for climate - perhaps with the exception of certain sub-grid models, but this is not very clear - the non-autonomy is not in the structural aspect of the dynamics itself, but it is due to some external time-varying input. A way to account for this fact is to introduce a separate structure for the input and a sort of autonomous dynamical system for the dynamics. The time-varying input is described either by a single function
ω : R Y → (Y is a set) and all its translations
Ω = θtω; t R { ∈ } where θtω : R Y is defined as →
(θtω)(r) = ω (t + r) or more generally by a set Ω and a group of transformations θt :Ω Ω, →
θ0 = Id
θt+s = θt θs. ◦
The dynamics is described by a family of continuous maps ϕt (ω): X X parametrized by elements ω Ω. → ∈ Definition 35 A dynamical system over the structure Ω, (θ ) is a family of continu- t t R ous maps ∈ ϕt (ω): X X → satisfying the cocycle properties
ϕt+s (ω) = ϕs (θtω) ϕt (ω) ◦ ϕ0 (ω) = Id 18 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS for all values of the parameters. When (Ω, , P) is a probability space, θt are measurable transformations and P is invari- ant under θ (inF such a case we say that Ω, , , (θ ) is a metric dynamical system), t P t t R and ϕ is a strongly measurable map, and the previousF identities∈ are asked to hold for -a.e. P ω, we say that ϕt (ω) is a Random Dynamical System (RDS).
We have been intentionally vague in qualifying whether the identies hold P-a.s. uni- formly in the time parameters or not. The easy case for the develpment of the theory is when they hold uniformly, which is however more diffi cult to check in examples; many results may be extended to the case when the null sets may depend on parameters. We address Chapter 3 for precise results.
Proposition 36 Given a dynamical system ϕ (ω) over the structure Ω, (θ ) , given t t t R ω Ω, the family of maps ∈ ∈ Uω (s, t): X X s t → ≤ defined by
Uω (s, t) = ϕt s (θsω) − is a non-autonomous dynamical system.
Proof. Clearly continuity holds as well as Uω (s, s) = Id. For s r t we have ≤ ≤
Uω (r, t) Uω (s, r) = ϕt r (θrω) ϕr s (θsω) ◦ − ◦ − = ϕt r (θr sθsω) ϕr s (θsω) − − ◦ − = ϕt s (θsω) . −
Let us make two different examples of metric dynamical system Ω, , , (θ ) . In P t t R the next section we write a detailed example of random dynamical systemF ϕ (ω). ∈ t Example 37 On the canonincal two-sided Wiener space defined in Example 5 consider the "shift"
θt (ω) = ω (t + ) ω (t) · − d for every ω C0 R; R . By known properties of Brownian motion it follows (θt)] P = P. Indeed, if (X∈) is a two-sided BM with law , we have t t R P ∈
θt (X )(r) = Xt+r Xt · − which is a new two-sided Brownian motion. 1.10. NON-AUTONOMOUS RDS (NARDS) 19
Example 38 On C R; Rd consider the ordinary shift θt (ω) = ω (t + ) . · Given a two-sided Brownian motion (X ) and two strictly positive numbers λ, σ, define t t R (Y ) as ∈ t t R ∈ t λ(t s) Yt = e− − σdXs Z−∞ which can be also defined by integration by parts, to avoid stochastic integrals. One can prove it is pathwise well defned and it is a stationary Gaussian process, solution of
dYt = λYtdt + σdXt. −
Call P its law; by stationarity, we have (θt) P = P for every t R. ] ∈ It is a simple exercise that we leave to the reader to formulate invariance and attraction in this new language. The theory in the case of RDS will be explicitly developed in Chapter 3.
1.10 Non-autonomous RDS (naRDS)
A RDS as defined above is a non-autonomous system. However, we shall call it an au- tonomous RDS. The attribute of autonomous refers to the fact that ϕt is time-independent. For applications to climatology the most interesting scheme would be to put into ω the fast varying sources of time-variation, stationary or with stationary increments, but con- sidering also slowly varying ones, which may be periodic (annual solar radiation) or have a trend (CO2). Thus we need a concept of non-autonomous Random Dynamical Systems (naRDS). The generic equation we have in mind has the form
∂tu = Au + B (u) + C (u, q (t)) + D (u) ω· (t) where q is the slowly varying term and ω· is white noise.
Definition 39 A non-autonomous Random Dynamical System (naRDS) is made of two input structures and a cocycle over their product. Precisely, the two input structures are
, (ϑ ) , Ω, , , (θ ) t t R P t t R Q ∈ F ∈ where , Ω are sets, (ϑ ) is a group of transformations of , Ω, , , (θ ) is a t t R P t t R metricQ dynamical system, to∈ which we associate the product spaceQ F ∈ Γ = Ω γ = (q, ω) Q × ∈ Y 20 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS and the group of transformations
Θt :Γ ΓΘt (q, ω) = (ϑtq, θtω) . → The cocycle over Γ, (Θ ) is a family of maps on a metric space (X, d) t t R ∈
ϕt (γ): X X → such that
ϕ0 = Id
ϕt+s (γ) = ϕs (Θtγ) ϕt (γ) ◦ for all values of the parameters.
In the finite dimensional case, this is a quite generic structure. Using the theory of stochasti flows, it is not necessary to restrict to special noise. Let us however illustrate by a simple example with additive noise the definition, to put hands in a concrete example. Let b : Rm Rd Rd continuous and satisfying × → m d b (a, x) b (a, y) L x y for all a R and x, y R . | − | ≤ | − | ∈ ∈ Let q C (R; Rm). Consider the non-autonomous stochastic equation ∈
dXt = b (q (t) ,X (t)) dt + dWt
d where Wt is a Brownian motion in R . This equation can be easily solved by usual stochastic methods. For the purpose of introducing the associated naRDS we interpret this equation pathwise: for every ω C R; Rd we consider the integral equation ∈ t x (t) = x0 + b (q (s) , x (s)) ds + ω (t) . Z0 By classical contraction principle one can easily prove it has a unique solution
q,ω,x d x 0 C R; R . ∈ Then we set m = C (R; R ) ϑtq = q (t + ) Q · d Ω = C R; R θtω = ω (t + ) ω (t) · − = Borel σ field of Ω, P = two-sided Wiener measure; F q,ω,x0 ϕt (q, ω)(x0) = x (t) . 1.10. NON-AUTONOMOUS RDS (NARDS) 21
In other words, withe the notation γ = (q, ω),
t ϕt (γ)(x0) = x0 + b (q (s) , ϕs (γ)(x0)) ds + ω (t) . Z0 In order to say that this example satisfies the abstract properties of a naRDS we have to check a few conditions. Group properties of shifts are obvious; property (θt)] P = P was already discussed above. We have to prove the cocycle property
ϕt+s (γ) = ϕs (Θtγ) ϕt (γ) . ◦ We have
t+s ϕt+s (γ)(x0) = x0 + b (q (r) , ϕr (γ)(x0)) dr + ω (t + s) 0 Z t = x0 + b (q (r) , ϕr (γ)(x0)) dr + ω (t) 0 t+s Z + b (q (r) , ϕr (γ)(x0)) dr + ω (t + s) ω (t) − Zt
t+s = ϕt (γ) + b (q (r) , ϕr (γ)(x0)) dr + (θtω)(s) t Z s r=t+u = ϕt (γ) + b ((ϑtq)(u) , ϕt+u (γ)(x0)) du + (θtω)(s) Z0 hence we see that z (s) := ϕt+s (γ)(x0) s 0 ≥ satisfies s z (s) = z0 + b (q∗ (u) , z (u)) du + ω∗ (s) Z0 where z0 = ϕt (γ), q∗ (u) = (ϑtq)(u), ω∗ (s) = (θtω)(s). By uniqueness of solutions to this equation, z (s) = ϕs (q∗, ω∗)(z0) . Collecting all identities and definitions,
ϕt+s (γ)(x0) = ϕs (q∗, ω∗)(z0)
= ϕs (ϑtq, θtω)(ϕt (γ)) which means precisely ϕt+s (γ) = ϕs (Θtγ) ϕt (γ). ◦ 22 CHAPTER 1. NON-AUTONOMOUS DYNAMICAL SYSTEMS
Remark 40 At the structural level (group and cocycle properties) there is nothing special in the additive noise. What is special is the possibility to solve, uniquely, the equation for all elements ω C R; Rd of the Wiener space, namely the pathwise solvability. Similar tricks hold only∈ for few equations, like
dXt = b (q (t) ,X (t)) dt + XtdWt.
However, in general there are two possibilities. One is the theory of rough paths, which at the price of some additional regularity of coeffi cients and a certain high level background in stochastic analysis, enables to solve "any" stochastic equation pathwise. One has to replace the Wiener space C R; Rd by the more structured space of (geometric) rough paths. This non trivial theory has the advantage to be fully pathwise and thus entirely analog to the previous additive noise example. The other possibility is to use the theory of stochastic x0 flows. That theory claims that, given the stochastic-process solutions Xt (ω), which at time t are equivalence classes (hence not pointwise uniquely defined in ω), there is a version of the family such that we can talk of the space-time trajectories
x0 (t.x0) X (ω) → t and, even more, we can do the same starting from an arbitrary time t0. Elaborating this concept with non trivial further elements, one can construct a naRDS (the technical diffi - culty is only in the stochastic part, not the non-autonomous one). Chapter 2
Generalities on RDS
2.1 Random dynamical systems
The theory of random dynamical systems develops with the introduction of probability instruments into classical deterministic models, to help representing uncertainties or sum- ming up complexity. After the first work by Ulam and Neumann ([?]) in 1945 showing this need for intersection of probability theory and dynamical systems, the interest for the sub- ject rises around the Eighties with the development of stochastic analysis and the discover of random and stochastic differential equations as sources for random transformations. In this chapter we introduce the definition of random dynamical system (also abbre- viated as RDS), and show how RDSs can be generated in continuous and discrete time. Then we present a finite dimensional model which exemplifies many properties of fluido- dynamical systems, and we outline the definition of the RDS associated to the stochastic Navier-Stokes equations. We end by giving the statement of the multiplicative ergodic theorem, which allows the definition of Lyapunov exponents for an RDS. In all chapters we will write T for the set of times, referring to the groups T = R, Z (two-sided time), or to the semigroups T = R+, N (one-sided time).
2.2 Definitions
The theory of random dynamical systems enlarges the classical theory of dynamical sys- tems, investigating those situations in which a noise is supposed to perturb the system.
The noise is formalized with a family of mappings (θ(t))t T of a measurable space (Ω, ) into itself, satisfying the following conditions: ∈ F
(i) the map (ω, t) θ(t)ω is measurable; 7→
(ii) θ(0) = idΩ, if 0 T; ∈ 23 24 CHAPTER 2. GENERALITIES ON RDS
(iii) θ(s + t) = θ(s) θ(t), for all s, t T (semiflow property). ◦ ∈
In this case the triple (Ω, , (θ(t))t T) is called a measurable dynamical system. In F ∈ addition, if P is a probability measure on (Ω, ) and if, for all t in T, the map θ(t) preserves 1 F the measure P (i. e. P (θ(t)− (F )) = P (F ) for all F ), then (Ω, ,P, (θ(t))t T) is said to be a metric dynamical system. ∈ F F ∈ Let us observe that condition (i) implies that, for each t T, the map θ(t):Ω Ω ∈ 7→ is measurable. Moreover, if T is a group, from conditions (ii) and (iii) we gain that each θ(t) is invertible, and θ(t) 1 = θ( t). − − In the sequel we will also write θt for θ(t). Let (Ω, ,P ) be a complete probability space, i.e. a probability space such that if F , P (FF) = 0 and E F , then E . ∈ F ⊂ ∈ F t Definition 41 A two-parameter filtration is a two-parameter family s s,t R, of sub-σ- {F }s ∈t algebra of , such that ≤ F (i) v t, for s u v t; Fu ⊂ Fs ≤ ≤ ≤ t+ u t t t t (ii) s := u>t s = s and s := u
Definition 42 A random dynamical system on a measurable space (X, ), over a metric B dynamical system (Ω, ,P, (θ(t))t T) is a measurable mapping F ∈ ϕ :(T Ω X, (T) ) (X, ), × × B ⊗ F ⊗ B → B satisfying the cocycle property:
ϕ(0, ω, ) = idX for all ω Ω(if 0 T), (2.2) · ∈ ∈ ϕ(t + s, ω, ) = ϕ(t, θ(s)ω, ϕ(s, ω, )) for all s, t T, ω Ω. (2.3) · · ∈ ∈ 2.2. DEFINITIONS 25
One can say that while a point x X is moved by the map ϕ(s, ω, ), ω is also shifted for a time s by θ, and θ(s)ω is the new∈ noise acting on ϕ(s, ω, x). · The map ϕ of the definition is also called a perfect cocycle, while, if condition (2.3) is weakened to hold for P -almost every ω, for fixed s and all t in T, then ϕ is said to be a crude cocycle, and a very crude cocycle in case (2.3) holds for each fixed s and t. We shall use the notation ϕ(t, ω) for the map ϕ(t, ω, ): X X. · →
Remark 43 Given a metric dynamical system (Ω, ,P, (θ(t))t T), with T = R or Z, we will sometimes consider random dynamical systemsF which are∈ defined for positive times only, being all theory extendible to this case.
Remark 44 If T is a group, by taking t = s in (2.3) and using (2.2), we find − idX = ϕ( s, θ(s)ω) ϕ(s, ω) for all ω, s, − ◦ while substituting t for s and θ(t)ω for ω in (2.3) one gains − idX = ϕ(t, ω) ϕ( t, θ(t)ω) for all ω, t. ◦ − This proves that the map ϕ(t, ω) is invertible and
1 ϕ(t, ω)− = ϕ( t, θ(t)ω) for all ω, t. − If X is a topological space and, for each ω Ω, the mapping (t, x) ϕ(t, ω, x) is continuous, then the random dynamical system is∈ called continuous or topological7→ . If X is a finite dimensional manifold, then ϕ is a Ck random dynamical system if it is continuous and for each (t, ω) T Ω the map x ϕ(t, ω)x is k-times differentiable with respect to x, and the derivatives∈ are× continuous in7→t and x.
Remark 45 Given a probability space (Ω, ,P ), and a P -preserving bijective map θ : Ω Ω, any family of continuous mappingsFψ(ω): X X such that (ω, x) ψ(ω)x is → → 7→ measurable generates a discrete random dynamical system ϕ : N Ω X X, by the formulas × × → n 1 ϕ(0, ω) = idX , ϕ(n, ω) = ψ(θ − ω) ψ(ω) for n 1. ◦ · · · ◦ ≥ In this case, the cocycle property can be rewritten as
ϕ(n + 1, ω)x = ϕ(n, θω)ψ(ω)x.
1 If ψ(ω)x is invertible for all ω, and the map (ω, x) ψ(ω)− x is measurable, then the random dynamical system can be estended to negative7→ times by
n 1 1 1 ϕ(n, ω) = ψ(θ ω)− ψ(θ− ω)− for n 1. ◦ · · · ◦ ≤ − Conversely, each random dynamical system ϕ over (Ω, , P, θ) with dicrete time T = N or F T = Z is fully described by its time-one map ψ(ω) = ϕ(1, ω): X X. → 26 CHAPTER 2. GENERALITIES ON RDS
Suppose we are given a random dynamical system ϕ on a Polish space (X, ), over B a metric dynamical system (Ω, ,P, (θ(t))t T). One can consider another measurable dy- namical system, associated to theF random dynamical∈ system ϕ:
Definition 46 The measurable map
Θ: T Ω X Ω X ×(t, ω,× x) → (θ(×t)ω, ϕ(t, ω, x)), 7→ is called the skew product (flow) of the metric dynamical system θ and the cocycle ϕ.
The family of mappings Θt = Θ(t, , ), t T, is a measurable dynamical system on (Ω X, ). The semiflow property· · is an∈ easy consequence of the flow and cocycle properties× F of ⊗θ Band ϕ:
Θ0 = idΩ X , × Θt+s(ω, x) = (θ(t + s)ω, ϕ(t + s, ω)x)
= (θ(t) θ(s)ω, ϕ(t, θ(s)ω)ϕ(s, ω)x) = Θt Θs(ω, x), ◦ ◦ for all t, s T, ω ω and x X. ∈ ∈ ∈ Deterministic dynamical systems are generated by ordinary differential equations; their role is played by stochastic differential equations in the case of random dynamical systems. An intermediate step is provided by considering random differential equations, i.e. families of ordinary differential equations with parameter ω, which can be solved “pathwise” for each fixed ω as a deterministic equation.
Example 47 (RDS from Random Differential Equations) n Let T = R, X = R , and let (Ω, ,P, (θ(t))t T) be a metric dynamical system. Consider F ∈ a measurable function f :Ω Rn Rn, and, for each fixed ω Ω, let the function n n × → ∈ fω : R R R be given by fω(t, x) = f(θtω, x). Suppose that fω is continuous, and locally Lipschitz× → in x. Then the solution of equation
x˙ t = f(θtω, xt) (2.4) exists globally on R for each ω Ω and is unique. The random dynamical system associated to the random differential equation∈ (2.4) is given by
t ϕ(t, ω, x) = x + f(θsω, ϕ(s, ω, x))ds. Z0 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 27
To prove the cocycle property, take s, t R, and suppose that s > 0, t > 0 (the other cases are analogous). Write ∈
t ϕ(t, θ(s)ω, ϕ(s, ω, x)) = ϕ(s, ω, x) + f(θu+sω, ϕ(u, θ(s)ω, ϕ(s, ω, x)))du 0 s Z = x + f(θuω, ϕ(u, ω, x))du 0 t+sZ + f(θzω, ϕ(z s, θsω, ϕ(s, ω, x)))dz. − Zs This proves that the function
ϕ(u, ω, x) 0 u s, ψ(u, ω, x) = ≤ ≤ ϕ(u s, θsω, ϕ(s, ω, x)) s u s + t, − ≤ ≤ satisfies t+s ψ(t + s, ω, x) = x + f(θuω, ψ(u, ω, x))du, Z0 and so by uniqueness
ϕ(t + s, ω, x) = ψ(t + s, ω, x) = ϕ(t, θ(s)ω, ϕ(s, ω, x)).
Definition 48 A mapping
φ : R R Rd Ω Rd × × × −→ (s, t, x, ω) φs,t(x, ω) 7→ is called a stochastic flow if for P -almost every ω Ω it satisfies: ∈
(i) φs,s(ω) = Id d for all s R; R ∈ (ii) φs,t(ω) = φu,t(ω) φs,u(ω) for all s, t, u R. ◦ ∈ d d If additionally φs,t(ω): R R is a homeomorphism (resp. is a k-times continuously → differentiable homeomorphism) for all s, t R and P -almost every ω, then φ is called a stochastic flow of homeomorphism (resp. of∈Ck-diffeomorphisms).
2.3 RDS from Stochastic Differential Equations
An important class of random dynamical systems is generated by stochastic differential equations. A complex general result (see [3], chapter 2.3) asserts the existence of RDS associated to any stochastic differential equation involving stochastic integrals of Kunita- type, i.e. integrals which generalize classical Itô and Stratonovich integrals, introduced 28 CHAPTER 2. GENERALITIES ON RDS by Kunita in [?]. Here we state a particular case of this theorem involving classical Itô integrals only. n n Consider the space Ω = 0(R, R ) of continuous functions ω from R to R , such that C ω(0) = 0. One can endow the space (R, Rn) with the topology given by the complete metric C ∞ 1 ω ω0 k d(ω, ω0) = k || − || , ω ω0 k = sup ω(t) ω0(t) , 2 1 + ω ω k || − || | − | k=1 0 k t k X || − || − ≤ ≤ and consider the induced topology on Ω. Denote by 0 the Borel σ-algebra on Ω. F Let (ξ(t))t R be the coordinate process on Ω, i.e. the process given by ξ(t, ω) = ω(t) ∈ for all ω Ω, t R. ∈ ∈ n n A continuous R -valued process (Bt)t 0 is a Brownian motion on R if B(0) = 0, for ≥ all 0 s t the variable Bt Bs is normally distributed with mean 0 and covariance ≤ ≤ − operator (t s)In, and for all m-uples 0 t1 ... tm the random variables − ≤ ≤ ≤
Bt1 ,Bt2 Bt1 ,...,Btm Btm 1 − − − are independent ((Bt)t 0 has independent increments). A natural extension of Brownian motion to negative times≥ is given by the following definition.
n Definition 49 A two-sided Brownian motion (Wt)t R in R is a continuous process with ∈ independent increments, such that W0 = 0 and, for each s, t R, the increment Wt Ws ∈ − is normally distributed with mean 0 and covariance matrix t s In. | − |
If (Wt)t R is a two-sided Brownian motion, we have in particular that the processes ∈ n (Wt)t 0 and (W t)t 0 are two independent Brownian motions on R . ≥ − ≥ The measure P on (Ω, ) such that the coordinate process (ξ(t))t R is a two-sided F ∈ Brownian motion is called (two-sided) Wiener measure. The process (ξ(t))t R is called standard two-sided Brownian motion. We denote by the completion of the σ∈-algebra 0 with respect to P . F F Define then the shift operators on Ω as
θt(ω)(s) = ω(s + t) ω(t), for s, t R. − ∈
The family (θt)t R clearly satisfies the semiflow property. The map (t, ω) θt(ω) is continuous, and then∈ measurable. 7→ The measure P is invariant with respect to the dynamical system (θt)t, as follows from the stationarity of the increments of the coordinate process: the law of the process
(ξ(t + s) ξ(s))t R equals the law of (ξ(t))t R, and then − ∈ ∈ 0 P (A) = P ω ω A = P ωt+ ωt A = P θt A for all A . { | · ∈ } { · − ∈ } { ∈ } ∈ F Let be the family of all subsets of P -null sets of 0: N F = A Ω: F 0 s.t. A F,P (F ) = 0 . N { ⊂ ∃ ∈ F ⊂ } 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 29
t Define a two-parameter filtration s s t as {F } ≤ t = σ Wu Wv : s v u t , for all s t. Fs { − ≤ ≤ ≤ } N ≤ _ Proposition 50 For each s t R the following identities hold: ≤ ∈ t+ t t u s = s = s− := σ( s u Proof. Consider the processes W Ws, for fixed s, and Wt W , for fixed t. The equalities t+ = t, t = t follow from· the− fact that for a d-dimensional− · strong Markov process Fs Fs Fs Fs+ with natural filtration ( t)t, the augmented filtration ( tN )t = ( t )t is right-continuous ([32], proposition 2.7.7).F The fact that the natural filtrationF ofF a∨Nd-dimensional adapted t t left-continuous process is left-continuous ([32], problem 2.7.1) implies that s− = s, and t t F F s = s. F − F For the last sentence, take u1, ..., un, v1, ..., vn R such that s vi ui t. Then for n ∈ ≤ ≤ ≤ each a R, B1, ..., Bn (R ), one has ∈ ∈ B 1 θ− ω Ω ω(u1) ω(v1) B1, ..., ω(un) ω(vn) Bn a { ∈ | − ∈ − ∈ } 1 = θ− ω ω(u1) ω(v1) B1, ..., ω(un) ω(vn) Bn { a | − ∈ − ∈ } = ω θaω(u1) θaω(v1) B1, ..., θaω(un) θaω(vn) Bn { | − ∈ − ∈ } t+a = ω ω(u1 + a) ω(v1 + a) B1, ..., ω(un + a) ω(vn + a) Bn . { | − ∈ − ∈ } ∈ Fs+a It is then clear that (2.1) holds. 0 t Hence (Ω, , s s t, (θ(t))t R,P ) is a filtered metric dynamical system. It is referred to as the canonicalF {F metric} ≤ dynamical∈ system describing Brownian motion. 0 Proposition 51 The dynamical system (Ω, , (θ(t))t R,P ) is ergodic. F ∈ Proof. It is a consequence of Kolmogorov’s zero-one law. Consider the two-parameter t filtration s s t defined above, and define the σ-algebra {F } ≤ ∞ = t R t∞. T ∩ ∈ F u z The independence of the σ-algebras s and t for all s < u t < z allows to apply Kolmogorov’s dichotomy and deduceF that theFσ-algebra is≤ degenerate, i.e. P (A) T ∞ ∈ 0, 1 for all A ∞. The conclusion follows by observing that every θ(t)-invariant set is contained{ } in ∈. T T ∞ 30 CHAPTER 2. GENERALITIES ON RDS To give the statement of the theorem, we have to introduce some spaces of functions on Rd. For each k Z+, and 0 δ 1, we denote by k,δ the space of functions f : Rd Rd which are k times∈ continuously≤ differentiable≤ and whoseC k-th derivative is locally δ-Hölder→ continuous. The space k,δ is a Fréchet space with the following seminorms: for each C compact convex subset K of Rd α f k,0;K = sup D f(x) , || || x K | | 0 α k ∈ ≤|X|≤ Dαf(x) Dαf(y) f k,δ;K = f k,0;K + sup | − δ |, 0 < δ 1. || || || || x,y K,x=y x y ≤ α =k ∈ 6 |X| | − | We then define the set k,δ as the space of functions in k,δ for which the norm Cb C f(x) α f k,0 = sup | | + sup D f(x) , || || d 1 + x d | | x R 1 α k x R ∈ | | ≤|X|≤ ∈ Dαf(x) Dαf(y) f k,δ = f k,0 + sup | − δ |, 0 < δ 1 || || || || x=y x y ≤ α =k 6 |X| | − | is finite. We briefly recall the definition of Itô and backward Itô integral. Consider a continuous u 2 process (Xu)u s, which is adapted to ( s )u s and such that supu [s,t] E[ Xu ] < for ≥ F ≥ ∈ | | ∞ a time t > s. For each partition σ = s = t0 ... tn = t of [s, t], consider the sum n 1 { ≤ ≤ } Iσ = k=0− Xtk (Wtk+1 Wtk ). The limit in probability of Iσ, as σ = supk tk tk 1 0, − t | | | − − | → exists and is called the Itô integral s XudWu of (Xu)u. For the definition of backward P 2 Itô integral, consider a continuous process (Xu)u s which satisfies supu [s,t] E[ Xu ] < R ≥ t ∈ | | ∞t for some t > s and such that the variable Xu is u-measurable (observe that the set u n 1F F decreases as u increases). Then the limit of k=0− Xtk+1 (Wtk+1 Wtk ) exists in probability − t as σ 0, and is called backward Itô integral of (Xu)u. It will be denoted by XudWu. | | → P s Recall that, if f , ..., f are Borel measurable maps from d into itself, then a solution 0 m R H of the stochastic differential equation m j d dXt = f0(Xt)dt + fj(Xt)dWt ,Xs = x R , (2.5) ∈ j=1 X d is a continuous R -valued adapted process (Xt)t s, such that P Xs = x = 1 and for each t > s the equality ≥ { } t m t j Xt = x + f0(Xu)du + fj(Xu)dWu s j=1 s Z X Z 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 31 holds almost surely. One says that the solution to (2.5) is unique if, given any two solutions (Xt)t s, (Yt)t s, then P Xt = Yt, t s = 1. ≥ ≥ { ≥ } For the following result on existence and uniqueness of solutions of stochastic differential equations see for example [?] (theorems 3.4.1, 4.7.3 and 4.7.4): m k,δ Theorem 52 Consider a standard Brownian motion (Wt)t R in R . Let f0 b , fj ∈ ∈ C ∈ k+1,δ, j = 0, ..., m, for some k 1 and some δ > 0. Suppose that the map Cb ≥ m d ∂fj f ∗ = f0 (fj)i 0 − ∂x j=1 i=1 i X X k,δ is in b . Then there exists a unique solution (φs,t(x))t s to the stochastic differential C ≥ equation (2.5) for any s R, x Rd. Moreover, the solution can be extended to a ∈ ∈ stochastic flow of diffeomorphisms (φs,t(ω, x))s,t R, such that for each s R the equality ∈ ∈ s m s j φs,t(x) = x f ∗(φs,u(x))du fj(φs,u(x))dW − 0 − u t j=1 t Z X I is satisfied for each x Rd and every t s. ∈ ≤ The theorem on generation of random dynamical systems from Itô stochastic differential equations is the following: m k,δ Theorem 53 Consider a standard Brownian motion (Wt)t R in R , and let f0 b , ∈ m d ∂fj k,δ k+1,δ ∈ C f ∗ = f0 (fj)i , fj , j = 0, ..., m, for some k 1 and some 0 − j=1 i=1 ∂xi ∈ Cb ∈ Cb ≥ δ > 0. Let (φs,t(ω, x))s,t R be the stochastic flow of diffeomorphisms associated to equation (2.5) accordingP toP theorem∈ (52). Then the map ϕ : R Rd Ω Rd × × −→ (t, x, ω) ϕ(t, ω, x) = φ0,t(x, ω). 7→ is a Ck random dynamical system over the filtered dynamical system describing Brownian motion. Moreover, ϕ has stationary independent increments, i.e. for all t0 t1 ... tn the random variables ≤ ≤ ≤ 1 1 1 ϕ(t1) ϕ(t0)− , ϕ(t2) ϕ(t1)− , . . . , ϕ(tn) ϕ(tn 1)− ◦ ◦ ◦ − are independent, and the law of ϕ(t + h) ϕ(t) 1 is independent of t, for each t, h. ◦ − 32 CHAPTER 2. GENERALITIES ON RDS 2.3.1 Additive noise We shall prove theorem (53) in a particular case, when the functions f1, ..., fm are constant. One speaks in this case of additive noise. The stochastic equation we consider is then of the form t Xt = x + b(Xu)du + Wt Ws, (2.6) − Zs with b belonging to k,δ for some k 1, δ > 0. We consider the stochastic flow of Cb ≥ diffeomorphisms (φs,t(ω, x))s,t R associated to this equation according to theorem (52), ∈ d and define ϕ(t, ω)x = φ0,t(x, ω) for all t R, x R and ω Ω. Note that in this simple case, the existence and uniqueness of solutions∈ of∈ equation (2.6)∈ can be proved also through a classical Cauchy theorem, for each fixed ω Ω. ∈ We show that ϕ satisfies the crude cocycle property: for each s R there exists a ∈ P -null set Ns such that ϕ(t + s, ω) = ϕ(t, θsω) ϕ(s, ω) for all t R, ω / Ns. (2.7) ◦ ∈ ∈ Observe first that (2.7) holds if and only if φ satisfies φs,s+t(ω) = φ0,t(θsω) for all t R, ω / Ns, (2.8) ∈ ∈ thanks to flow property of the map φ. Indeed the cocycle property for ϕ rewrites as φ0,s+t(ω) = φ0,t(θsω) φ0,s(ω). ◦ Compose both sides with φs,0(ω), and find φ0,s+t(ω) φs,0(ω) = φs,s+t(ω) = φ0,t(θsω), as ◦ wanted. Conversely, if φs,s+t(ω) = φ0,t(θsω), then ϕ(t + s, ω) = φ0,s+t(ω) = φs,s+t(ω) φ0,s(ω) = φ0,t(θsω) φ0,s(ω), ◦ ◦ and (2.7) holds. Let us prove (2.8). Consider the case t > 0. For each x Rd, s R, we have ∈ ∈ s+t φs,s+t(x) = x + b(φs,u(x))du + Ws+t Ws s − Z t = x + b(φs,s+u(x))du + Ws+t Ws − Z0 almost surely. Moreover, for almost every ω we have t φ0,t(θsω, x) = x + b(φ0,u(θsω, x))du + (θsω)(t) (θsω)(0) 0 − Z t = x + b(φ0,u(θsω, x))du + Ws+t(ω) Ws(ω). − Z0 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 33 Then the uniqueness of the solution for equation (2.6), and the continuity on x, imply the existence of a P -null set Ns depending on s such that (2.8) holds. The case t < 0 is analogous. Moreover, there exists a perfect cocycle which is indistinguishable from ϕ, according to a general result (theorem 1.3.2 of [3]). The system (Ω, , (θt)t R, P, ϕ) is called the random dynamical system associated to equation (2.6). F ∈ 1 1 t Observe that ϕ(t) ϕ(s) equals φ0,t φ− = φ0,t φs,0 = φs,t, which is a -measurable ◦ − ◦ 0,s ◦ Fs random variable. Moreover, it follows by (2.8) that φt,t+h = φ0,h θt almost surely for each 1 ◦ t, h, and then the law of ϕ(t + h) ϕ(t)− does not depend on t, and ϕ is an RDS with stationary independent increments.◦ Example 54 (One-dimensional Ornstein-Uhlenbeck process) Consider the equation on R dXt = αXtdt + σdWt,Xt0 = x0 R. (2.9) − ∈ Its solution, given by the Ornstein-Uhlenbeck process (see appendix 6.3) t α(t t0) α(t s) Xt = e− − x0 + e− − σdWs, Zt0 can be rewritten, through an integration by parts formula (proposition 3.12 in [?]), for every ω Ω, as ∈ t α(t t0) α(t s) Xt(ω) = e− − x0 + σω(t) αe− − σω(s)ds. − Zt0 Therefore the random dynamical system associated to (2.9) is t αt α(t s) ϕ(t, ω)x = e− x + σω(t) αe− − σω(s)ds. − Z0 2.3.2 A finite dimensional model We consider a finite dimensional model which covers many important examples, for instance the finite dimensional approximation of the Navier-Stokes equations (see paragraph 2.5). We consider the stochastic equation in Rn dXt + (AXt + B(Xt,Xt))dt = fdt + QdWt, t t0, (2.10) ≥ p where A is an n n symmetric matrix satisfyng Ax, x λ x 2 for each x Rn, B is a bilinear continuous× mapping with the property h i ≥ | | ∈ n B(y, x), x = 0 x, y R , (2.11) h i ∀ ∈ 34 CHAPTER 2. GENERALITIES ON RDS Q is a positive semidefinite symmetric matrix, and Wt is a two-sided Brownian motion over t the canonical metric dynamical system (Ω, , s s t, (θ(t))t R,P ). We shall consider t0 - random variables as initial conditions. TheF equation{F } ≤ in integral∈ form is then F t Xt = η (AXs + B(Xs,Xs) f)ds + QWt QWt0 . (2.12) − t0 − − Z p p To define the random dynamical system associated to (2.10), one could solve, for each fixed ω, a system of ordinary differential equation. We choose instead a probabilistic approach for the following existence and uniqueness result: Proposition 55 For every t -measurable random variable η, equation (2.12) admits a F 0 unique solution (Xt)t, which is continuous and adapted to ( t)t. F 1 1 Proof. For uniqueness, suppose (Xt )t, (Xt )t are two solutions of equation (2.10), and let 1 2 CB be a constant such that B(x, y) CB x y . Then the difference Vt = X X satisfies ≤ | || | t − t dVt 1 2 = AVt B(X ,Vt) B(Vt,X ). dt − − t − t Taking the scalar product with Vt, and using property (2.11) we find dVt 2 ,Vt = AVt,Vt B(Vt,X ),Vt , h dt i −h i − h t i and consequently, by the inequality on A we have 2 d Vt 2 2 | | CB Vt X . dt ≤ | | | t | The Gronwall lemma gives then t 2 CB X ds 2 2 t0 s Vt Vt0 e | | . (2.13) | | ≤ | | R Uniqueness of the solution follows. Existence of a solution can be proved by a truncation procedure as follows. Observe first that the initial condition η can be supposed to be bounded. This can be 2 seen by defining for each m N the set Am = ω Ω: η(ω) n , and a random variable ∈ { C∈ | | ≤ } ηm as equal to η on Am, and equal to 0 on Am, and considering for each m the unique m solution (Yt )t 0 of equation (2.10) with initial condition ηm. The process Y ∞ defined as m ≥ Yt∞ = Yt on Am is then the unique solution of equation (2.12). Fix a T > t0, and a constant C such that η C. For each m C, consider a Lipschitz | | ≤ ≥ n continuous function Bm on such that Bm(x) = B(x, x) for every x R with x m. Then the equation ∈ | | ≤ dXt = ( AXt Bm(Xt,Xt) + f)dt + QdWt, − − p 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 35 m with initial condition η, has a unique adapted solution (Xt )t, as a consequence of theorem (52). For each m > C, define the stopping time m τm = inf t0 t T : X = m T. { ≤ ≤ | t | } ∧ m On the stochastic interval [t0, τm], Xt is a solution of the original equation, as t τm m ∧ m m Xt τm = η + ( AXs Bm(Xs ) + f)ds + QWt τm QWt0 (2.14) ∧ − − ∧ − Z0 t τm p p ∧ m m m = η + ( AXs B(Xs ,Xs ) + f)ds + QWt τm QWt0 . 0 − − ∧ − Z p p k m Moreover, if k is greater than m, then the processes Xt and Xt are equal on [t0, τm] with m probability 1. A process X∞ is then defined on [t0, supm>C τm] by X∞ = Xt on [t0, τm], for every m > C. To estimate supm>C τm, apply Itô formula to (2.14), use the fact that B(y, x), x = 0, and find h i t τm t τm m 2 ∧ m m 2 ∧ m Xt τm + 2 AXs ,Xs ds = η + 2 Xs , f ds | ∧ | h i | | h i Z0 Z0 t τm ∧ m + 2 Xs , QdWs + TrQ(t τm). 0 h i ∧ Z p The coercivity of A and the inequality Xm, f Xm f imply that h s i ≤ | s || | t τm m 2 2 ∧ m Xt τm η + 2m(t τm) f + 2 Xs , QdWs + TrQ(t τm). | ∧ | ≤ | | ∧ | | 0 h i ∧ Z p Then m 2 2 E sup Xt τm E[ η ] + 2mT f + T TrQ, t [t0,T ] | ∧ | ≤ | | | | " ∈ # and in particular m 2 E I τm Example 56 A simplified GOY model (Gledzer, Ohkitani, Yamada model) A system which is included in the described setting is the following. Consider, for each N N, by the finite dimensional stochastic system ∈ u 1 = u0 = uN = uN+1 = 0, − 2 1 1 dun + νknundt =kn un 1un+1 un+1un+2 + un 1un 2 dt 4 − − 8 − − n + σndWt for n = 1, ..., N, n n where kn = 2 , ν > 0, and (Wt )t R are independent two-sided Brownian motions, for 0 n N. ∈ ≤It is≤ a stochastic version of a simplification of the GOY model, a system belonging to the class of shell models, which are designed to capture many aspects of fluid dynamics, and are structured as simplified versions of the Navier-Stokes equations, by considering some variants of their Fourier approssimation. Observe that the velocity on each single cell depends directly only by velocities in the 2 nearest and next-nearest neighbour shells, and that the energy ( un ) is preserved when n | | ν = σn = 0 (namely without viscosity and external force). P 2.3.3 n-point motion It is sometimes useful, for the study of the evolution of a dynamical system, to consider the analysis of n copies of the system, starting at different points, for example when focusing on sensitiveness of the system to initial conditions. In the stochastic settings described so far, we are interested in the study of the evolution of a random dynamical system for each fixed realization ω of the noise. Denote by ϕ the random dynamical system generated by d d n the SDE (2.5) or (2.10). For each n-uple x1, ..., xn R , we call the (R ) -valued process ∈ (t, ω) (ϕ(t, ω)x1, ..., ϕ(t, ω)xn) (2.15) 7→ the n-point motion associated to the considered equation. For example, if ϕ is the random dynamical system associated to the stochastic equation (2.6), then the n-point motion rises from the system of stochastic equations t Xt = x1 + s b(Xu)du + Wt Ws t − Xt = x2 + b(Xu)du + Wt Ws Rs − ... R t Xt = xn + b(Xu)du + Wt Ws; s − in particular, the same noise acts on allR components. The n-point motion (2.15) is a Markov process (for the definition see appendix 6.3). For the proof we will need the following lemma: 2.3. RDS FROM STOCHASTIC DIFFERENTIAL EQUATIONS 37 Lemma 57 Let (Ω, ,P ) be a probability space, and , be two independent sub-σ-fields of . Let X be a -measurableA random variable withF valuesG on a measurable space (E, ), A G E and let ψ : E Ω R be measurable with respect to , and such that the map ω ψ(X(ω), ω×) is integrable.→ Define Ψ(x) := E[ψ(x, )]. ThenE ⊗ F 7→ · E[ψ(X, ) ] = Ψ X. · |G ◦ Proof. Take ψ of the form ψ(x, ω) = h(x)f(ω), where f is -measurable. Then F E[ψ(X, ) ] = E[h(X)f( ) ] = h(X)E[f] = Ψ X. · |G · |G ◦ Conclusion follows by a classical monotone class argument. Proposition 58 The n-point motion associated to equations (2.5) or (2.10) is a Markov t process with respect to the filtration ( 0)t 0, with transition probabilities given by F ≥ (n) P ((x1, ...xn),B) = P ω Ω:(ϕ(t, ω)x1, ..., ϕ(t, ω)xn) B , t { ∈ ∈ } for t R+, B (Rnd). ∈ ∈ B Proof. Take B (Rnd), and define ψ : Rnd Ω R by ∈ B × → ψ(x, ω) = IB((ϕ(t, θsω)x1, ..., ϕ(t, θsω)xn)). Then s s E[IB(ϕ(s + t, ω)x1, ..., ϕ(s + t, ω)xn) ] = E[ψ((ϕ(s, )x1, ..., ϕ(s, )xn), ) ]; |F0 · · · |F0 s nd s+t s+t s moreover ϕ(s, )x is 0 -measurable, and ψ is (R ) s -measurable. As s and 0 are independent,· weF can apply lemma (57) andB obtain⊗ F F F s E[IB(ϕ(s + t, ω)x1, ..., ϕ(s + t, ω)xn) ] = Ψ (ϕ(s, )x1, ..., ϕ(s, )xn), |F0 ◦ · · (n) where Ψ(x) = E[IB(ϕ(t, ω)x1, ..., ϕ(t, ω)xn)] = Pt (x, B), i.e. (2.15) is a Markov process (n) associated to the transition functions (Pt )t R+ . (n) ∈ nd We still denote by (Pt )t R+ the Markov semigroup of operators on Bb(R ) associated (n∈) nd nd to the transition function Pt defined above: for each f Bb(R ), x = (x1, ..., xn) R , we have ∈ ∈ (n) (n) Pt f(x) = f(y)Pt (x, dy) = f((ϕ(t, ω)x1, ..., ϕ(t, ω)xn))P (dω). ZX ZΩ (n) As a consequence of continuity on initial data, the Markov semigroup (Pt )t R+ is Feller. ∈ 38 CHAPTER 2. GENERALITIES ON RDS 2.4 RDS as products of i.i.d. random mappings We saw in remark 45 that a discrete random dynamical system can be generated by com- positions of mappings of the form ψ(θn ). The RDS for which these random variables are independent and identically distributed· have been deeply studied, for example by Kifer [33]. They are also referred to as RDS with independent increments. Here we want to briefly describe how they are related to Markov chains, while in chapter ?? we will show why they are interesting, together with RDS generated by SDE, from the point of view of random invariant measures. The proof of this result follows very closely that of proposition 58: Proposition 59 Let ϕ be a discrete dynamical system over a metric dynamical system (Ω, , P, θ), given by the composition ϕ(n, ω) = ψn(ω) ψn 1(ω) ψ0(ω) of i.i.d. F n ◦ − ◦ · · · ◦ random mappings ψn = ψ θ . Then for each n-uple x1, ..., xn the n-point motion ◦ (k, ω) (ϕ(k, ω)x1, ..., ϕ(k, ω)xn) 7→ is a homogeneous Markov chain with transition probability (n) P ((x1, ..., xn),B) = P ω Ω:(ψ(ω)x1, ..., ψ(ω)xn) B . (2.16) { ∈ ∈ } Moreover, if the random dynamical system is continuous, the Markov chain is Feller. 2.5 An example in infinite dimensions: the stochastic Navier-Stokes equations We consider the Navier-Stokes equations for a viscous incompressible fluid in two-space dimensions. We are given an open bounded subset Ω of R2, with boudary Γ. The equations for the velocity vector u and the pressure p, for a fluid with constant density equal to 1, are ∂u + (u )u ν∆u + p = f ∂t (2.17) divu = 0·, ∇ − ∇ where ν > 0 represents the viscosity of the fluid, and f is the density of force applied to the fluid. The first equation expresses the momentum conservation, the second expresses the mass conservation. Two kinds of boundary conditions, as well as an initial condition u(x, 0) = u0(x), are usually considered. The first one u Γ=0 | corresponds to the boundary Γ being a solid at rest. The second has no physical interpre- tation, but presents some more affordable aspects. It can be considered in case the domain Ω1 is a square, say Ω1 = (0, l) (0, l), and consists in imposing × 2 u(x + lei, t) = u(x, t) x R , t > 0. ∀ ∈ ∀ 2.5. AN EXAMPLE IN INFINITE DIMENSIONS:THE STOCHASTIC NAVIER-STOKES EQUATIONS39 In this case one imposes also that the integral of the velocity u over Ω equals zero. For this to be verified, it is suffi cient that u0 and f have zero-integral over Ω, as follows by integrating the equations on Ω and using the periodicity property. Before giving the stochastic version of these equations, we describe how they can be formulated in a abstract form. We give a brief sketching of the whole problem, and we refer to [?], [?] for a complete description of the setting and details. We denote by Hm(Ω) the Sobolev spaces of functions in L2(Ω), whose derivatives of order less or equal to m are in L2(Ω). Hm(Ω) is a Hilbert space with scalar product α α (u, v)m = (D u, D v), α m | X|≤ α α α α where (D u, D v) = Ω D u(x)D v(x)dx. We write Hm (Ω ) for the space of restrictions on Ω of Ω -periodic functions in per 1R 1 1 Hm(Rn), which belong to Hm(S) for each bounded open set S. If X is a space of functions defined on Ω, we denote with X the space X2 with the ˙ product structure, with X the set of the functions u X such that Ω udx = 0, and with X the functions in X which are zero on Γ. ∈ 0 R The following spaces are useful for establishing the mathematical setting of the equa- tions: 1 0 V = u H0(Ω), divu = 0 ,H = u H0(Ω), divu = 0 { ∈ } { ∈ } for the case of still boundary, and ˙ 1 ˙ 0 V = u Hper(Ω), divu = 0 ,H = u Hper(Ω), divu = 0 { ∈ } { ∈ } for the case of boundary periodic functions. On both cases V is a Hilbert space with scalar α α 1/2 product ((u, v)) = α =1(D u, D v) and norm u = ((u, u)) . Consider the bilinear form | | || || P 2 ∂u ∂v a(u, v) = i , i , ∂x ∂x i,j=1 j j X and the trilinear form b given by n ∂v b(u, v, w) = u j w dx. i ∂x j i,j Ω i X Z A weak form of the Navier-Stokes equations (2.17) can be given by taking the scalar product of the equations with a function v V . The pressure disappears and one finds ∈ d (u, v) + νa(u, v) + b(u, u, v) = (f, v), (2.18) dt 40 CHAPTER 2. GENERALITIES ON RDS The continuity of a and b guarantees the existence of a linear operator A and a bilinear operator B on V V such that (Au, v) = a(u, v), (B(u, v), w) = b(u, v, w) for all u, v, w V . × ∈ Equation (2.18) can then be written as an equation in V 0: du + νAu + B(u, u) = f. (2.19) dt This equation admits a unique solution u in C([0,T ]; H) L2(0,T ; V ) for each given f ∩ and u0 in H, for each T > 0. Moreover, one can prove that the mapping u0 u(t) 2 7→ is continuous from H into D(A), and u belongs to C([0,T ]; V ) L (0,T ; D(A)) if u0 is taken in V . This result on existence and continuity of solutions allows∩ the definition of a semigroup of continuous operators S(t): u0 u(t), t 0 7→ ≥ which define the dynamical system associated to Navier-Stokes equations. A stochastic version of equations (2.19), with additive noise, is given by the equation m du + (νAu + B(u, u))dt = fdt + φjdWj(t), (2.20) j=1 X where φ1, . . . , φm are functions in H and W1, ..., Wm are independent two-sided Brownian motions. To show how a random dynamical system can be associated to (2.20), consider the solution z of the Orstein-Uhlenbeck equation m dz = Azdt + φjdWj(t). j=1 X Then, if u is a solution of (2.20), the process v(t) = u(t) z(t) satisfies formally − dv + (Av + B(v, v) + B(v, z) + B(z, v) + B(z, z))dt = fdt. In this form, the equation can be solved pathwise for almost every ω. As can be done for equation (2.19), one can show that for each initial condition v0 H and for each ∈ T > 0 there exists a unique solution vu0 ( , ω) C([0,T ]; H) L2(0,T ; V ), and belonging 2 · ∈ ∩ to C([0,T ]; V ) L (0,T ; D(A)) if v0 is in V . Moreover the map v0 v(t, ω) is continuous for all t > 0. Then∩ one can consider the map 7→ R+ Ω H H × × −→ u (t, ω, u0) ϕ(t, ω)u0 = v 0 (t, ω) + z(t, ω). 7→ As in paragraph (2.3.1), one can show that the well-posedness of equation (2.20) implies the cocycle property, and then ϕ is a random dynamical system, defined fot t 0, over the canonical two-sided metric dynamical system describing Brownian motion. ≥ 2.6. MULTIPLICATIVE ERGODIC THEOREM AND LYAPUNOV EXPONENTS 41 2.6 Multiplicative ergodic theorem and Lyapunov exponents An important result of the theory of random dynamical systems is the so-called Multi- plicative Ergodic Theorem, which allows the definition of Lyapunov exponents for RDS. Its full statements and the proof of all its variants can be found in [3], Part II. We give the statement first for the case of a linear cocycle Φ on Rd, i.e. an RDS over a metric dynamical system (Ω, ,P, (θt)t T), such that Φ(t, ω) is linear for each t T, ω Ω. F ∈ ∈ ∈ Theorem 60 Suppose that Φ satisfies + 1 + 1 1 sup log Φ(t, ) L , sup log Φ(t, )− L . 0 t 1 || · || ∈ 0 t 1 || · || ∈ ≤ ≤ ≤ ≤ Then there exists an invariant set Ω˜ such that P (Ω)˜ = 1 and for each ω Ω˜ the limit 1/2t λp(ω)(ω) λ1(ω)∈ Ψ(ω) = limt (Φ(t, ω)∗Φ(t, ω)) exists. Call e < < e the different →∞ ··· eigenvalues of Ψ(ω), and Up(ω)(ω),...,U1(ω) the corresponding eigenspaces, with di(ω) = dim Ui(ω). Then the functions ω p(ω), ω λi(ω), ω Ui(ω) and ω di(ω) are measurable, and they satisfy 7→ 7→ 7→ 7→ p(θ(t)ω) = p(ω) for all t T, ∈ λi(θ(t)ω) = λi(ω), di(θ(t)ω) = di(ω), t, 1 i p(ω). ∀ ≤ ≤ For each x Rd 0 , the limit ∈ \{ } 1 λ(ω, x) = lim log Φ(t, ω)x t t →∞ || || exists and is called a Lyapunov exponent for Φ. Moreover λ(ω, x) = λi(ω) if and only if x Vi(ω) Vi+1(ω), where ∈ \ Vi(ω) = U Ui(ω) for 1 i p(ω),V = 0 , p(ω) ⊕ · · · ⊕ ≤ ≤ p(ω)+1 { } d and λ(θ(t)ω, Φ(t, ω)x) = λ(ω, x), for all x R 0 , Φ(t, ω)Vi(ω) = Vi(θ(t)ω) for all 1 i p(ω). ∈ \{ } ≤ ≤ For each ω Ω˜, Rd splits as ∈ p(ω) d R = Ek(ω), k=1 M where the Ek are random subspaces with dimension dk, given by Ek = Vi Vp−(ω)+k 1, ∪ − where Vi−, 1 i p−(ω) = p(ω) is the family of subspaces associated to Φ(t, ω). ≤ ≤ ˜ In case (Ω, ,P, (θt)t T) is ergodic, then the map ω p(ω) is constant on Ω, and λi, F ∈ 7→ di are constant on ω Ω:˜ p(ω) 1 , for i = 1, ..., d. { ∈ ≥ } 42 CHAPTER 2. GENERALITIES ON RDS We will now give the statement of the theorem for manifolds. Consider a C1 RDS ϕ on a d-dimensional Riemannian manifold M, over the metric DS (Ω, ,P, (θt)t T), with F ∈ associated skew product (Θ(t))t T, and let ρ be an invariant measure for the one-point motion. Denote by TM the fiber∈ boundle over M. By differentiating the identity ϕ(t + s, ω) = ϕ(t, θsω) ϕ(s, ω), one finds that the differential ◦ T ϕ(t, ω):TM TM → (x, v) (ϕ(t, ω)x, Dϕ(t, ω, x)v) 7→ is a continuous cocycle over (θt)t T. ∈ Theorem 61 Assume that T ϕ satisfies + 1 sup log T ϕ(t, ω, x) x,ϕ(t,ω)x L (P ), (2.21) 0 t 1 || || ∈ ≤ ≤ + 1 1 sup log T ϕ(t, ω, x)− ϕ(t,ω)x,x L (P ), (2.22) 0 t 1 || || ∈ ≤ ≤ where is the norm of the differential as a linear mapping from TxM to T M. ||·||x,ϕ(t,ω)x ϕ(t,ω)x Then there exist Θ-invariant random variables p, di, λi which satisfy p(ω,x) 1 p(ω, x) d, λ1(ω, x) > > λ (ω, x) > , di(ω, x) = d, ≤ ≤ ··· p(ω,x) −∞ i=1 X for (ω, x) in a Θ-invariant set A Ω M of full measure, and such that the tangent space ⊂ × TxM decomposes as p(ω,x) TxM = Ek(ω, x) k=i M for each (ω, x) A, where Ek are random subspaces of dimension di satisfying ∈ T ϕ(t, ω, x)Ek(ω, x) = Ek(Θ(t)(ω, x)), and 1 lim log T ϕ(t, ω, x)v ϕ(t,ω)x = λi(ω, x) if and only if v Ei(ω, x) 0 . t t →±∞ || || ∈ \{ } The λi are called Lyapunov exponents of ϕ under P ρ. If ρ is ergodic, then p, di and λi are constant. × Chapter 3 Random Attractors and Random Invariant Measures 3.1 Attractors for random dynamical systems In Chapter 1 we have already investigated attractors for non-autonomous systems in a quite satisfactory manner. The results move directly to RDS, working ω-wise. Only for comparison of exposition, thus, we repeat from scratch some elements of the theory in the language of RDS. We are given a random dynamical system ϕ over a metric dynamical system (Ω, ,P, (θt)t T) on a Polish space (X, d). F ∈ In the random setting, the analogous of the ω-limit set is defined for a random set K as the random set ΩK (ω) = ϕ(t, θ tω)K(θ tω). − − s 0 t s \≥ [≥ For every ω Ω, ΩK (ω) is the set of points y X such that there exist a sequence ∈ ∈ tn and a sequence xn in K(θ tn ω) such that ϕ(tn, θ tn ω)xn y as n . With → ∞ − − → → ∞ this observation, it is easy to verify that ΩK is forward invariant for the RDS ϕ, in the sense of the definition given in paragraph 3.3.1. Let y be in ΩK (ω), and let (tn)n and (xn)n K(θ tn ω) be such that ϕ(tn, θ tn ω)xn y. Then we find, by the cocycle property ⊂ − − → ϕ(t, ω)y = lim ϕ(t, ω)ϕ(tn, θ tn ω)xn = lim ϕ(t, θtn θ tn ω)ϕ(tn, θ tn ω)xn n − n − − →∞ →∞ = lim ϕ(t + tn, θ tn ω)xn = lim ϕ(t + tn, θ tn tθtω)xn n − n − − →∞ →∞ = lim ϕ(sn, θ sn θtω)xn, n − →∞ with sn = t + tn. As xn is in K(θ tn ) = K(θ sn θtω), one gains ϕ(t, ω)y ΩK (θtω). There are several ways to generalize− the notion− of attraction when taking∈ into account noise perturbations. The following one is the most classical. 43 44CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES Definition 62 A random set A attracts another random set B if d(ϕ(t, θ tω)B(θ tω),A(ω)) 0 as t (3.1) − − → → ∞ for P -almost every ω Ω. ∈ The kind of attraction presented in the definition is sometimes referred to as pullback attraction. A weaker form of convergence is attraction in probability, according to which convergence (3.1) takes place in probability, i. e. for every > 0 P d(ϕ(t, θ tω)B(θ tω),A(ω)) > 0 as t . { − − } → → ∞ In many situations, the pullback approach turns out to be more appropriate then the notion of forward attraction, according to which a random set A attracts another random set B if convergence d(ϕ(t, ω)B(ω),A(θtω)) 0 → takes place for P -almost every ω as t . The following example shows that→ the ∞ two kinds of convergence are different. Example 63 Consider a metric dynamical system (Ω, , P, θ), a random variable f :Ω F → R+ and a constant α, with 0 α 1. The map ψ :Ω R R given by ≤ ≤ × → f(θω) ψ(ω, x) = α x f(ω) defines a discrete random dynamical system ϕ as described in remark 45. By an induction argument one sees that f(θnω) ϕ(n, ω)x = αn x. f(ω) Consider a random set B and take A(ω) = 0 . We want to see under which condition the set A is forward or pullback attracting for B{.} For this purpose, we will use the result of the proposition below, which can be found in [3] (proposition 4.1.3): The proposition implies that the quantity 1 f(θnω) log f(θnω) 1 B(ω) log αn B(ω) = log α + + log | | n f(ω) | | n n f(ω) converges to log α < 0 or to as n , and A is forward attracting if and only if 1 n ∞ → ∞ n log f(θ ω) converges to zero. As for pullback attraction, one can consider n 1 n f(ω) n log f(ω) 1 B(θ− ω) log α B(θ− ω) = log α + + log | |, n f(θ nω)| | n n f(θ nω) − − n 1 B(θ− ω) and A is pullback attracting if and only if log | n | converges to zero. n f(θ− ω) 3.1. ATTRACTORS FOR RANDOM DYNAMICAL SYSTEMS 45 Proposition 64 Let (Ω, ,P, (θt)t T) be a metric dynamical system and let g :Ω R be a random variable. Then F ∈ → 1 lim sup g(θtω) 0, for P -almost every ω. t t ∈ { ∞} →∞ If one considers convergence in probability instead of almost sure convergence, then the two definitions coincide, thanks to the θ-invariance of P , which implies P d(ϕ(t, θ tω)B(θ tω),A(ω)) > = P d(ϕ(t, ω)B(ω),A(θtω)) > . { − − } { } Definition 65 A random compact set A is a random attractor for a random set B if (i) A is invariant; (ii) A attracts B. The definition of absorbing set in random context is given for the pullback point of view. Definition 66 A random set K absorbs a random set B, if for P -almost every ω Ω ∈ there exists a time t0 (which may depend on ω) such that ϕ(t, θ tω)B(θ tω) K(ω) for all t t0. − − ⊂ ≥ The following result is the analogous of theorem 116. Theorem 67 Suppose that there exists a compact random set K which absorbs a random set B. Then the Ω-limit A of B is an attractor for B. Proof. Let t0 be such that ϕ(t, θ tω)B(θ tω) K(ω) for all t t0. Then − − ⊂ ≥ ΩB(ω) ϕ(t, θ tω)B(θ tω) K(ω), ⊂ − − ⊂ s t0 t s [≥ \≥ which implies that ΩB(ω) is compact. To prove invariance, it is suffi cient to show that ΩB(θtω) ϕ(t, ω)ΩB(ω). Take x ΩB(θtω), then ⊂ ∈ x = lim ϕ(tn, θ tn θtω)xn = lim ϕ(t, ω)ϕ(tn t, θ tn+tω)xn, n − n − →∞ →∞ − for some sequence tn and (xn) B(θ tn+tω). The sequence ϕ(tn t, θ tn+tω)xn is eventually contained→ in ∞K(ω), and then⊂ it− admits a subsequence converging− − to a point y ΩB(ω). By continuity of ϕ(t, ω) we find x = ϕ(t, ω)y ϕ(t, ω)ΩB(ω). Suppose ∈ ∈ now that A does not attract B. Then there exist a δ > 0, a sequence tn and → ∞ a sequence xn B(θ tn ω) such that d(ϕ(tn, θ tn ω)xn, ΩB(ω)) δ for all n. For n ∈ − − ≥ 46CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES suffi ciently big we have (ϕ(tn, θ tn ω)xn) K(ω), and then it possesses a convergent − ⊂ subsequence (ϕ(tnk , θ tn ω)xnk )k. The limit point belongs to ΩB(ω), thus contradicting − k d(ϕ(tnk , θ tn ω)xnk , ΩB(ω)) δ. − k ≥ The notion of global attractor for random dynamical systems is more delicate that the deterministic one. Following the paper by Crauel and Flandoli ([?]), we say that an invariant random compact set is a global or universal attractor if it attracts all bounded deterministic sets in X. The following result could be found in the same paper. Theorem 68 Suppose that there exists a compact random set K which absorbs every bounded deterministic set B X. Then there exists a global attractor A for ϕ, given by the -measurable random⊂ set F − A(ω) = ΩB(ω). B X [⊂ 3.1.1 Attractor for the finite dimensional approximation of SNSE We give the proof of existence of an attractor for equation (2.10), following the proof for the stochastic Navier-Stokes equations, due to Crauel and Flandoli ([?]). This finite dimen- sional version exemplifies all the techniques which are involved in the infinite-dimensional case, for the proof of dissipativity. The instruments combine those of the determinist case, such as the estimate of energy through continuity inequalities and Gronwall lemma, and some others typical of the stochastic case, for instance the pullback approach and the aux- iliary Ornstein-Uhlenbeck process. The aspect of compactness needs a further effort for the full Navier-Stokes equations; nevertheless, the procedure strictly follows that of the deterministic case, for whom we refer to [?]. We want to show that there exists a bounded random set which attracts all bounded deterministic sets, i.e. for P -almost every ω there exists a random ball B(0, r(ω)) such that for each bounded subset B of Rn, there exists a time T such that ϕ(t, θ tω)B B(0, r(ω)), for all t T. − ⊂ ≥ t,x t,x As ϕ(t, θ tω)x = X0− (ω), we have to estimate the modulus of X0− (ω). − On this purpose, consider an auxiliary stochastic differential equation dZt + (AZt + αZt)dt = QdWt, where α is a positive constant, which will be specifiedp later. As recalled in appendix 6.3, a stationary solution of this equation is given by the Ornstein-Uhlenbeck process t (A+α)(t s) Zt = e− − QdWs. Z−∞ p 3.1. ATTRACTORS FOR RANDOM DYNAMICAL SYSTEMS 47 If we write shortly Xt for the solution of equation (2.10), the process given by the difference Yt = Xt Zt satisfies the random equation − dYt + AYt + B(Yt + Zt,Yt + Zt) = αZt + f. dt − Taking the scalar product with Yt, one finds 2 1 d Yt | | = AYt,Yt B(Yt + Zt,Yt + Zt),Yt αZt,Yt + f, Yt . 2 dt −h i − h i − h i h i Using the property of A and B one gains the inequality 2 1 d Yt 2 | | + λ Yt B(Yt + Zt,Zt),Yt αZt,Yt + f, Yt 2 dt | | ≤ −h i − h i h i 2 2 C Yt Zt + C Yt Zt + α Yt Zt + f Yt . ≤ | | | | | || | | || | | || | 1 2 1 2 C 2 Apply now the inequality ab a + b , with a = √2 Yt , b = ( Zt + α Zt + f ), ≤ 2 2 | | √2 | | | | | | where 0 < < λ, and find 2 1 d Yt 2 2 2 C 2 2 | | + λ Yt C Yt Zt + Yt + ( Zt + α Zt + f ) , 2 dt | | ≤ | | | | | | | | | | | | which implies 2 d Yt 2 2 2 | | + (λ1 C1 Zt ) Yt C1( Zt + α Zt + f ) , dt − | | | | ≤ | | | | | | for some constants λ1 and C1. Using a differential version of the Gronwall lemma, proved 2 t by differentiating Yt exp (λ1 C1 Zs )ds , one finds | | t0 − | | R t t t t0,x 2 2 t (λ1 C1 Zs )ds (λ1 C1 Zu )du 2 2 Y x e− 0 − | | + e− s − | | C1( Zs + α Zs + f ) ds | t | ≤ | | | | | | | | R Zt0 R t,x and the inequality for Y0− 0 t,x 2 2 t(λ1 C1 Zs )ds Y0− x e− − − | | | | ≤ | | R 0 0 (λ1 C1 Zu )du 2 2 + e− s − | | C1( Zs + α Zs + f ) ds. (3.2) t R | | | | | | Z− Recall now that the law of large numbers for stationary processes (theorem (128)) implies that 1 0 lim Zs ds = E[Z0]. t t | | →−∞ − Zt 48CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES λ1 λ1 Then if α is such that C1E[Z0] , then we have λ1 + C1E[Z0] , and ≤ 2 − ≤ − 2 0 1 λ1 lim (λ1 C1 Zs )ds = λ1 + C1E[Z0] . t t − | | − ≤ − 2 →−∞ Zt This implies that 0 (λ1 C1 Zs )ds lim e− t − | | = 0, t − →∞ R which, together with the fact that 2 2 ( Zt + αZt + f ) lim | | | | = 0 t tβ →−∞ for almost every ω, for β suffi ciently big, gives 0 0 (λ1 C1 Zu )du 2 2 e− s − | | C1( Zs + α Zs + f ) ds < . R | | | | | | ∞ Z−∞ The application of the above estimates to (3.2) gives the existence, for almost every ω, of a constant C(ω) such that, for each bounded set B, t,x 2 sup Y0− (ω) C(ω) x B | | ≤ ∈ for t suffi ciently big. Then, for almost every ω, there exists an R(ω) such that, for each bounded set B there exists a tB such that t,x 2 t,x+Z0(ω) 2 2 sup X0− (ω) sup C( Y0− (ω) + Z0(ω) ) D(ω) x B | | ≤ x B | | | | ≤ ∈ ∈ for t tB, as wanted. ≥ 3.2 Random Measures Let (Ω, ,P ) be a probability space, and (X, d) be a Polish space, i.e. a complete separable F metric space, with Borel σ-algebra . Denote by Bb(X) (resp. Cb(X)) the space of real bounded measurable (resp. continuous)B functions on X. In this setting a random measure is a map µ : Ω [0, 1] B × → (B, ω) µω(B), 7→ such that (i) for P -almost every ω Ω, B µω(B) is a probability measure; ∈ 7→ (ii) for each B , the map ω µω(B) is measurable. ∈ B 7→ 3.3. RANDOM MEASURES AND RANDOM DYNAMICAL SYSTEMS 49 The random measure µ will also be indicated as µ = (µω)ω Ω, and considered as a ∈ family of measures on (X, ), indexed on Ω. Imposing condition (ii) to the family (µω)ω Ω B ∈ is equivalent to asking that, given any function f Cb(X), the map ∈ ω µω(f) = f(x)µω(x) 7→ ZX is a random variable (see appendix 6.2). Such families of measures are also called Markov kernels or transition probabilities. One of the reasons for referring to them with a different name is the necessity, in RDS theory, of emphasizing the asymmetric roles of the spaces Ω and X, the first representing the possible noises and the second supporting the dynamic. See [14] for a complete overview of random measures on Polish spaces. 3.3 Random measures and random dynamical systems Let ϕ be a random dynamical system on a Polish space (X, ), over a metric dynamical B system (Ω, ,P, (θ(t))t T), and let µ be a random measure on X Ω. F ∈ × Definition 69 The random measure µ = (µω)ω Ω is said to be invariant for the random ∈ dynamical system ϕ if, for all t T, one has ∈ 1 E[ϕ(t, )µ (A) θ(t)− ] = µθ(t) (A), for all A . (3.3) · · | F · ∈ B 1 If θ is measurably invertible (for example if T is a group), then θ(t)− = and µ is F F invariant if and only if for all t T ∈ ϕ(t, ω)µω = µ for P -almost every ω Ω. (3.4) θ(t)ω ∈ Example 70 Suppose there exists a random variable x0 :Ω X such that → ϕ(t, ω)x0(ω) = x0(θtω) for P -almost every ω Ω. (3.5) ∈ Then the random measure µω = δx0(ω) is invariant. It is called a random Dirac measure. Example 71 Consider the random dynamical system ϕ of example 54. For almost every ω, the integral 0 αs x0(ω) = αe σω(s)ds − Z−∞ Wt is convergent, as a consequence of the fact that, for almost every ω, t converges to 0 as t goes to infinity (see for example [32], problem 9.3). We show that x0 satisfies (3.5), and 50CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES then the random measure δx0(ω) is invariant for ϕ. We have t αt α(t s) ϕ(t, ω)x0(ω) = e− x0(ω) + σω(t) αe− − σω(s)ds − 0 0 Z t α(t s) α(t s) = αe− − σω(s)ds + σω(t) αe− − σω(s)ds − − 0 Z−∞t Z α(t s) = αe− − σω(s)ds + σω(t) − Z −∞0 0 = αeαsσω(t + s)ds + σω(t) αeαsds − Z Z −∞0 −∞ αs = αe σ(ω(t + s) ω(s))ds = x0(θtω). − − Z−∞ Consider the probability measure λ on (Ω X, ), given by × F ⊗ B λ(A) = P (dω) µω(dx)IA(ω, x), for each A . (3.6) ∈ F ⊗ B ZΩ ZX We have the following proposition. Proposition 72 The random measure µ = (µω)ω Ω is invariant for the RDS ϕ if and only if the measure λ on (Ω X, ) given by (3.6)∈ is invariant for the skew product associated to ϕ. × F ⊗ B Proof. Observe that, for each F , B , ∈ F ∈ B (Θtλ)(F B) = λ Θt F B × { ∈ × } = P (dω) µω(dx)I Θt F B (ω, x) { ∈ × } ZΩ ZX = P (dω) µω(dx)I θt F,ϕ(t, ) B (ω, x) { ∈ · ·∈ } ZΩ ZX = P (dω) µω(dx)I ϕ(t, ) B 1 { · ·∈ } Zθ(t)− F ZX = P (dω) µω x ϕ(t, ω)x B 1 { | ∈ } Zθ(t)− F ZX = P (dω) (ϕ(t, ω)µω)(B), (3.7) 1 Zθ(t)− F ZX and λ(F B) = P (dω)IF (ω)µω(B) = P (dω)IF (θ(t)ω)µω(B) × ZΩ ZΩ = P (dω)µθ(t)ω(B). (3.8) 1 Zθ(t)− F 3.3. RANDOM MEASURES AND RANDOM DYNAMICAL SYSTEMS 51 Suppose then that (3.3) holds: then clearly (3.7) equals (3.8), and Θtλ = λ. Conversely, suppose that λ is invariant under Θt, then for each B and t T, the function ∈ B ∈ ω µθ(t)ω(B) is a version of the conditional expectation of the function ω ϕ(t, ω)µω(B), 7→ 1 7→ with respect to θ(t)− . Since is countably generated, one can find an exceptional set which fits all B , andF (3.3) holds.B ∈ B We write P (Ω X) for the set of probability measures λ on (Ω X, ) of the form P × × F ⊗B (3.6) for some random measure µ. Observe that Θt maps P (Ω X) into itself. For each P × λ P (Ω X), the random measure µ such that (3.6) holds is P -almost surely unique ([3],∈ Pproposition× 1.4.3). It is called the factorization of λ. We call P (ϕ) the subset of P (Ω X) of measures λ for which the associated random measure µ isI invariant. P × 1 Define the space L (Ω, b(X)) of functions f :Ω b(X) such that the map (ω, x) P C → C 7→ f(ω)(x) is measurable, and the integral Ω supx X f(ω)(x) dP is finite. One can endow ∈ | | the space P (Ω X) with the topology of weak convergence, i.e. the smallest topology P × R 1 such that the maps µ µ(f) = Ω X fdµ are continuous for each f LP (Ω, b(X)), 7→ × ∈ C µ P (Ω X). ∈ P × R 1 Consider the action of the skew product (Θt)t of ϕ on functions f belonging to LP (Ω, b(X)) 1 C given by Θtf = f Θt. Observe that Θtf belongs to LP (Ω, b(X)) whenever f is in 1 ◦ C LP (Ω, b(X)): for each t, the map (ω, x) f(θtω)(ϕ(t, ω)x) is measurable, continuous in x for eachC fixed ω and 7→ sup (Θtf)(ω)(x) dP = sup f(θtω)(ϕ(t, ω)x) dP (3.9) Ω x X | | Ω x X | | Z ∈ Z ∈ sup f(θtω)(x) dP < . (3.10) ≤ Ω x X | | ∞ Z ∈ Proposition 73 If ϕ is a continuous random dynamical system on a Polish space X, the map µ Θtµ on P (Ω X) is affi ne and continuous, and the set P (ϕ) is convex and closed. 7→ P × I Proof. The first property is obvious. For the second, suppose (µα) is a net converging to 1 α α α µ, i.e., for each f LP (Ω, b(X)), µ (f) converges to µ(f). Then (Θtµ )(f) = µ (Θtf) ∈ C 1 → µ(Θtf) = (Θtµ)(f), as Θtf belongs to L (Ω, b(X)). P C The second part follows from proposition (72), as the set P (ϕ) concides with the set I of fixed points of the map µ Θtµ on P (Ω X). 7→ P × Existence of measures in P (ϕ) can sometimes be proved through a Krylov-Bogolyubov argument, as described in theI following proposition: Proposition 74 Let ϕ be a continuous random dynamical system on a Polish space X, with continuous time T, and let ν be in P (Ω X). For each T T, T > 0, consider the P × ∈ measure µT defined by the means 1 T µT (A) = (Θtν)(A), A . (3.11) T ∀ ∈ F ⊗ B Z0 52CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES Then every limit point of (µT )T for T , in the topology of weak convergence, is in → ∞ P (ϕ). I Proof. Let (Tk)k be a sequence such that Tk and µTk µ weakly. For each t > 0, 1 → ∞ → we show that (Θtµ)(f) = µ(f) for all f L (Ω, b(X)). The equality Θtµ = µ then follows ∈ P C by the fact that all functions of the form IA B, with A and B closed belonging to , × 1 ∈ F B can be approximated by decreasing sequences in LP (Ω, b(X)). 1 1C T Observe that, if f is in LP (Ω, b(X)), then µT f = T 0 ν(Θtf)dt. For each t > 0, we have then that C R 1 Tk Tk (ΘtµT )(f) µT (f) = ν(Θt+sf)ds ν(Θsf)ds | k − k | T − k Z0 Z0 Tk+t Tk 1 = ν(Θsf)ds ν(Θsf)ds T − k Zt Z0 Tk+t t 1 = ν(Θsf)ds ν(Θsf)ds T − k ZTk Z0 2t f 1 0 as k . L (Ω, b(X)) ≤ Tk || || P C → → ∞ This convergence, together with the continuity of the map µ Θtµ, imply that Θtµ = µ, 7→ for each t > 0. If t is in T and t < 0, then the invariance of Θt follows by the equality Θtµ = ΘtΘ tµ. The existence− of limit points for the sequence (3.11), and then of random invariant measures for ϕ, can be established through an anologous of Prohorov theorem for random measures. Definition 75 A set of measures Γ P (Ω X) is said to be tight if for every > 0 there ⊂ P × exists a compact set C X such that, for each λ Γ, λ(Ω C) 1 , or equivalently ⊂ ∈ × ≥ − P (dω)µω(C) 1 , if µω is the factorization of λ. Ω ≥ − R The following theorem is due to Crauel ([14], theorem 4.4): Theorem 76 If Γ P (Ω X) is tight, then every sequence (µn)n N Γ admits a convergent subsequence.⊂ P × ∈ ⊂ If ϕ is a continuous random dynamical system on a compact metric space X, ran- dom invariant measures always exist. Denote by LP∞(Ω, (X)) the set of P -essentially bounded measurable functions with values on the set of signedM measures (X) on (X, ) of finite total variation. The set of random measures on X, which is aM closed subsetB of LP∞(Ω, (X)), is compact by the Banach-Alaoglu’s theorem. The set P (Ω X) can be identifiedM with the set of random measures, through the uniqueness ofP the factorization.× We know by proposition (73) that the maps (Θt)t are a family of commuting affi ne maps 3.3. RANDOM MEASURES AND RANDOM DYNAMICAL SYSTEMS 53 on P (Ω X), and the existence of random invariant measures can be then proved by recurringP × to the Theorem 77 (Markov-Kakutani fixed point theorem) Let S be a nonempty compact and convex subset of a normed linear space X. If is a commuting family of affi ne maps on S, then there exists an x S such that x = f(Hx) for all f . ∈ ∈ H 3.3.1 Support of a random measure We denote by supp(σ) the support of a measure σ on (X, ). B Definition 78 Consider a random measure µ on X Ω. Denote by Nµ the P -null set × outside of which µω is a probability measure. The support C of µ is the map on Ω with values on the subsets of X, given by c supp(µω) if ω Nµ, ω Cω = ∈ 7→ X if ω Nµ, ∈ A map A :Ω (X) → P ω Aω, 7→ is called a random closed (open, compact) set if Aω is closed (open, compact) for each ω Ω, and the map ω d(x, Aω) is measurable for each x X. ∈ 7→ ∈ Proposition 79 The support of a random measure is a closed random set. Proof. It is suffi cient to prove that for each δ > 0 the set ω Ω d(ω, Cω) < δ belongs to . One has { ∈ | } F ω Ω d(ω, Cω) < δ = ω Ω Cω B(x, δ) = { ∈ | } { ∈ | ∩c 6 ∅} = Nµ ω N : supp(µω) B(x, δ) = ∪ { ∈ µ ∩ 6 ∅} c = Nµ ω N : µω(B(x, δ)) > 0 , ∪ { ∈ µ } which belongs to by definition of random measure. F Consider now a continuous RDS ϕ over a metric DS (Ω, ,P, (θ(t))t T). We say that a random set C is forward (resp. backward) invariant for theF RDS ϕ if for∈ each t > 0 (resp. t < 0) ϕ(t, x)Cω C for P -almost every ω Ω. ⊂ θ(t)ω ∈ C is called invariant if for each t T ∈ ϕ(t, ω)Cω = C for P -almost every ω Ω. θ(t)ω ∈ 54CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES Proposition 80 If µ is a random invariant measure for the continuous RDS ϕ, then the support C of µ is forward invariant if θ is invertible, and invariant if time is two-sided. Proof. We have that ϕ(t, ω)Cω supp(ϕ(t, ω)µω) = supp(µ ) = C , ⊂ θ(t)ω θ(t)ω with equality if time is two-sided. Denote by P (C) the set of measures on Ω X supported on a random set C. Observe P × 1 1 that C is (forward) invariant for ϕ if and only if C Θ− C (C = Θ− C) for t > 0, where ⊂ t t Θt is the skew product associated to ϕ. This implies that if K is an invariant compact random set, the set P (K) is invariant for Θt. If X is compact, then P (K) is convex and compact and oneP can apply the Markov-Kakutani fixed point theoremP to prove the existence of random invariant measures supported by K. If X is not compact, the existence of a random invariant measures supported on K still holds, with a more delicate proof, which can be found in [14] (corollary 6.13). 3.3.2 Markov measures Let ϕ be a random dynamical system with times R+, over a metric dynamical system (Ω, ,P, (θ(t))t R), and we suppose that the one point motion associated to ϕ is a Markov F ∈ process with transition probabilities (Pt)t R+ . Markov measures are random measures∈ with particular measurability properties, which are useful for establishing a corrispondence between random invariant measures for a ran- dom dynamical system ϕ and invariant measures for the associated Markov semigroup. Define the σ-algebras + = σ ω ϕ(t, θsω, x): x X, t, s 0 , F { 7→ ∈ ≥ } − = σ ω ϕ(t, θ sω, x): x X, 0 t s F { 7→ − ∈ ≤ ≤ } representing respectively the future and the past of the RDS ϕ. Random measures which are measurable with respect to −, are called Markov random measures. Next theorem gives a motivation for this definition.F Proposition 81 Let µ = (µω)ω Ω be a Markov random invariant measure for the random dynamical system ϕ, and suppose∈ that + and are independent. Then ρ = P µ is an F F − invariant measure for the Markov semigroup Pt associated to ϕ. Proof. Take f Bb(X) and t T. Then ∈ ∈ (ρPt)[f] = ρ[Pt(f)] = ρ(dx) Pt(x, dy)f(y) ZX ZX = P µ(dx) P (dω)f(ϕ(t, ω, x)). ZX ZΩ 3.4. FROM INVARIANT MEASURES TO INVARIANT RANDOM MEASURES 55 Apply the last part of proposition (119) in appendix 6.2, with = +, and h = f(ϕ(t, , )), and find G F · · (ρPt)[f] = P (dω) µω(dy)f(ϕ(t, ω, y)). ZΩ ZX Using the invariance of µ with respect to ϕ, and that of P with respect to θ, we can conclude (ρPt)[f] = P (dω) µθ(t)ω(dy)f(y) ZΩ ZX = P (dω) µω(dy)f(y) = ρ[f]. ZΩ ZX Existence of invariant Markov measures supported by invariant random compact set is established by the following theorem, which can be found in [14]: Theorem 82 If K is a forward invariant random compact set which is measurable with respect to , then there exists an invariant Markov measure supported by K. F − Observe for istance that, in the setting of example 70, if x0 is -measurable, then µ F − is a Markov random invariant measure, supported on x0(ω) ω Ω . { | ∈ } 3.4 From invariant measures to invariant random measures In this paragraph we consider a continuous RDS ϕ generated by a stochastic differential t equation as in Chapter 1, over the canonical metric dynamical system (Ω, , s s t, (θ(t))t R,P ). We want to present a result on convergence of random measures whichF leads{F } to≤ the con-∈ struction of a random invariant measure for ϕ, starting from an invariant measure for the one-point motion Pt. The same theorem holds for a discrete random dynamical system on a Polish space (X, ), given by the product of i.i.d. random mappings. The result has been studied in severalB versions for istance by Le Jan [35], Kunita [?] and Baxendale [5]. Suppose ρ is an invariant measure for the Markov semigroup associated to the one-point motion of ϕ, i.e. ρ(dx)Pt(x, B) = ρ(B) x X,B , t R ∀ ∈ ∈ B ∈ ZX or more compactly ρPt = ρ for each t R, ∈ with Pt(x, B) = P ω Ω: ϕ(t, ω)x B for each x X, B , t R. { ∈ + ∈ } (t) ∈ (t) ∈ B ∈ Define, for each t R , a random measure µ = (µω )ω Ω on X, by ∈ ∈ µ(t) = ϕ(t, θ( t)ω, )ρ. ω − · This is a kind of pullback of the measure ρ. For each function f in b(X), the process (t) 0 C (t) 1 (µ (f))t R+ is adapted to ( t)t R+ . Moreover, each random variable µ (f) is in L (Ω). ∈ F− ∈ 56CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES (t) Proposition 83 For each function f in b(X), the process (µ (f))t R+ is a martingale 0 C ∈ with respect to the filtration t 0 t< . {F− } ≤ ∞ Proof. Given s, t R+, recalling the definition of a RDS we can write ∈ µ(t+s)f(ω) = f(ϕ(t + s, θ( t s)ω, x))ρ(dx) − − ZX = f(ϕ(t, θ( t)ω, ) ϕ(s, θ( t s)ω, x))ρ(dx) − · ◦ − − ZX = f(ϕ(t, θ( t)ω, x))[ϕ(s, θ( t s)ω, )ρ](dx) − − − · ZX = f(ϕ(t, θ( t)ω, x))λω(dx), − ZX 0 where λ = (ϕ(s, θ( t s)ω, )ρ)ω Ω. The map ω f(ϕ(t, θ( t)ω, x)) is t-measurable, − − · ∈ 0 7→ − F− while ω ϕ(s, θ( t s)ω, x) is t-independent. Then we can apply part (ii) of propo- sition (119)7→ to find− − F− (t+s) 0 E[µ f t] = P λ(dx)f(ϕ(t, θ( t) , x)). |F− − · ZX One observes that, if A , then ∈ B P λ(A) = P (dω)[ϕ(s, θ( t s)ω, )ρ](A) − − · ZΩ = P (dω) ρ(dx)IA(ϕ(s, θ( t s)ω, x)) − − ZΩ ZX = ρ(dx) P (dω)IA(ϕ(s, ω, x)) ZX ZΩ = ρ(dx)Ps(x, A) = (ρPs)(A), ZX and proves the equality P λ = ρPs, which gives, by invariance of ρ (t+s) 0 E[µ f t] = f(ϕ(t, θ( t) , x))(ρPs)(dx) |F− − · ZX = f(ϕ(t, θ( t) , x))ρ(dx) = µ(t)f. − · ZX Observe that, if M is the martingale of the theorem, for each p 1, we have ≥ p p (t) p p E[ Mt ] = E[ µ (f) ] = E f(ϕ(t, θ( t) , x))ρ(dx) f , | | | | − · ≤ || ||∞ ZX in particular M is uniformly integrable. We can then apply this theorem on convergence of right-continuous martingales (see [32], theorem 3.15, for the proof): 3.4. FROM INVARIANT MEASURES TO INVARIANT RANDOM MEASURES 57 Theorem 84 On a probability space (Ω, , ( t)0 t< ,P ), let (Xt)0 t< be a right-continuous + F F ≤ ∞ ≤ ∞ submartingale such that supt 0 E[Xt ] < . Then the limit X (ω) = limt Xt(ω) exists for almost every ω Ω, and ≥X is in L1∞. ∞ →∞ ∈ ∞ One can conclude that (t) µ (f) converges a.s. for all f Cb(X). (3.12) ∈ Next question is whether this kind of convergence implies the existence of a random measure (µω)ω Ω such that ∈ (t) µ converges to µω weakly for almost all ω Ω. (3.13) ω ∈ Such a random measure µ is called the statistical equilibrium of the random dynamical system. Condition (3.13), being clearly necessary for condition (3.12), has been shown to be suffi cient if the space X is Radon, in a recent paper by Berti, Pratelli and Rigo [?]. A metric space is said to be Radon if every Borel probability measure on X is tight, i.e., taken any probability measure µ on (X, ), for each > 0 there exists a compact subset B Cµ, X such that µ(Cµ,) 1 . Since⊂ every Polish space≥ is Radon− (see [8], theorem 1.4), we can conclude that there exists a random measure µ such that (3.13) is satisfied. Remark 85 More generally, Berti, Pratelli and Rigo proved that for conditions (3.12) and (t) (3.13) to be equivalent it is suffi cient that the family of measures (λ )t 0 defined by ≥ (t) (t) (t) λ (A) = P (dω) µ (dx)IA(ω, x) = P [µ IA], for each A ω ∈ F ⊗ B ZΩ ZX is tight (definition (75)). This condition is also easy to verify in our case: observe that, (t) for each f Cb(X), the expected value of µ f is given by: ∈ P [µ(t)f] = P (dω) ρ(dx)f(ϕ(t, θ( t)ω, x)) (3.14) − ZΩ ZX = ρ(dx) P (dω)f(ϕ(t, θ( t)ω, x)) − ZX ZΩ = ρ(dx) P (dω)f(ϕ(t, ω, x)) ZX ZΩ = ρ(dx)Ptf(x) = ρ[f]. ZX Given any > 0, take a compact set C X such that ρ(C) 1 , and use (3.14) with (t) ⊂ ≥ − f = IC , finding P µ (C) = ρ(C) 1 . ≥ − 58CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES For completeness, we give also the original proof of Kunita for the existence of a random measure µ such that (3.13) holds, in case X = Rn. The martingale µ(t)f converges almost n surely for each f in Cb(R ). In order to find an exceptional null set which fits every n n f Cb(R ), consider a countable dense subset D of Cb(R ). Then there exists a subset Ω ∈ (t) of Ω of full measure such that µω f converges for all ω Ω and f D. For each ω Ω, ∈ ∈ n ∈ define a positive linear functional µ∞ on the linear subspace of Cb(R ) generated by D, as D (t) µω∞f := lim µω f, for all f . t →∞ ∈ D n We have µω∞ f Cb(R ) for all f . Then we can apply the Hahn-Banach theorem | | ≤ || || ∈ D n n and extend µω∞ to a positive linear functional on Cb(R ), such that µω∞ f Cb(R ) for n | | ≤ || || all f Cb(R ). For each ω Ω, the Riesz theorem now gives a measure µω satisfying ∈ ∈ (t) µωf = µ f. The convergence µω (Ω) µω(Ω) implies that µω is a probability measure. ω∞ → To see that µ = (µω)ω Ω is a random measure, observe that, for each f Cb(X), µ(f) is 0 ∈ ∈ pointwise limit of -measurable functions; ω µω(f) is then measurable with respect to 0 , and µ is aF−∞ Markovian random measure.7→ F−∞ n We can now prove convergence (3.13). Let f be in Cb(R ), and let (fm)m N be a ∈ sequence in such that fm f n 0 as m . Then we have: D || − ||Cb(R ) → → ∞ (t) (t) (t) fdµ fdµω fdµ fmdµ ω − ≤ ω − ω ZX ZX ZX ZX (t) + fmdµ fmdµω + fmdµω fdµω ω − − ZX ZX ZX ZX (t) fmdµ fmdµω + 2 fm f n . ≤ ω − || − ||Cb(R ) ZX ZX n For each > 0, take m0 such that fm0 f Cb(R ) < /4, and and t0 such that for each t t0 (t) || − || (t) ≥ fm dµω fm dµω < /2. Then for each t t0 we have fdµω fdµω < X 0 − X 0 ≥ X − X as wanted. R R R R To prove invariance of µ, observe that ϕ(t, ω)µω is the limit of ϕ(t, ω)ϕ(s, θ( s)ω)ρ = ϕ(s + t, θ( s)ω)ρ − − for s ; by taking u = s + t one sees that this limit equals → ∞ lim ϕ(u, θ(t u)ω)ρ = lim ϕ(u, θ( u)θ(t)ω)ρ = µθ(t)ω, u u →∞ − →∞ − as desired. From (3.14) it follows in particular that P µ = ρ. 3.4. FROM INVARIANT MEASURES TO INVARIANT RANDOM MEASURES 59 Example 86 Consider again the random dynamical system ϕ of examples 54 and 71. Observe that, for each t 0, x R and almost every ω Ω ≥ ∈ ∈ t αt α(t s) ϕ(t, θ tω)x = e− x σω( t) αe− − σ(ω(s t) ω( t))ds − − − − 0 − − − Z t αt α(t s) αt = e− x σω( t) αe− − σω(s t)ds + σω( t)(1 e− ) − − − 0 − − − Z 0 αt αt αs = e− x σω( t)e− αe σω(s)ds, − − − t Z− 0 αs which converges to αe σω(s)ds = x0(ω) for almost every ω. Consequently, for each − −∞ probability measure λ on (R, (R)), the pullbacks ϕ(t, θ tω)λ converge to δx0(ω) as t , R − for almost every ω. In particularB this holds for the unique invariant measure of the Markov→ ∞ semigroup associated to ϕ (see appendix 6.3), and (δx0(ω))ω Ω is the statistical equilibrium of the system. ∈ The results of Section 3.3.1 imply that the set of invariant random measures supported on the attractor is non-void. Remark 87 In the setting of this paragraph, if a global attractor A exists, then the statis- tical equilibrium (µω)ω Ω is supported on A. ∈ Proof. Let ω be such that ϕ(t, θ tω)ρ converges weakly to µω. For each n 1, consider − 1 ≥ the closed set Un = x X : d(x, A(ω)) . Given any > 0, let B X be a { ∈ ≤ n } ⊂ compact set such that ρ(B) 1 . For every n 1, if t is suffi ciently big, we have ≥ − ≥ ϕ(t, θ tω)B Un, by the definition of attractor. Then, for every n 1 − ⊂ ≥ µω(Un) lim sup(ϕ(t, θ tω)ρ)(Un) = lim sup ρ x X ϕ(t, θ tω)x Un ≥ t − t { ∈ | − ∈ } →∞ →∞ ρ(B) 1 , ≥ ≥ − and consequently µω(Un) = 1 for all n. Since Un A(ω), we have µω(A(ω)) = limn µω(Un) = 1. ↓ →∞ 60CHAPTER 3. RANDOM ATTRACTORS AND RANDOM INVARIANT MEASURES Chapter 4 Synchronization ...... 61 62 CHAPTER 4. SYNCHRONIZATION Chapter 5 Physical measures 5.1 On the concept of physical measure In the previous chapters we introduced the concept of random dynamical system associated to stochastic differential equations rising from fluiddynamics models, focusing our attention on some properties which characterize their asymptotic behaviour, when considering the special form of pullback ϕ(t, θ tω). In particular, we highlighted the existence of the statistical equilibrium, as almost− sure limit obtained by starting the random dynamical system in the past, with random initial conditions distributed according to ρ, the invariant measure for the Markov semigroup. For systems showing sensitiveness to initial conditions, it is natural to search for a limit behaviour which could be shared by typical trajectories of the system. If f is continuous and bounded, for almost every fixed ω, the convergence f(ϕ(t, θ tω, x))ρ(dx) = µω(f) (5.1) − ZX says that the behaviour of a random trajectory, chosen according to the measure ρ, is asymptotically described by the measure µω. We ask what happens when a different measure is chosen to describe a typical trajectory. We want to see whether a convergence to the statistical equilibrium like (5.1) could take place when the integral is taken with respect to another measure. For example, a convergence of this kind for initial conditions distributed according to the Lebesgue measure could be interesting from a physical point of view, as the Lebesgue measure is usually thought of as the measure describing observable events. In the next Chapter we will try to see if in some cases the statistical equilibrium possesses this physical property. We will see how the two-point motion could provide a useful instrument in the analysis of this property, also in view of some results which hold in compact manifold settings. 63 64 CHAPTER 5. PHYSICAL MEASURES We devote the rest of this chapter to a discussion on convergence properties which can be considered as outstanding for having a physical interpretation, and for pointing out some measures as physically relevant, in deterministic and stochastic settings. 5.2 Physical properties in deterministic settings We present the usual properties which are considered to detect measures having a physical meaning, for a deterministic system. Consider a system described by a map f on a manifold M, and suppose that this system possesses an attractor A, i.e. a compact invariant set on which the dynamics asympotically lies, and which supports all the invariant measures. An experimenter who is interested in the asymptotic behaviour of the system, and wants to study the value assumed by an observable ϕ on A, should start the process at some initial n condition x0 and measure the value of ϕ along the orbit (f (x0))n untill a certain finite N time N, and assume the quantity ϕ(f (x0)) as an approximation of the searched one. In order to recover the average value of ϕ on the attractor, an approach which is sometimes fruitful consists in considering the temporal averages of the observable ϕ along the orbits: if 1 N 1 N n n the sequence N n=1 δf (x0) converges to µ as N , then the means N n=1 ϕ(f (x0)) could be assumed as approximations of the integral→ of∞ϕ on A. Note that this convergence is that describedP in (5.1), when disregarding the temporal means and takingPρ as the Dirac delta. In the simplest case the Lebesgue measure, or a measure which is absolutely continuous with respect to Lebesgue measure, is invariant and ergodic, and plays this physical role, as a consequence of the Birkoff ergodic theorem. On the other hand, in many physical systems, such those arising in fluidodynamic studies, volume is not preserved, and orbits eventually approach sets which are small in the sense of Lebesgue measure. For dissipative systems a natural or dynamically relevant measure is then expected to be concentrated on a set of Lebesgue measure zero. These observations motivate the search for physical measures, i.e. measures having the property that for almost every initial condition in the sense of Lebesgue measure the temporal mean along the orbit equals the spatial mean, as one usually considers properties that hold on positive Lebesgue measure sets as the observable properties of the system. Let us precise this usual definition of physical measure for deterministic settings. Sup- pose f (resp. (St)t 0) is a discrete (resp. continuous) dynamical system on a finite di- mensional Riemannian≥ manifold X, and µ is a Borel probability measure on X, which is invariant for f (for (St)t 0). Denote by m the Riemannian measure on X, which will be referred to also as Lebesgue≥ measure on X. Definition 88 A point x X is ρ-generic if the convergence ∈ n 1 1 − ϕ(f i(x)) ϕ(x)ρ(dx), as n n → → ∞ i=0 X X Z 5.2. PHYSICAL PROPERTIES IN DETERMINISTIC SETTINGS 65 in the discrete case, or the convergence 1 T ϕ(St(x))dt ϕ(x)ρ(dx), as T (5.2) T → → ∞ Z0 ZX in the continuous case take place, for every bounded continuous function ϕ on X. The basin of ρ is the set B(ρ) of generic points for ρ. Then ρ is called a physical measure if ρ(B(ρ)) > 0 and m(B(ρ)) > 0. Observe that, according to this definition, a physical measure is necessarily ergodic. Physical measures are related to another important class of invariant measures, the so- called SRB measures, introduced by Sinai, Ruelle and Bowen in the Seventies. We give a brief presentation of their properties; for a survey on the subject we refer to [51]. Physical measures and SRB measures One can roughly say that SRB measures are those measures which are smooth along un- stable directions. Questions surrounding these measures, from their definition to their existence, are quite delicate, and are releted to important concepts like entropy and Lya- punov exponents. We briefly recall the involved definitions, without details, and then sum up the relations between SRB measures and physical measures. In Chapter 1 we gave definition of Lyapunov exponents for a random dynamical system; we summarize it here for clarity for the particular case of a deterministic setting. We consider a C2 diffeomorphism f of a compact Riemannian manifold (M, d) into itself. For each x M, and each v TxM, consider the limits ∈ ∈ 1 lim log Df n(v) . (5.3) n n x →±∞ || || If these limits exist and coincide, and if µ is an f-invariant probability measure on M, then, according to a theorem due to Oseledec, for each x M the tangent space TxM (i) ∈ decomposes as the sum of subspaces TxM = iEx , such that the limit (5.3) is constant (i) ⊗ on each subspace Ex . The values λi(x) of these limits are called Lyapunov exponents of the system. If µ is ergodic, then the map x λi(x) is constant for µ-almost every x. 7→ To give the definition of entropy hµ of the dynamical system f, given an invariant measure µ, consider the following notations. If α and β are two measurable partitions of 1 the manifold M, write α β for the joint partition A B : A α, B β , and f − α 1 ∨ { ∩ ∈ ∈ } for the partition f − A : A α . Define the entropy H of the partition α = A1, ...An n{ ∈ } 1 n{ 1 1 } as H(α) = i=1 µ(Ai) log µ(Ai), and define hµ(f) = supα limn H i=1− f − α , − →∞ n where the supremum is taken over all measurable partitions of M. P W For each x M, the unstable manifold at x is given by ∈ U 1 n n Wx = y M lim sup log d(f − (x), f − (y)) < 0 . ∈ | n n →∞ 66 CHAPTER 5. PHYSICAL MEASURES We say that a measurable partition α of M is subordinate to the unstable manifolds if, for U every x M, the element α(x) of α containing x is a subset of Wx . An f-invariant Borel probability∈ measure µ on M is said to be an SRB measure, or to have absolutely contin- uous conditional measures on unstable manifolds if f has a positive Lyapunov exponent a.e. (such that the unstable manifolds are defined) and for every measurable partition α U subordinate to W the measure µ α(x) is absolutely continuous with respect to m α(x), i.e. the Lebesgue measure restricted to| α(x), for µ-almost every x. | In case µ is an SRB measure for f, the inequality relating metric entropy hµ(f) of f to Lyapunov exponents of the system hµ(f) λidim(Ei)dµ ≤ M λ >0 Z Xi proved by Ruelle (1978) can be replaced by the equality. The condition is also suffi cient in case µ is an invariant measure for f with a positive Lyapunov exponent a.e. Under some conditions one can prove that SRB measures are physical measures. An important example of such a situation is that of systems possessing an ergodic SRB measure with no zero Lyapunov exponents. Example 89 Consider the differential equation on R dx = x3. dt The solution starting from x is given by x if x = 0, for 0 t 1 , f t(x) = √1 2x2t 6 ≤ ≤ 2x2 0 − if x = 0, for t 0. ( ≥ The origin is invariant, and the delta at the origin is the only invariant measure. A t 1 convergence of the form (5.2) cannot hold, as f (x) for t 2x2 . The delta at the origin can neither be considered an SRB measure, as→ the ∞ Lyapunov→ exponent is null (some authors admit such a degenerate case and consider this system as a counterexample for SRB measures being physical measures). Existence of physical measures In some simple examples the asymptotic behaviour is easy to characterize. Consider for example the system of equations in R2: x˙ = λx y˙ = −µy, − 5.2. PHYSICAL PROPERTIES IN DETERMINISTIC SETTINGS 67 with λ, µ > 0. The solution is given by the flow x xe λt S = − . t y ye µt − The origin is the unique non-trivial invariant set, and the global attractor. The unique (and then ergodic) invariant measure is the Dirac delta at the origin. It is also the physical measure for the system, as if ϕ is continuous and bounded, then for every > 0 there exists a T such that ϕ(St(x, y)) ϕ(0, 0) for every t T , and temporal averages converges to the value at| the origin aswell.− | ≤ ≥ In other simple situations periodic orbits act as attractors. For the system of equations on R2 θ˙ = 1 ρ˙ = ρ + 1, − the solution is given by t θ θ + t f = 1 ρ ρ 1 − , − et for all t 0, the global attractor is the unit circle ρ = 1 . An other≥ well understood class of dynamical systems{ is} that of gradient systems. Con- sider the dynamical system on an open subset U of Rn, generated by the equations dx = V, dt −∇ where V is a function in C2(U). Suppose that the system has isolated fixed points, i. e. points where V = 0. Then these points attract the bounded solutions of the system, as a consequence∇ of the fact that the derivative of V along the solutions in negative. As an x4 x2 example, for V (x) = 4 2 the fixed points are 0 and 1; the attractor is the set 1, 1 , while the origin is unstable.− ± {− } To observe a more complex behaviour, consider the discrete dynamical system generated by the so-called tent map: 2x if 0 x 1 , f(x) = 2 2 2x if 1 ≤ x ≤ 1 − 2 ≤ ≤ on the interval [0, 1]. To see how this map acts, consider the binary expansion x = j j αj2 of numbers in x [0, 1]. Then if α1 = 0, f(x) = αj+12 ; if α1 = 1, j 1 − ∈ j 1 − then≥ f(x) = (1 α )2 j. The tent map is one of the simple maps≥ exhibiting chaos, P j 1 j+1 − P in the sense of sensitive≥ − dependence on initial conditions. With n iterations of the map, all relations betweenP two points which are less then 1/2n far from each other are lost. Observe that any rational number eventually reaches a periodic orbit. The system has an infinite number of distinct periodic orbits, and then it admits infinite invariant 68 CHAPTER 5. PHYSICAL MEASURES measures. The Lebesgue measure is also invariant: the pre-image of an interval [a, b] contained in [0, 1/2] or in [1/2, 1] is given by the union of two intervals, each possessing half of the mass of the original interval. The Lyapunov exponent of the system is ln 2; the Lebesgue measure is an ergodic SRB measure, and then it is a physical measure. The situation becomes more diffi cult when one consider the properties of the Lorenz system (example (117)), which is the paradigm of the system showing chaos. In their long-time behaviour, the orbits of the Lorenz system settles on the attractor, showing a motion which sensitively depends on the initial condition. Lorenz noted for example that successive pairs of maxima of z dispose on the plane on a tent shape, resembling to that produced by the example we have just considered. However, numerical computations for the Lorenz system gave evidence of the existence of a physical measure; its existence and uniqueness has been recently proved by Araujo et al. [1]. No results are available for other systems modelling fluidbehaviour, such as the Navier- Stokes equations, for whom the question is still open. The proof of existence of a physical measure has always proved to be a hard task. It has been achieved firstly by Sinai, Ruelle and Bowen for Axiom A attractors. An attractor Λ for a diffeomorphism f of a compact Riemannian manifold M is an Axiom A attractor if f is uniformly hyperbolic on Λ, i.e. for each x Λ the tangent bundle on x splits continuously as the sum of two Df-invariant subspaces∈ Eu(x) Es(x), and there exist λ > 0 and 0 < ρ < 1 such that ⊕ Df n(x)v λρn v for v Es(x), | | ≤ | | ∈ n n u Df − (x)v λρ v for v E (x). | | ≤ | | ∈ In this case Df is said to be uniformly expandig on Eu and uniformly contracting on Es. Existence of a physical measure was proved later for systems satisfying weaker hyper- bolicity conditions, for uniformly expanding maps (maps f on Rimannian manifold M for 1 1 whom there exists a σ such that Df − σ for all x M) and for the Hï¿2 non map (the first example of non-uniformly| hyperbolic| ≤ system having∈ an SRB measure). Another relevant example is that of unimodal maps, well understood one-dymensional systems gen- erated by maps ft of the interval [ 1, 1] given by − 2 ft(x) = (1 + t)x + t, (5.4) − woth t [ 1, 1]. These maps admit a unique attractor, which is either a periodic orbit, a Cantor∈ set− or a cycle of intervals. Moreover, for Lebesgue-almost every t [ 1, 1], the ∈ − map ft possesses a physical measure. 5.3 What is known for stochastic systems The question of convergence of time averages, which in deterministic settings is far from being completely understood, is surprisingly less problematic in the stochastic setting. 5.4. SOME RESULTS VIA TWO-POINT MOTION 69 Suppose that (Xt)t is a Markov process on a Polish space (X, ), with transition prob- B abilities (Pt(x, ))x X,t 0. Suppose that the probability measures (Pt(x, )) are equivalent · ∈ ≥ · for all t > 0, x X, and that µ is an invariant measure for (Pt)t. Then, for each Borel ∈ function ϕ : X R with ϕ dµ < the convergence → X | | ∞ R T 1 x lim ϕ(Xt )dt = ϕdµ; (5.5) T T →∞ Z0 ZX hold for every x X. The proof of this theorem can be found in [2] (theorem 5.1). It can be applied to the stochastic∈ Navier Stokes equations (paragraph 2.5) under usual hypothesis of nondegeneracy of the noise. On the same hypothesis, denoting by V the total variation of a measure, one can prove that for every Borel measure ν on X|| · || νPt µ V 0 as t , (5.6) || − || → → ∞ see [46]. In particular, for every B we have ∈ B νPt(B) µ(B) as t . → → ∞ Moreover, both conditions (5.5) and (5.6) imply that the invariant measure µ is unique. A convergence for almost every ω as (5.1), on the other hand, has not be fully investi- gated. The proposal is then to focus on the study of the statistical equilibrium. 5.4 Some results via two-point motion In this chapter we consider the settings of paragraph 3.4 and we study the relationships between the two-point motion and the statistical equilibrium, with the intent of analyzing the property of convergence to the statistical equilibrium we presented in the last Chapter: ϕ(t, θ tω)λ converges weakly to µω, for almost every ω, (5.7) − when λ is a probability measure. 5.5 Statistical equilibrium and two-point motion (2) For the random dynamical system ϕ of paragraph 3.4, we denote by (Pt)t R+ and (Pt )t R+ the Markov semigroups associated respectively to the one and the two-point∈ motion.∈ We consider a measure ρ which is invariant for the Markov semigroup Pt, and for each t 0 (t) ≥ we define the random measure µω = ϕ(t, θ tω)ρ. We showed that there exists a random − 70 CHAPTER 5. PHYSICAL MEASURES (t) measure µ = (µω)ω Ω such that, for almost every ω Ω, µω converges weakly to µω as t . Denote by µ∈ the measure on (X X, (X ∈X)) given by µ = P (µ µ), i.e. → ∞ × B × ⊗ µ(A B) = P (dω)µω(A)µω(B). × ZΩ According to our notations on semigroups, described in the appendices, the measure (ρ ⊗ ρ)P (2) on (X X, (X X)) is given by t × B × (2) (2) (ρ ρ)Pt (A) = ρ(dx)ρ(dy)Pt ((x, y),A), for all A (X X). ⊗ X X ∈ B × Z × The following lemma shows how it is possible to find an invariant measure for the semigroup (2) Pt of the two-point motion, using that of the one-point. It can be found in [5] (proposition 2.6). (2) Proposition 90 The measures (ρ ρ)Pt converge weakly to µ for t , and µ is (2) ⊗ → ∞ invariant for Pt , for all t. Proof. For each g Bb(X X) one has, for the θ( t)-invariance of P and the definition of µ(t): ∈ × − (2) ((ρ ρ)Pt )(g) = ρ ρ(dx, dy) P (dω)g(ϕ(t, ω, x), ϕ(t, ω, y)) ⊗ X X ⊗ Ω Z × Z = ρ ρ(dx, dy) P (dω)g(ϕ(t, θ( t)ω, x), ϕ(t, θ( t)ω, y)) X X ⊗ Ω − − Z × Z (t) (t) = P (dω) µω µω (dx, dy)g(x, y) Ω X X ⊗ Z Z × (t) Weak convergence of µω to µω and dominated convergence theorem give ((ρ ρ)P (2))(g) P ((µ µ)g) = (P (µ µ))(g), ⊗ t → ⊗ ⊗ which is the first part of the statement. The second follows easily from the first: for each g Mb(X X) we have ∈ × ((P (µ µ))P (2))(g) = (P (µ µ))(P (2)g) = lim ((ρ ρ)P (2))(P (2)g) t t s s t ⊗ ⊗ →∞ ⊗ = lim (ρ ρ)(P (2) g) = lim ((ρ ρ)P (2) )(g) s s+t s s+t →∞ ⊗ →∞ ⊗ = (P (µ µ))(g). ⊗ 5.5. STATISTICAL EQUILIBRIUM AND TWO-POINT MOTION 71 For each x, y X, t R define the random variable ∈ ∈ rt(x, y) = d(ϕ(t, , x), ϕ(t, , y)). · · Let ∆ be the diagonal of X X and write Xˆ for X X ∆. For each δ > 0, let × × \ Uδ = (x, y) Xˆ : d(x, y) > δ . Next proposition, due again to Baxendale [5] (propositions { ∈ } 2.6 and 2.8), derives information on µω and on the two-point motion from an hypotesis on µ. Proposition 91 If µ(Xˆ) = 0 then: (i) µω consists P -almost surely of a single atom; (ii) there exists a sequence of times tn such that, for (ρ ρ)-almost every (x, y) → ∞ ⊗ ∈ X X, rt (x, y) 0 in probability as n ; × n → → ∞ (iii) every invariant probability measure for the two-point motion on Xˆ is singular with respect to ρ ρ. ⊗ If µ(Xˆ) > 0 then: (i) if µ(∆) = 0, then µω is atomless almost surely; (ii) there exists a stationary probability measure σ for the two-point motion on Xˆ; (iii) for σ-almost every (x, y) Xˆ ∈ P rt(x, y) 0 as t = 0. { → → ∞} ˆ ˆ Proof. First part: (i) X X P (dω)µω µω(X) = 0 implies µω µω(X) = 0 P -almost surely. For such ω the identity× ⊗ ⊗ R 0 = µω(dx) µω(dy)IXˆ (x, y) = µω(dx) µω(dy)IX x (y) \{ } ZX ZX ZX ZX implies that for µω-almost every x µω is concentrated on x , i.e. µ is concentrated on a single point. { } (ii) We have to prove that there exists a sequence tn such that, for each δ > 0, → ∞ P rt (x, y) > δ 0 as n . For each δ one has { n } → → ∞ ρ ρ(dx, dy)P rt(x, y) > δ X X ⊗ { } Z × = P (dω) ρ ρ(dx, dy)IUδ (ϕ(t, ω, x), ϕ(t, ω, y)) Ω X X ⊗ Z Z × 72 CHAPTER 5. PHYSICAL MEASURES (2) = (ρ ρ)P (Uδ) 0, ⊗ t −→ 1 and then P rt(x, y) > δ 0 as t in L (ρ ρ). For each δ one can extract a { } → → ∞ ⊗ sequence tn(δ) such that P r (x, y) > δ 0 as n for (ρ ρ)-almost every → ∞ { tn(δ) } → → ∞ ⊗ (x, y) X X. For each n take tn such that P rt (x, y) > 1/n < 1/n for (x, y) in a set ∈ × { n } An of plenty (ρ ρ)-measure. Then, given any δ > 0 and > 0, for each n > max 1/, 1/δ one has ⊗ { } P rt (x, y) > δ P rt (x, y) > 1/n < 1/n < { n } ≤ { n } for every (x, y) An. ∈ n (iii) Let σ be an invariant probability measure for the two-point motion on Xˆ, then σ T decomposes as σ = σ1 + σ2, where σ1 and σ2 are respectively absolutely continuous and singular with respect to (ρ ρ) ˆ . Take tn as in the previous point. Then, for each δ > 0, ⊗ |X P rtn (x, y) > δ σ1(dx, dy) 0 as n . X X { } → → ∞ Z × Write ˆ σ1(X) = P rtn (x, y) > δ σ1(dx, dy) + P rtn (x, y) δ σ1(dx, dy), X X { } X X { ≤ } Z × Z × then ˆ σ1(X) P rtn (x, y) > δ σ1(dx, dy) − X X { } Z × ˆ = P rtn (x, y) δ σ1(dx, dy) σ(X Uδ) X X { ≤ } ≤ \ Z × and as n goes to and δ goes to zero one finds σ1(Xˆ) = 0. ∞ Second part: (i) The assumption implies µω µω(∆) = 0 P -almost surely, and then ⊗ almost every µω cannot have an atom. (ii) Define a probability measure σ on Xˆ as µ(A) σ(A) = for all A ,A X.ˆ µ(Xˆ) ∈ B ⊂ The invariance of ∆ for the two-point motion implies that, for each x ∆, P (2)(x, A) = ∈ t P (2)(x, A ∆) for every A , and then, if A Xˆ, t ∩ ∈ B ⊂ (2) (2) 1 (2) σPt (A) = Pt (x, A)σ(dx) = Pt (x, A)µ(dx) ˆ µ(Xˆ) ˆ ZX ZX 1 (2) µ(A) = Pt (x, A)µ(dx) = = σ(A), µ(Xˆ) X X µ(Xˆ) Z × 5.5. STATISTICAL EQUILIBRIUM AND TWO-POINT MOTION 73 where we used the invariance of µ with respect to the two-point motion. (2) ˆ (iii) Let σ be as above. Then, using its invariance with respect to Pt on X, one finds P rt(x, y) 0 as t σ(dx, dy) ˆ { → → ∞} ZX lim lim inf P rt(x, y) < δ σ(dx, dy) ≤ ˆ δ 0 t { } ZX → →∞ lim lim inf P (ϕ(t, x), ϕ(t, y)) Xˆ Uδ σ(dx, dy) ≤ δ 0 t ˆ { ∈ \ } → →∞ ZX (2) = lim lim inf Pt I Xˆ U (x, y)σ(dx, dy) δ 0 t ˆ δ → →∞ ZX { \ } = lim(σ(Xˆ) σ(Uδ)) = 0. δ 0 − → Remark 92 From the previous proposition it follows that, in case µ(Xˆ) = 0, we have that µω = δ for some random variable x0, and then for each A (M) x0(ω) ∈ B ϕ(t, θ( t)ω)ρ(A) 1 as t if x0(ω) A,˚ − → → ∞ ∈ and ϕ(t, θ( t)ω)ρ(A) 0 as t if x0(ω) / A. − → → ∞ ∈ Consequently, if X is a Riemannian manifold, and ρ has a strictly positive density with respect to the Riemannian measure m, we have for each Borel set A ϕ(t, θ( t)ω)(m A) m(A)δ as t . − | → x0(ω) → ∞ Next theorem, following the idea of [19] (lemma 15), proves a convergence of the desired type, when a proper ergodic hipothesis on the two-point motion is satisfied. Theorem 93 Suppose ϕ is a RDS whose one point motions form a Markov family. Let ρ be invariant for Pt and let λ be a probability measure on (X, ). Let µ = (µω)ω Ω be (t) B ∈ defined by (3.13) above, and let νω be given by ν(t) = ϕ(t, θ( t)ω, )λ. (5.8) ω − · Suppose that given any couple of measures ν1, ν2 ρ, λ , the measure ν1 ν2 satisfies ∈ { } ⊗ ν1 ν2(∆) = 0, where ∆ is the diagonal of X X, and suppose the following ergodic property⊗ of the two-point motion is satisfied: × (2) (ν1 ν2)P converges weakly to P (µ µ) as t . ⊗ t ⊗ → ∞ Then, given a function f b(X), ∈ C E[ (ν(t) µ)(f) 2] 0 as t . | − | → → ∞ 74 CHAPTER 5. PHYSICAL MEASURES Proof. Since (ν(t) µ)(f) (ν(t) µ(t))(f) + (µ(t) µ)(f) , and E[ (µ(t) µ)(f) 2] 0, it is suffi cient| to show− that| ≤ | − | | − | | − | → E[ (ν(t) µ(t))(f) 2] 0. | − | → For all ω Ω we have ∈ 2 (ν(t)f(ω) µ(t)f(ω))2 = ν(t)(dx)f(x) µ(t)(dx)f(x) − ω − ω ZX ZX 2 = λ(dx)f(ϕ(t, θ( t)ω, x)) ρ(dx)f(ϕ(t, θ( t)ω, x) − − − ZX ZX = λ λ(dx, dy)f(ϕ(t, θ( t)ω, x))f(ϕ(t, θ( t)ω, y)) X X ⊗ − − Z × + ρ ρ(dx, dy)f(ϕ(t, θ( t)ω, x))f(ϕ(t, θ( t)ω, y)) X X ⊗ − − Z × 2 λ ρ(dx, dy)f(ϕ(t, θ( t)ω, x))f(ϕ(t, θ( t)ω, y)). − X X ⊗ − − Z × By taking the expected value, denoting with h the function in Cb(X X) given by h(x, y) = (f(x), f(y)) for all x, y in X, we find: × E[ (ν(t) µ(t))(f) 2] = (λ λ)(P (2)h) + (ρ ρ)(P (2)h) 2(λ ρ)(P (2)h), | − | ⊗ t ⊗ t − ⊗ t (2) and we can conclude by using the assumption on Pt . We give also an exponential version of the previous result. For any positive measurable function f 1, the f-norm µ f of a measure µ is defined as ≥ || || µ f = sup g(x)µ(dx) . || || g f | |≤ Z Theorem 94 In the hypotesis of the previous theorem, assume that there exist a measur- able function V 1, and positive constants C, d and δ, such that, for each initial condition x X X ∆,≥ ∈ × \ (2) δt P (x, ) P (µ µ) V C(d + V (x))e− . || t · − ⊗ || ≤ Then, for every measure σ ρ, λ , every f b(X), with (f, f) V , and every δ < δ, ∈ { } ∈ C | | ≤ 0 there exists a random variable ω Cδ0 > 0 such that 7→ ω δ0 δ0n (ϕ(n, θ( n)ω, )σ µω)f Cω (d + (σ σ)V + (σ ρ)V )e− , for n N. | − · − | ≤ ⊗ ⊗ ∈ 5.5. STATISTICAL EQUILIBRIUM AND TWO-POINT MOTION 75 Proof. Take σ ρ, λ , and f b(X), with (f, f) V . As µ is invariant for ϕ, we can write ∈ { } ∈ C | | ≤ ϕ(t, θ( t)ω, )µθ( t)ω = µω, − · − and then ((ϕ(t, θ( t)ω, )σ)f(ω) µf(ω))2 − · − 2 = σ(dx)f(ϕ(t, θ( t)ω, x)) µθ( t)ω(dx)f(ϕ(t, θ( t)ω, x)) − − − − ZX ZX = σ σ(dx, dy)f(ϕ(t, θ( t)ω, x))f(ϕ(t, θ( t)ω, y)) X X ⊗ − − Z × + µω µω(dx, dy)f(x)f(y) X X ⊗ Z × 2 σ µθ( t)ω(dx, dy)f(ϕ(t, θ( t)ω, x))f(ϕ(t, θ( t)ω, y)). − X X ⊗ − − − Z × By passing to the expected value, the markovianity of µ and independence of ω ϕ(t, ω, x) from , together with point (ii) of proposition (119) imply: 7→ F − P (dω) µω(dy) σ(dx)f(ϕ(t, ω, x))f(ϕ(t, ω, y)) ZΩ ZX ZX = ρ(dy) P (dω) σ(dx)f(ϕ(t, ω, x))f(ϕ(t, ω, y)) ZX ZΩ ZX = (ρ σ)P (2)g, ⊗ t where g = (f, f), and where we used that P µ = ρ. Then E[ (ϕ(t, θ( t)ω, )σ µ)(f) 2] =(σ σ)(P (2)g) + P (µ µ)g | − · − | ⊗ t ⊗ 2(σ ρ)(P (2)g) − ⊗ t ((σ σ)(P (2)g) P (µ µ)g) ≤ ⊗ t − ⊗ + 2(P (µ µ)g (σ ρ)(P (2)g)). ⊗ − ⊗ t 76 CHAPTER 5. PHYSICAL MEASURES Observe also that (σ σ)(P (2)g) P (µ µ)g + 2 (σ ρ)(P (2)g) P (µ µ)g | ⊗ t − ⊗ | | ⊗ t − ⊗ | = P (2)g P (µ µ)g d(σ σ) t − ⊗ ⊗ ZX X × + 2 P (2)g P (µ µ)g d(σ ρ ) t − ⊗ ⊗ ZX X × P (2)g P (µ µ)g d(σ σ) + 2 P (2)g P (µ µ)g d(σ ρ) ≤ t − ⊗ ⊗ t − ⊗ ⊗ Z δt Z δt C( d + (σ σ)V )e− + 2C(d + (σ ρ) V )e− ≤ ⊗ ⊗ δt C0− ≤ for some constant C0 > 0. Consider then, for n N and δ0 < δ, the set ∈ δ n An = ω Ω: (ϕ(n, θ( n)ω, )σ µω)f C0− 0 . { ∈ | − · − | ≥ } We have E[ (ϕ(n, θ( n)ω, )σ µ)(f) 2] (δ δ0)n P (An) | − δ n· − | e− − , ≤ C0− 0 ≤ and then we can conclude by Borel-Cantelli lemma. 5.6 Results on compact manifolds In this section we want to describe some situations in which hypothesis of theorem 93 are satisfied. The setting is that of the works of Baxendale [5] and Baxendale and Stroock [6]. We consider the stochastic Stratonovich differential equation t d t k φs,t(x) = x + X0(φs,u(x))du + Xk(φs,u(x)) dW (5.9) ◦ u s k=1 s Z X Z on a connected, smooth, compact Riemannian manifold M, of finite dimension, over the t canonical metric dynamical system (Ω, , s s t, (θ(t))t R,P ), where X0, X1,... , Xd are F {F } ≤ ∈ d smooth vector fields on M, and W = (W1, ..., Wd) is a Brownian motion on R . The solution (φs,t(ω, x))s,t R of equation (5.9) is a family of random variables such that, for ∈ each C∞ function with compact support h, equation t d t k h(φs,t(x)) = h(x) + X0(h(φs,u(x)))du + Xk(h(φs,u(x))) dW ◦ u s k=1 s Z X Z 5.6. RESULTS ON COMPACT MANIFOLDS 77 is satisfied. If M = Rd, the equation can be written as an Itô stochastic differential equation as t d t k φs,t(x) = x + X0(φs,u(x))du + Xk(φs,u(x))dWu , s k=1 s Z X Z 1 d where X0 is the vector field X0 = X0 + 2 k=1(DXk)Xk. We do not give details of theory of stochastic differential equation on manifolds, for whom we refer to [22]. A random P dynamical system ϕ can be associated to the solution (φs,t(ω, x))s,t R through an analogous of theorem (53) for compact manifolds ([3], theorem 2.3.42). ∈ We denote by TM the tangent bundle of M, and by SM the unit sphere bundle v TM : v = 1 . For each vector field X on M, we denote by TX the vector field on TM{ ∈given by| | the differential} of X, and with X˜ the vector field on SM given by TX(v) X˜(v) = for each v SM. TX(v) ∈ | | X(2) stands for the vector field on Mˆ given, for each (x, y) Mˆ , by ∈ (2) X (x, y) = (X(x),X(y)) TxM TyM T M.ˆ ∈ × ' (x,y) We call m the Riemannian measure on M, and we assume that ρ is an ergodic measure for the one-point motion on M. In particular E[ϕ(t, )ρ] = ρ. A useful instrument for the analysis of the statistical equilibrium and of the ergodic· properties of the two-point motion is provided by the Lyapunov exponents of the system. For each x M we can ∈ consider the random linear mapping Dϕ(t, ω)x, from TxM to Tϕ(t,ω)xM. Hypothesis 2.21 is satisfied as M is a compact manifold, and we can consider the Lyapunov exponents λ1 λ2 λN associated to Dϕ(t, ω)x. ≥ ≥ · · · ≥ In the case λ1 < 0, it is possible to characterize the statistical equilibrium as described by the following result of Le Jan ([35], proposition 2): Proposition 95 If λ1 < 0, then µω consists P -almost surely of a finite number of point measures of the same mass. An analogous of this proposition has been given by Masmoudi and Young ([41]) in infinite dimensions, and has been applied to randomly forced Navier-Stokes equations in two dimensions. Proposition 96 Suppose that λ1 < 0 and (i) there is no compact non-empty set C Mˆ which is invariant for the two-point motion; ⊂ (ii) the Lie algebra generated by X0, ..., Xd on v SM equals TvSM for each v = 0; ∈ 6 e e 78 CHAPTER 5. PHYSICAL MEASURES v,u (iii) if xt is the solution of the stochastic differential equation on SM dxv,u d t = X˜ (xv,u) + u (t)X˜ (xv,u), dt 0 t k k t k=1 X with initial condition v SM and control u C([0, ), Rd), then the set ∈ ∈ ∞ v,u d xt : t 0, u C([0, ), R ) { ≥ ∈ ∞ } is dense in SM. Then µ(Mˆ ) = 0 and µω consists of a single atom P -almost surely. In particular, in the setting of the previous proposition, we can apply remark 92 and deduce that the statistical equilibrium is a physical measure. The analogous of proposition 96 for positive first Lyapunov exponent is the following: Proposition 97 Suppose that λ1 > 0 and that hypothesis (ii) and (iii) of proposition (96) hold true. Then µ(Mˆ ) = 1 and µω is atomless P -almost surely. We give a result on ergodicity of the two-point motion for λ1 > 0, as stated in [6]: Theorem 98 Suppose that λ1 > 0 and (i) there is no compact non-empty set C Mˆ which is invariant for the two-point motion; ⊂ (ii) the Lie algebra generated by TX1, ..., T Xd on v TM equals TvTM for each v = 0. ∈ 6 Then for all (x, y) Mˆ and all bounded measurable functions f : Mˆ R ∈ → 1 t f(ϕ(s, ω)x, ϕ(s, ω)y)ds fdµ as t t → ˆ → ∞ Z0 ZM for P -almost every ω, and µ is the unique invariant probability measure for the two-point motion on Mˆ . Remark 99 Hypothesis (i) is implied, as a consequence of the Stroock and Varadhan Support theorem (theorem (106)), by the existence, for each z Mˆ , of a control u ∈ ∈ C([0, ); Rd) such that the solution of equation ∞ dx d t = X(2)(x ) + X(2)(x )ui dt 0 t i t t i=1 X starting from z gets at arbitrary small distance from the diagonal. (2) (2) ˆ Hypothesis (ii) implies that Lie(X1 , ..., Xd ) = TzM, which is a key point in the proof of ergodicity. 5.7. CONCLUSION AND PROGRAM 79 On the hypothesis of theorem (98), theorem (93) applies. Note that the ergodicity of the two-point motion, in the hypothesis of positive leading Lyapunov exponent, follows from a controllability property for the two-point motion and from a condition on the Lie algebra generated by the lift of the vector fields of the noise X1, ..., Xd on TM, X0 being excluded. Other results could be given on the two-point motion and statistical equilibrium in terms of the sum λΣ of all Lyapunov exponents (Le Jan [35], proposition 1): Theorem 100 (i) If λΣ < 0, then µω is singular with respect to ρ for P -almost every ω Ω; ∈ (ii) if ρ is equivalent to m, then λΣ 0, and λΣ = 0 if and only if ρ is invariant for ϕ. ≤ If the hypothesis of proposition 97 are satisfied, one can show that ρ has a positive smooth density with respect to m (see [6]). In this case, as a consequence of the previous theorem, either λΣ < 0, and µω is atomless and singular with respect to ρ P -almost surely, or λΣ = 0, and µω = ρ P -almost surely. Next result deals with the case λ1 = 0. Theorem 101 Suppose that λ1 = 0 and that (i) and (ii) of theorem (98) hold. Then there is no invariant measure for the two-point motion on Mˆ , µ(Mˆ ) = 0 and µω consists of a single atom almost surely. 5.7 Conclusion and program We ask whether a property as (5.7) holds for random dynamical systems arising from stochastic differential equations of fluid dynamics. With the intent of verifying this con- vergence to the statistical equilibrium (µω)ω Ω, with a result as that of theorem 93, we are led to the analisys of the two-point motion associated∈ to the random dynamical system. A first effort should be devoted to the study of the invariant measure µ for the two- point, to determine the role of the diagonal in the asymptotic behaviour of the system. We have in fact from remark 92 that if µ(Xˆ) = 0, then the desired convergence of the pullbacks of the Lebesgue measure to the statistical equilibrium holds. According to proposition 96, the condition µ(Xˆ) = 0 is garanteed by the negativity of the leading Lyapunov exponent, together with the presence of a suffi cient noise. In this case, however, the physical measure consists of a random Dirac delta: this is a rather peculiar behaviour which is not expected in turbulent flows. More attention, in the theory of fluid dynamics, is given to the case of positive leading Lyapunov exponent. If λ1 > 0, then, in the compact manifold setting, in presence of suffi cient noise (propo- sition 97 and theorem 98), the diagonal has null measure with respect to µ, which is the 80 CHAPTER 5. PHYSICAL MEASURES unique invariant probability measure for the two-point motion on Mˆ , thus allowing the application of theorems such as 93 and 94. An extension of this kind of procedure to systems in Rn, although seeming feasible under some dissipativity properties like those exhibited by fluid dynamic equations, shows some obstructions. The analysis of the Lyapunov exponents is a question of huge and unfaceable diffi culties, both from mathematical and numerical points of view. We restrict then the attention on the aspect of ergodicity of the two-point motion. In the next Chapter we will consider a simple three-dimensional model inspired by fluid dynamics, and show that, under some conditions on the noise, properties on regularity and transitivity of the systems hold. These results are not decisive, but are likely to go in the right direction. 5.8 On the study of the one and the two-point motion This chapter sums up some considerations derived in studying the two-point motion asso- ciated to a stochastic differential equation, outside the diagonal. We consider a particular case of the simplified Goy model presented in example (56), for N = 4, i.e. the three dimensional model du1 = ( νλ1u1 2u2u3) dt + σ1dW1 − − du2 = ( νλ2u2 + u1u3) dt + σ2dW2 (5.10) − du3 = ( νλ3u3 + u1u2) dt + σ3dW3 − where σ = 0, λ , i = 1, 2, 3, and ν are given constants, (W (t)) , i = 1, 2, 3 are i i i t R independent6 two-sided Brownian motions. The system could be written∈ in compact form as du + (νAu + B(u, u)) dt = QdW (5.11) 3 3 2 with A = i=1 λiUi, Q = i=1 σi Ui, where Ui is the p3 3 matrix with entries (Ui)αβ = 3 3 3 × δiαδiβ and B (., .): R R R is the bilinear mapping P × P→ u2v3 + u3v2 B (u, v) = u3v1 − u2v1 − having the property B (u, v) , v 3 = 0 (5.12) h iR for every u, v R3. ∈ The associated two-point motion is the stochastic differential equation in R6 du + (νAu + B(u, u)) dt = √QdW (5.13) dv + (νAv + B(v, v)) dt = √QdW with initial condition u(0) = u0, v(0) = v0, u0 = v0. Existence and uniqueness of solution have been6 proved in proposition (55). 5.9. ERGODICITY 81 5.8.1 Existence of an invariant measure The existence of invariant measures follows by a Krylov-Bogoliubov argument: Proposition 102 If νλi > 0 for i = 1, 2, 3, then there is at least one invariant measure for equation 5.10. Proof. Let α = min (νλi) . i=1,2,3 Apply Itô formula to 2, and using (5.12) find | · | 1 1 d u 2 + ν Au, u dt = u, QdW + TrQdt. 2 | | h i 2 D p E Then 1 1 d u 2 + α u 2 dt u, QdW + TrQdt. 2 | | | | ≤ 2 D E By taking the expected value one obtains that pE u(t) 2 satisfies the equation | | dE u(t) 2 | | = 2αE u(t) 2 + TrQ. dt − | | d 2α 2 Computing the derivative dt e− E u(t) , and integrating the result on [0, t], one proves this Gronwall-like inequality: | | 2 2αt 2 TrQ E u (t) e− E u (0) + C, | | ≤ | | 2α ≤ h i h i 2 which implies that is a Lyapunov function for (Pt)t. Then the conclusion follows from theorem 125. | · | This argument can be reproposed for the two-point motion, too; alternatively, existence of an invariant measure for the two-point motion can be derived from the existence of an invariant measure for the one-point through proposition 90. 5.9 Ergodicity One way to prove uniqueness of the invariant measure is provided by theorem 123, according to which ergodicity is garanteed by regularity of the transition semigroup (Pt)t R+ at some ∈ time t0. Proposition 120 splits the question of regularity into these of Strong Feller property and irreducibility. These two properties are often the key ideas in the proof of ergodicity for a huge number of systems, and rise up in many and deeply studied variants. For istance, the problem of ergodicity in presence of a degenerate noise has been faced for the finite- dimensional approximations of the Navier-Stokes equations, by E and Mattingly [20] for 82 CHAPTER 5. PHYSICAL MEASURES the two-dimensional case, and by Romito [?] in three dimensions. They were able to prove that the Galerkin truncations of the equations in Fourier form admit a unique invariant measure as soon as the noise forces few small Fourier modes. In these papers and in other works dealing with degenerate noise (for example [42]), the problem of ergodicity is parted into the proof of regularity of transition probabilities, and the search for a proper form of irreducibility. In next subsections we will briefly recall how these two properties are related respec- tively to hypoellipticity of the generator of the Markov process and controllability of an associated deterministic system, we show how they can be verified at first for the one- point of the considered system, and then we try to analyze the problems araising in the applications of these techniques to the two-point. 5.9.1 Strong Feller property In order to prove that Pt regularizes bounded functions, one can show that the semigroup has a regular density with respect to Lebesgue measure. A useful criterion to obtain the existence of the density is given by the Hörmander condition. Given two vector fields V and W , we denote by [V,W ] the Lie bracket of V and W , and we define by induction 0 n+1 n adW V = V , adW V = [W, adW V ]. Definition 103 The family of vector fields Xj 0 j m satisfies Hörmander condition if n { } ≤ ≤ the Lie algebra generated by the set ad Xk : n 0, 1 k r has maximal rank at { X0 ≥ ≤ ≤ } every point x Rd. ∈ Observe that the vector fields Xk, with 1 k r, and [Xi,Xj], [Xi, [Xj,Xh]] etc., with 0 i, j, h r, are contained in L. ≤ ≤ ≤ ≤ Consider now the following stochastic differential equation in Rd: d dxt = b(xt)dt + σ(xt)dWt, x0 R , (5.14) ∈ n n n m n where b : R R , σ = (σij)0 i n,0 j m : R R R are C∞ maps, which are → ≤ ≤ ≤ ≤ → × 1 m uniformly bounded together with their derivatives, and Wt = (Wt , ..., Wt ) is a Brownian motion on Rm. Define the vector fields n m n n 1 ∂σik ∂ ∂ X0 = bi σjk ,Xk = σik , 1 k m. (5.15) − 2 ∂xj ∂xi ∂xi ≤ ≤ i=1 k=1 j=1 i=1 X X X X Theorem 104 If the family of vector fields Xj 0 j m satisfies Hörmander condition, { } ≤ ≤ then Markov semigroup Pt admits densities pt, which are C∞. Proofs of this result could be found for example in [47], and in [?]. Here we show only how the theorem can be generalized to the case in which b, σ and their derivatives are only locally bounded, as this matter seems not to have been clarified in literature. 5.9. ERGODICITY 83 Proposition 105 Suppose that the Hörmander condition is satisfied for the well-posed stochastic equation in Rn dXt = b(Xt)dt + σ(Xt)dWt, (5.16) n n n m n where b : R R and σ : R R R are C∞ maps, with locally bounded derivatives. → → × Then the Markov semigroup (Pt(x, ))t associated to the equation is Strong Feller. · x Proof. We denote by Xt the solution of the stochastic equation (5.16), starting from x Rn. ∈ n For each R > 0, let ϕR : R [[0, 1] a C∞ function satisfying ϕR(z) = 1 for z R, → (R) n n (R) n n r | | ≤ ϕR(z) = 0 for z R + 1, and let b : R R and σ : R R R be given by (R) | | ≥ (R) → → × b (x) = b(x)ϕR(x), σ (x) = σ(x)ϕR(x). The stochastic equation (R) (R) (R) (R) (R) dXt = b (Xt )dt + σ (Xt )dWt (5.17) has C∞ coeffi cients, with globally bounded derivatives, and the associated Markov semi- (R) (R),x group (Pt (x, ))t is Strong Feller in B(0,R). We write Xt for the solution of the equation (5.17),· starting from x. n n Let Γ be a Borel subset of R , and x0 be a point in R , and fix a real number T > 0. We want to show that, if x x0, then → PT (x, Γ) PT (x0, Γ). → n Take < T , and call ϕ the function on Bb(R ) given by ϕ(x) = PT (x, Γ), for all n − x R . Then, the Chapman-Kolmogorov equation implies that PT (x, Γ) = PT IΓ(x) = ∈ P(x, ϕ) = Pϕ(x). Take R > 2 x0 . Define the random variable | | τ (R) = inf t 0 : Xx R . x { ≥ | t | ≥ } For each function f Bb(X), we have ∈ (R) (R),x x P (x, f) P(x, f) = E[f(X ) f(X )] | − | | − | (R),x x (R),x x = E I (R) f(X ) f(X ) + E I (R) f(X ) f(X ) τx τx > { ≤ } − { } − h i h i (R),x x (R) = E I (R) f(X ) f(X ) 2 f P τx τx { ≤ } − ≤ || ||∞ { ≤ } h i Then we can write (R) PT (x, Γ) PT (x0, Γ) = P(x, ϕ) P(x0, ϕ) P(x, ϕ) P (x, ϕ) | − | | − | ≤ | − | (R) (R) (R) + P (x, ϕ) P (x0, ϕ) + P (x0, ϕ) P(x0, ϕ) | − | | − | 84 CHAPTER 5. PHYSICAL MEASURES (R) (R) (R) (R) 2P τ + P (x, Γ) P (x0, Γ) + 2P τ . ≤ { x ≤ } | T − T | { x0 ≤ } For every 0 > 0, we have to show the existence of a δ > 0 such that, if x x0 < δ, (R) | − | then PT (x, Γ) PT (x0, Γ) < . For the Strong Feller property of P , there exists a δ1 | − | 0 such that, for each x in B(x0, δ1), one has (R) (R) P (x, Γ) P (x0, Γ) < 0/3. | T − T | (R),x Recalling that Xt verifies t t (R),x (R) (R),x (R) (R),x Xt = x + b (Xs )ds + σ (Xs )dWs Z0 Z0 t (R) (R),x and observing that, if is suffi ciently small, we have x + b (Xs )ds R/2 for 0 ≤ 0 t , we deduce, by invoking Doob’sinequality on submartingales ≤ ≤ R (R) (R),x P τx P sup Xt R { ≤ } ≤ 0 t | | ≥ ≤ ≤ t (R) (R),x P sup σ (Xs )dWs R/2 ≤ 0 t 0 ≥ ≤ ≤ Z 2 (R ) (R),x E 0 σ (Xs )dWs ≤ R (R/2)2 (R) (R),x 2 E σ (Xs ) ds 0 | | CR. ≤ hR (R/2)2 i ≤ (R) (R) Therefore can be chosen in such a way that 2P τx and 2P τx are controlled { ≤ } { 0 ≤ } by 0/3, and the thesis follows. For the one-point (5.10), Hörmander condition is clearly satisfied, as X1 = (σ1, 0, 0), X2 = (0, σ2, 0) and X3 = (0, 0, σ3), with σi = 0 for i = 1, 2, 3. 6 The two-point motion The proof of smoothness of densities of transition probabilities in the cited works comes through the application of this criterion, and does not seem to show particular diffi culties; it essentially reduces to some more or less complex computations. The same seems to be true for the two-point motion associated to (5.10), on the set M = R6 (x, x) x R3 , as some computations performed with a mathematical software show. The\{ system| ∈ to be} considered is c 5.9. ERGODICITY 85 du1 = ( νλ1u1 2u2u3) dt + σ1dW1 − − du2 = ( νλ2u2 + u1u3) dt + σ2dW2 − du3 = ( νλ3u3 + u1u2) dt + σ3dW3 − dv1 = ( νλ1v1 2v2v3) dt + σ1dW1 − − dv2 = ( νλ2v2 + v1v3) dt + σ2dW2 − dv3 = ( νλ3v3 + v1v2) dt + σ3dW3; − the drift vector X0 and the vectors X1, X2, X3, associated to the noise are then 3 ∂ ∂ ∂ ∂ X0 = νλi ui + vi 2 u2u3 + v2v3 − ∂u ∂v − ∂u ∂v i=1 i i 1 1 X ∂ ∂ ∂ ∂ + u u + v v + u u + v v , 1 3 ∂u 1 3 ∂v 1 2 ∂u 1 2 ∂v 2 2 3 3 and ∂ ∂ X = σ + , for i = 1, 2, 3. i i ∂u ∂v i i One can see that the determinant of the matrix X1 X2 X3 [X0,X1][X0,X2][X0,X3] 2 2 2 is given by 4σ σ σ (v 1 u1)(v2 u2)(v3 u3). This proves that the Lie algebra L associ- 1 2 3 − − − ated to the vector fields X0, X1, X2, X3 as described above has full rank outside the set 3 i=1 ui = vi . To prove the condition on these remaining points, one can verify that the vectors{ } S [X0, [X0, [X0,Xk]]], [X0, [Xi, [X0,Xj]]] k,i,j 1,2,3 ,i=j { } ∈{ } 6 are independent everywhere outside the diagonal. 5.9.2 Irreducibility To establish the irreducibility of a Markov semigroup Pt, one has to study the quantities Pt(x, A), with A . They can be interpreted as the probability to reach the set A in time t starting from∈ Bx. This idea finds a formalization in what follows. The question of irreducibility of the Markov semigroup associated to a stochastic differential equation as (5.14), is reduced to a problem of control of a deterministic system, through the celebrated Stroock-Varadhan Support Theorem (see [48]). Consider the control system m dxt i d = X0(xt) + Xi(xt)ut, x0 R , (5.18) dt ∈ i=1 X where u is a control function varying in the set of piecewise continuous functions with m (x,u) U values in R . We denote by t xt the solution of this system associated to the control 7→ 86 CHAPTER 5. PHYSICAL MEASURES u, and starting from x. One says that a point y X is accessible from x in time t if there (x,u) ∈ exists a control u such that xt = y. Denote by At(x) the set of accessible point from x in time t. ∈ U Theorem 106 If X0, ..., Xm are the vector fields associated to the stochastic equation (5.14) according to (5.15), Pt is the Markov semigroup associated to the solutions of the same equation, and At is the set of accessible point from x in time t for the control system (5.18), then suppPt(x, ) = At(x). · d d Remark 107 For system (5.10), one can show that At(x) = R for each x R and each d ∈ 1 t > 0, and then the Markov semigroup Pt is irreducible. For each y R , take a path ψ ∈ C in Rd such that ψ(0) = x, ψ(t) = y. Then ψ is a solution of the control system associated 1 dψ to u = √Q− ( νAψ B(ψ, ψ)). dt − − Even when this strong controllability property does not hold, the question of unique- ness is faced by considering a form of irreducibility; if the solutions of the control system associated to the stochastic differential equation does not reach the whole space in an ar- bitrary time, starting from an arbitrary point, weaker forms of control could be suffi cient for the proof of ergodicity, when added to a regularity property for the semigroup. A used criterion originates by considering the Markov chain obtained by sampling the solution of the stochastic differential equation at dicrete time intervals, and using the Markov chain theory to prove that this chain admits a unique invariant measure. The conclusions rely on a result of Harris, according to which a Markov chain (Xn)n admits a unique invariant measure if there exists a measure m on (Rn) such that, if m(E) > 0 n B for E (R ), then P Xn E infinitely often X0 = x0 = 1 for every initial condition ∈ B { ∈ | } x0. The role of m is played by the restriction of the Lebesgue measure to a neighboorhood of the origin. When considering a two-point motion, this criterion is not of immediate application, as the origin seems not to have this privileged role. The criteria usually used for degenerate noise systems, although seeming promising, need a non obvious revision in order to be applicable to the two-point. Even if known suffi cient conditions for ergodicity has not been verified for the two-point motion associated to system (5.10), we can assert that a form of controllability holds, when considering an additional multiplicative noise. This property provides a first step in the understanding of the ergodic properties of the two-point motion, also in view of remark (99). We present this result together with some elements of control theory. A controllability property for the two-point motion We consider the control system x + Ax˜ + B(x, x) = √Qu + 3 Γ xv 0 i=1 i i (5.19) y + Ay˜ + B(y, y) = √Qu + 3 Γ yv , 0 Pi=1 i i P 5.9. ERGODICITY 87 3 ˜ 3 where Γi is the matrix i=1 γiUi, with γi = 0, A is the matrix i=1 liUi, with li = 1 2 3 6 νλi + 2 γi , i = 1, 2, 3, and u : [0,T ] R , vi : [0,T ] R, i = 1, 2, 3, are the controls. This is the control system (5.18)P associated→ to the two-point→ motion (5.13)P with an additional 3 multiplicative noise i=1 Γixdβi. We prove that given any points (x0, y0), (x1, y1) M, for any > 0 there exists a time P ∈ t, and controls u, vi, i = 1, 2, 3 such that the solution (x, y) to equation (5.19) with initial condition (x0, y0) and controls u, vi verifies (x(t), y(t)) c(x1, y1) < . We call this property approximate controllability, and we say that| a system− is approximatively| controllable if it satisfies this property. As this first lemma shows, we introduced the multiplicative noise in order to nullify the effects of the linear part. The motivation will be clear in the sequel. Lemma 108 The control system (5.19) is approximatively controllable on M if the reduced control system x + B(x, x) = √Qu c 0 (5.20) y + B(y, y) = √Qu 0 is approximatively controllable on M. 3 Proof. Given T > 0, > 0, x0,c y0, x1, y1 R with x0 = y0 and x1 = y1, choose vi constant such that ∈ 6 6 3 Ax˜ = Γixvi i=1 X for every x R3 and choose (this is possible by the assumption on (5.20)) u : [0,T ] ∈ → R3 such that the solution (x, y) to (5.20) on [0,T ] with control u and initial condition (x0, y0) has the property (x(T ), y(T )) (x1, y1) . Then (x, y) is also solution to (5.19) | − | ≤ on [0,T ] with the chosen controls u and vi and initial condition (x0, y0), and it satisfies (x(T ), y(T )) (x1, v1) . Thus system (5.19) is approximatively controllable. | We have reduced− the| ≤ study of the controllability to system (5.20). We then relate the controllability of this system to that of a system in R3. Lemma 109 System (5.20) approximatively is controllable on M if the auxiliary 3-dimensional system w0 + B(w, w) + B(w, η) + B(η, w) =c 0 (5.21) is approximatively controllable on R3 0 , where w : [0,T ] R3 is the solution, η : \{ } → [0,T ] R3 is the control. → 3 Proof. Given > 0, x0, y0, x1, y1 R with x0 = y0 and x1 = y1, set w0 := x0 y0, ∈ 6 6 3 − w1 := x1 y1. By assumption, there is a continuous η : [0,T 0] R such that the solution − 3 → w : [0,T 0] R of equation (5.21) with control η and initial condition w0 has the property → wT w1 /2. | 0 − | ≤ 88 CHAPTER 5. PHYSICAL MEASURES Now we show that we can replace η with a continuously differentiable control such that η(0) = y0. Let CB be a constant such that B(u, v) CB u v , and M a constant | | ≤ | || | such that T 0 η(s)ds M. For each 0 < δ < T , consider a continuously differentiable | 0 | ≤ 0 T 0 T 0 control ηδ suchR that ηδ(0) = y0, 0 ηδ(s)ds M and 0 ηδ(s) η(s) ds δ. Let wδ be the solution of (5.21) with control| η and initial| ≤ condition |x y−, and| write≤ f(x, y) for R δ R 0 0 B(x, x) B(x, y) B(y, x). Observe that, if w is a solution of− (5.21) with control η such − − − that T 0 η(s)ds M, then, by property (5.12) we have | 0 | ≤ R 2 1 d w 2 | | = w, w0 = B(w, η), w CB η w , 2 dt h i h i ≤ | || | and by the Gronwall lemma we find that w is bounded. As a consequence, if w and z are solutions of (5.21) relative to control functions η, η0 whose integrals on [0,T 0] are bounded, then there exists a constant C such that f(w, η) f(z, η0) C w z + C η η0 . | − | ≤ | − | | − | Write w0 w0 = f(wδ, ηδ) f(w, η) = f(wδ, ηδ) f(wδ, η) + f(wδ, η) f(w, η); δ − − − − we have then T 0 T 0 wδ(T 0) w(T 0) f(wδ, ηδ) f(wδ, η) ds + f(wδ, η) f(w, η) ds | − | ≤ | − | | − | Z0 Z0 T 0 T 0 C ηδ(s) η(s) ds + C wδ(t) w(t) ds. ≤ | − | | − | Z0 Z0 By the Gronwall lemma we conclude that wδ(T 0) is suffi ciently close to w(T 0) if δ is suffi - ciently small. Now, for the single equation y0 + B(y, y) = Qu (5.22) 3 p there exists a u : [0,T 0] R such that its solution y is equal to the function η. u is simply 1/2 → defined by u = Q− (η0 + B(η, η)). If x is the solution to equation (5.22) with initial condition x0, then the function x y − satisfies equation (5.21) with initial condition w0 and control η, as a simple calculation shows. By uniqueness of the solution we have that x y equals w. As a consequence, we have that − (xT yT ) (x1 y1) /2. (5.23) | 0 − 0 − − | ≤ 5.9. ERGODICITY 89 Consider now a constant control u∗ in system (5.20), with initial condition (xT 0 , yT 0 ) at time T 0. We have, at any time T > T 0 T xT = xT + T T 0 Qu∗ B(xs, xs)ds 0 − − ZT 0 p T yT = yT + T T 0 Qu∗ B(ys, ys)ds 0 − − ZT 0 p and thus (xT , yT ) (x1, y1) | − | xT + T T 0 Qu∗ x1 + yT + T T 0 Qu∗ y1 ≤ 0 − − 0 − − T T p p + B(xs, xs) ds + B(ys, y s) ds. | | | | ZT 0 ZT 0 From (5.23) and the continuity of B we have (xT , yT ) (x1, y1) 2 xT + T T 0 Qu∗ x1 + /2 | − | ≤ 0 − − + C T T sup p x 2 + y 2 . B 0 t t − t [T 0,T ] | | | | ∈ We claim that we may choose T and u∗ so that T T 0 Qu∗ = x1 xT . (5.24) − − 0 and p (xT , yT ) (x1, y1) , | − | ≤ thus concluding the proof. Condition (5.24) implies that 2 2 (xT , yT ) (x1, y1) /2 + CB T T 0 sup xt + yt , | − | ≤ − t [T 0,T ] | | | | ∈ so it is suffi cient to prove that there is a constant C∗ > 0 such that for every choice of T T suffi ciently small and u so that (5.24) is true we have − 0 ∗ 2 2 sup xt + yt C∗. t [T ,T ] | | | | ≤ ∈ 0 This is true since condition (5.24) implies that for t [T ,T ] ∈ 0 t 2 xt D + CB xs ds | | ≤ | | ZT 0 90 CHAPTER 5. PHYSICAL MEASURES t 2 2 2 with D = xT + x1 xT ; if g(t) = D + CB xs ds, then g0(t) = xt g(t) , | 0 | | − 0 | T 0 | | | | ≤ and consequently g(T ) 2D if T 1/2D. Then there exists a T such that xt ≤ ≤ R | | ≤ 2( xT 0 + x1 xT 0 ) for every t [T 0,T ], and similarly for yt . | For| system| − (5.21)| we are able∈ to prove a stronger form| of| controllability. We consider some definitions and results of control theory which could be found in [31]. Suppose we are given a system of the form x˙ = f(x, η), x X Rn, with η varying in the set U of piecewise continuous controls. If f is suffi ciently∈ regular,⊂ the system has a unique solution η starting from an x0 X, which we denote by γ (x0). The orbit of x0 is the set ∈ t ηk η1 Orb(x0) = γt γt k 1, η1, ..., ηk U, t1, ..., tk R , { k ◦ · · · ◦ 1 | ≥ ∈ ∈ } while the reachable set from x0 is the subset of the orbit of x0 given by ηk η1 (x0) = γ γ k 1, η1, ..., ηk U, t1, ..., tk 0 . R { tk ◦ · · · ◦ t1 | ≥ ∈ ≥ } 1 The inclusion (x0) Orb(x0) holds if f is of class C in the first variable and continuous in theR second⊂ one ([31], proposition 4.4). Moreover, the two sets concide if the system verifies a condition of time reversibility (proposition 4.3). The condition which is considered by Jakubczyk is the following: the system is feedback time-reversible if there exists a function (x, η) v(x, η) U, which is continuous in η and of class C1 in x, such that 7→ ∈ f(x, η) = f(x, v(x, η)), (x, η) X U. − ∀ ∈ × The idea is that by using the controls v(x, η) we are able to move backwards in time along trajectories. If (x) = X for all x X, then we say that the system is controllable. We want to R ∈ show that system (5.21) is controllable on M, and this will conclude that system (5.19) is approximatively controllable. Call the smallest linear space of vectorc fields on Rn which contains the fields f( , u), with u LU, and which is closed under Lie brackets. The following theorem could be found· in [31] (theorem∈ 3.10 and proposition 4.5). Theorem 110 If X is connected, the system is time reversible, and the dimension dim (x) equals n for each x X, then (x) = X for any x X. L ∈ R ∈ The argument could be summarized as follows. The condition dim (x) = n must be L verified in a neighborhood V of x0, by regularity of f, and there should exist a control η1 η1 such that f(x0, η1) = 0 (otherwise dim (x0) = 0). The trajectory γt (x0) describes then a one dimensional submanifold6 of X. SupposeL now there exists a largest natural number k < n such that V contains a submanifold S of dimension k with parametrization of the ηk ηk 1 η1 form (t , ..., t ) γ γ − γ (x ). The assumption that dim (x) = n in all V , 1 k tk tk 1 t 0 and the fact that7→ the Lie◦ bracket− ◦ · · · of ◦ vector fields tangent to a submanifoldL is still tangent to this submanifold, give the existence of a point y S and a control η such that f(y, η ) ∈ 0 0 5.9. ERGODICITY 91 is not contained in the tangent space to S at y. This allow to find a k + 1-submanifold of X, by using the trajectories as for S, contradicting the assumption k < n. Accessibility from x0 implies the existence of a point x1 cointained in the reachable set (x0), together with a neighborhood W of x1. Feedback time-reversibility allows then to R bring W back in time, and prove that x0 is contained in the interior of (x0) (proposition 4.5, again in [31]). It follows that, if dim (x) = n for all x, each orbitR of the system is open, and we can conclude by connectednessL of X that (x) = X for all x. R Lemma 111 The auxiliary control system (5.21) is controllable on R3 0 . \{ } Proof. Our system is easily seen to be feedback time-reversible. Consider the function v(x, η) = x η. Then − f(x, v(x, u)) = B(x, x) B(x, x η) B(x η, x) − − − − − = B(x, x) + B(x, η) + B(η, x) = f(x, u). − Time-reversibility was not possible in presence of the linear term Ax˜ . To see that system 3 5.21 is accessible from any x R 0 , let us observe that contains the vectors vi = ∈ \{ } L B(w, ηi) + B(ηi, w), i = 1, 2, 3, with η1 = (1, 0, 0), η2 = (0, 1, 0), η3 = (0, 0, 1). The matrix 0 2w3 2w2 (v1 v2 v3) = w3 0 w1 | | − − w2 w1 0 − − has determinant d = 4w1w2w3, which means that the rank of is maximum at least where the polynomial d is not null. To verify that the condition for accessibilityL holds even in the zeros of d, we have to calculate some Lie brackets. Consider the vectors z1 = [v2, v3], z2 = [v1, v3], z3 = [v1, v2]; it can be proved that the matrices (v1 v2 z1), (v2 v3 z2), (v1 v3 z3) | | | | | | 2 2 2 have determinants respectively 8w2w , 4w w3 and 4w1w ; only the sets A1 = w2 = 3 1 2 { 0, w3 = 0 , A2 = w1 = 0, w3 = 0 and A3 = w1 = 0, w2 = 0 remain to be controlled. It } { } { } is suffi cient then to take h = [v1, [v2, v3]], k = [v2, [v1, v3]], and calculate the determinants of (v2 v3 k), (v1 v3 h), (v1 v2 h), | | | | | | 3 3 3 which are respectively 4w1 on A1, 8w2 on A2 and 8w3 on A3. We conclude with theorem (110). − − − 92 CHAPTER 5. PHYSICAL MEASURES Chapter 6 Appendix 6.1 Attractors for deterministic systems Attractors are those sets on which the orbits of a dynamical system settle as time goes to infinity. To give a rigorous definition, consider a continuous dynamical system (S(t))t R+ on a metric space (X, d), i.e. a family of continuous operators from X into itself, satisfying∈ the semigroup properties S(0) = Id, S(t + s) = S(t) S(s) for all s, t 0. We will denote by d(x, A) the distance between a point x X◦ and a subset A ≥X, given by ∈ ⊂ d(x, A) = infy A d(x, y). Recall that a subset A X is said to be invariant for the ∈ ⊂ dynamical system (S(t))t if S(t)A = A for all t T. Global orbits provide examples of invariant sets. Other examples are given by ω-limit∈sets: given a subset A of X, the ω-limit ω(A) of A is defined as ω(A) = S(t)A. s 0 t s \≥ [≥ A point x X belongs to ω(A) if and only if there exist a sequence (xn) of elements of A ∈ and a sequence of times tn going to infinity such that S(tn)xn x as n . → → ∞ Proposition 112 If A X is non-void, and there exists a t > 0 such that S(t)A 0 t t0 is relatively compact in X⊂, then ω(A) is non-empty, invariant and compact. ≥ S Proof. The set t s S(t)A is non-void for every s 0. For s t0, the sets t s S(t)A ≥ ≥ ≥ ≥ are closed sets contained in S(t)A, which is compact. They are then compact and S t t0 S their intersection ω(A) is non-void≥ and compact. To see that ω(A) is invariant, take y S(t)ω(A); then y = S(t)Sx for some x ω(A). As observed before, x is the limit of ∈ ∈ a sequence S(tn)xn, with xn in A, and then y is the limit of S(t + tn)xn, and belongs to ω(A). On the other hand, for a point x belonging to ω(A) take two sequence tn and xn as before. For tn t t0, we can extract subsequences tnk and xnk such that S(tnk t)xnk converges to a −y ≥H, as k . We have that y belongs to ω(A) and − ∈ → ∞ S(tn )xn = S(t)S(tn t)xn S(t)y as k , k k k − k → → ∞ 93 94 CHAPTER 6. APPENDIX and then x = S(t)y S(t)ω(A). If the condition∈ on the maps S(t) assumed in the proposition is satisfied for each bounded set A, the family (S(t))t is said to be eventually uniformly compact. The following definition describes what we mean with convergence to an attracting set. With d(B0,B1) we denote the semidistance between the two sets B0, B1 X, given by ⊂ d(B0,B1) = sup inf d(x, y). x B0 y B1 ∈ ∈ Definition 113 A set A attracts a bounded set B X if it attracts uniformly the points of B, i. e. ⊂ d(S(t)B,A) 0 as t . → → ∞ Definition 114 A subset A X is an attractor for the dynamical system (St)t R+ if ⊂ ∈ (i) A is invariant; (ii) there exists an open neighborhood U of A such that A attracts all bounded subsets of U. If A is an attractor, the biggest open set satisfying condition (ii) of the definition is called basin of attraction of A. If A is a compact attractor attracting all bounded subsets of X, then A is called the global or universal attractor for the dynamical system. In this case the basin of attraction is the whole X. Observe that if A is a global attractor, then it contains all bounded invariant subsets of X: if B is bounded and invariant, then d(B,A) = d(S(t)B,A) 0 d(B,A) = 0 B A. → ⇒ ⇒ ⊂ The global attractor must then be unique. The following definition is useful for establishing the existence of attractors. Definition 115 Let B be a subset of X, and U be an open set containing X. B is said to be absorbing in U if for each bounded subset B0 of U there exists a t0 such that S(t)B0 B ⊂ for each t t0. ≥ Systems possessing a compact set which absorbs all bounded subsets of X are called dissipative. The existence of a global attractor A implies the existence of an absorbing set for the whole X: take > 0 and consider the set V = x AB(x, ). For each bounded subset ∪ ∈ B of X, the convergence of d(S(t)B,A) to zero implies the existence of a tB such that d(S(t)B,A) /2 for all t tB, and V is an absorbing set. ≤ ≥ 6.1. ATTRACTORS FOR DETERMINISTIC SYSTEMS 95 Theorem 116 Let the family (S(t))t be eventually uniformly compact. Let U be an open subset of X, and let B be absorbing in U. Then the ω-limit A of B is a compact attractor, attracting the bounded subsets of U. Proof. Proposition (112) implies that A is non-void, invariant and compact. We prove that A attracts the bounded subsets of U. Suppose B0 is a bounded subset of U such that the convergence d(S(t)B,A) does not hold. Then there exist a δ > 0 and a sequence tn such that → ∞ d(S(tn)B,A) δ > 0 n. ≥ ∀ This implies the existence of a sequence (bn) of points in B0 such that d(S(tn)bn,A) δ/2 > 0 n. ≥ ∀ B is absorbing in U, and then S(tn)bn belongs to B for tn suffi ciently big, say tn t. The ≥ sequence (S(tn)bn) admits a converging subsequence (S(tnk )bnk ). Let x be the limit of this subsequence. Then x = lim S(tn )bn = lim S(tn t)S(t)bn ω(B) = A, k k k k k − k ∈ →∞ →∞ which gives a contradiction. Example 117 The Lorenz system. The dynamical system generated by the equations dx = ax + ay dt − dy = bx y xz dt − − dz = xy cz dt − with a, b, c > 0, is dissipative, and then possesses a global attractor. We show that theorem 116 applies by proving that there exists a ball centered at (0, 0, a + b) which is an absorbing set for all bouded subset of R3. Consider the function V (x, y, z) = x2 + y2 + (z a b)2, − − and its derivative with respect to time along the orbits of the system. We have dV (x(t), y(t), z(t)) dx dy dz = 2x(t) + 2y(t) + 2(z(t) a b) dt dt dt − − dt = 2ax(t)2 2y(t)2 2cz(t)2 + 2acz(t) + 2bcz(t) − − − = 2ax(t)2 2y(t)2 c(z(t) a b)2 − − − − − cz(t)2 + c(a + b)2. − By taking α = min 2a, 2, c we find { } dV αV + c(a + b)2, dt ≤ − and we can conclude with the Gronwall lemma. 96 CHAPTER 6. APPENDIX The dynamical system associated to the Navier-Stokes equations (2.17), with the boudary conditions described in paragraph (2.5) possesses a global attractor, whose Hausdorff di- mension is bounded by a constant which depends on the physics on the fluid ([?]). 6.2 Kernels Given two measurable spaces (E, ) and (F, ), consider a family µ = (µx)x E of prob- ability measures on (F, ), indexedE on E. If,F for any bounded measurable function∈ f on F (F, ), the map µ(f): E R defined by F → µ(f)(x) = f(y)µx(dy) ZF is measurable, then µ is called a (Markov) kernel from (E, ) to (F, ). Equivalently, µ is E F a kernel if for any A , the map x µx(A) is measurable. If is a sub-σ-algebra∈ F of , one7→ says that µ is measurable with respect to (resp. independentG from) if, for anyE bounded measurable function f on (F, ), the map µ(f) is -measurable (resp.G -independent). F GIf P is a measure onG(E, ), one can associate to the kernel µ its mean with rispect to P , i.e. the measure P µ on (F,E ) given by F P µ(A) = µx(A)P (dx), for all A . ∈ F ZE From this definition one gains that, for any A , the identity ∈ F (P µ)(A) = P [µIA] holds. By approssimating bounded measurable functions with simple ones, and applying dominated convergence theorem, one proves the duality relation for each bounded f on (F, ): F P µ[f] = P [µ(f)]. Proposition 118 If µ = (µx)x E is a kernel from a measurable space (E, ) to another ∈ E measurable space (F, ), then, for any bounded measurable function h : E F R, the map F × → x h(x, y)µx(dy) 7→ ZF is measurable. Proof. Let be the space of bounded measurable function h : E F R for which the thesis holds.H Then is a monotone vector space, containing constant× → functions and H characteristics functions IA of sets A . One conclude with a classical monotone class argument. ∈ E ⊗ F 6.2. KERNELS 97 Proposition 119 Given two measurable spaces (E, ) and (F, ), let µ = (µx)x E be kernel from (E, ) to (F, ), and let be a sub-σ-algebraE of F. If h is any bounded∈ measurable functionE on E F F , one has:G E × (i) if µ is -independent, and x f(x, y) is -measurable for all y F , then, for all A G 7→ G ∈ ∈ G P (dx) µx(dy)f(x, y) = P (dx) P µ(dy)f(x, y), ZA ZF ZA ZF or equivalently P f( , y)µ (dy) = f( , y)P µ(dy); (6.1) F · · G F · Z Z (ii) if µ is -independent, and x f(x, y) is -measurable for all x X, then, for all A G 7→ G ∈ ∈ G P (dx) µx(dy)f(x, y) = P (dx) µx(dy) f(z, y)P (dz), ZA ZF ZA ZF ZE or equivalently P f( , y)µ (dy) = µ (dy) f(z, y)P (dz). (6.2) F · · G F · E Z Z Z In both cases P (dx) µx(dy)f(x, y) = P µ(dy) P (dx)f(x, y). (6.3) ZE ZF ZF ZE Proof. Let be the set of bounded measurable functions h : E F R satisfying the theses of (i).H We first prove that contains the characteristic functions× h→of subsets of E F H × of the form A B, with A , B . One can write F f(x, y)µx(dy) = IA(x)µIB(x), and then, by taking× the conditional∈ E expextation∈ F with respect to , using the -independence R of µ and the -measurability of h G G G P [IA(µIB) ] = IAP [µIB] = IA(P µ)IB = IA B( , y)P µ(dy). |G × · ZF Conclusion follows again by recurring to monotone class theorems. For part (ii), a similar argument holds: by taking h = IA B, with A , B , one obtains × ∈ E ∈ F P f( , y)µ (dy) = µIBP [IA] = µx(dy) P (dz)IA(y)IB(z). F · · G F E Z Z Z Equality (6.3) follows by passing to the mean value in both (6.1) and (6.2). 98 CHAPTER 6. APPENDIX 6.3 Markov semigroups Let (X, d) be a Polish space, with Borel σ-algebra , and denote by Bb(X) and Cb(X) respectively the set of bounded measurable functionsB and bounded continuous functions on X. Let T = R+. A family (Pt)t T of markovian kernels from (X, ) to itself is a (markovian) transition function if the following∈ two conditions are satisfied:B (1) P0(x, B) = δx(B), for all x X, B ; ∈ ∈ B (2) Ps+t(x, B) = Ps(y, B)Pt(x, dy), for all s, t T, B . X ∈ ∈ B R Any transition function (Pt)t T defines a semigroup of linear operators on Bb(X), still ∈ denoted by Pt, by the formula Ptf(x) = f(y)Pt(x, dy); ZX it is called the Markovian semigroup associated to the transition function Pt. Properties (1) and (2) can be rewritten as P0 = I (identity on Bb(X)),Pt+s = Pt Ps. ◦ A Markovian semigroup (Pt)t T is said to be stochastically continuous if, for each ∈ f Cb(X) ∈ lim Ptf = f. t 0 → We call (Pt)t T a Feller semigroup if, for each t 0 and each bounded and continuous ∈ ≥ function ϕ on X, we have Ptϕ Cb(X), and a Strong Feller semigroup at a time t0 > 0, if ∈ Pt0 ϕ is bounded and continuous whenever ϕ is bounded and measurable. By the semigroup property, we have that for each t > t0 Ptf = Pt0 (Pt t0 f) − and then if Pt is Strong Feller at t0, than it is Strong Feller at each time t t0.A ≥ Markovian semigroup Pt is irreducible at a time t0 > 0 if for all x X and every open set A ∈ ∈ B Pt0 (x, A) > 0. Again, if Pt is irreducible at t0, then it is irreducible at each time t > t0. This result is due to Khas’minskii: Proposition 120 If (Pt)t T is Strong Feller at time t0 > 0 and irreducible at time s0 > 0, ∈ then all probabilities Pt +s (x, ), x X are mutually equivalent. 0 0 · ∈ 6.4. EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES 99 Proof. Let x0 X and A be such that Pt +s (x0,A) > 0. We have to show that ∈ ∈ B 0 0 Pt +s (x, A) > 0, for each x X. We can write 0 0 ∈ Pt0+s0 (x0,A) = Ps0 (x0, dy)Pt0 (y, A) > 0, ZX and then there exists a y0 such that Pt0 (y0,A) > 0. As the semigroup is Strong Feller at t0, the function y Pt (y, A) is continuous, and there must be an r0 such that Pt (y, A) > 0 7→ 0 0 for each y B(y0, r0). This implies ∈ Pt +s (x, A) = Ps (x, dy)Pt (y, A) Ps (x, dy)Pt (y, A) > 0 0 0 0 0 ≥ 0 0 ZX ZB(y0,r0) for each x X, as Ps (x, B(y0, r0)) > 0. ∈ 0 If all probabilities Pz0 (x, ), x X, are mutually equivalent, then the semigroup (Pt)t T · ∈ ∈ is said to be regular at time z0. 6.4 Existence and uniqueness of invariant measures A probability measure λ on (X, ) is said to be invariant with respect to the semigroup B Pt if + λPt = λ for all t R . ∈ A set A is said to be invariant for Pt if ∈ B PtIA = IA. An invariant measure µ for Pt is said to be ergodic if for any invariant Borel set A either µ(A) = 0 or µ(A) = 1. Theorem 121 If µ is the unique invariant probability measure for Pt, then it is ergodic. Proof. Suppose µ is not ergodic, then there exists an invariant set A such that µ(A) = 0 = 1 µ(A). This allows to consider the measure λ defined by ∈ B 6 6 − µ(B A) λ(B) = ∩ , B . µ(A) ∀ ∈ B We show that λ is invariant. We have, for each B , ∈ B 1 λP (B) = λ(dx)P (x, B) = µ(dx)P (x, B) t t µ(A) t ZX ZA 1 1 c = µ(dx)Pt(x, B A) + µ(dx)Pt(x, B A ). µ(A) ∩ µ(A) ∩ ZA ZA 100 CHAPTER 6. APPENDIX Invariance of A implies Pt(x, A) = IA(x), and then c if x A, then Pt(x, A ) = 0, ∈ if x / A, then Pt(x, A) = 0. ∈ which gives respectively c µ(dx)Pt(x, B A ) = 0, µ(dx)Pt(x, B A) = 0. ∩ c ∩ ZA ZA We conclude that 1 λPt(B) = µ(dx)Pt(x, B A) µ(A) ∩ ZA 1 = µ(dx)Pt(x, B A) µ(A) ∩ ZX µ(B A) = ∩ = λ(B), µ(A) which contradicts the uniqueness of µ. The following results could be found in [18] (proposition 3.2.5): Lemma 122 Let X be a Polish space, and let (Pt)t 0 be a stochastically continuous ≥ Markovian semigroup. If µ and ν are ergodic measures for (Pt)t 0, then either µ = ν or µ and ν are singular. ≥ Theorem 123 Let (Pt)t 0 be a stochastically continuous semigroup on a Polish space ≥ (X, ). It (Pt)t 0 is t0-regular for some t0 > 0, then (Pt)t 0 has at most one invariant measure.B ≥ ≥ Proof. If µ is an invariant measure for Pt, then for every A ∈ B µ(A) = Pt(x, A)µ(dx) for all t > 0. (6.4) ZX Then, since Pt is t0-regular, if Pt (x, A) = 0 then Pt (y, A) = 0 for all y X, and µ(A) = 0. 0 0 ∈ On the other hand, if µ(A) = 0, then by (6.4) we have that Pt0 (x, A) = 0 for µ-almost every x X, and the regularity of Pt implies that Pt (x, A) = 0 for all x X. We have proved ∈ 0 ∈ that µ is equivalent to Pt (x, ). Moreover, if A is an invariant set for Pt such that µ(A) > 0, 0 · then from Pt IA = IA we have that Pt (x, A) = 1 if x A. Since all probabilities Pt (x, ) 0 0 ∈ 0 · are equivalent, we have Pt0 (x, A) = 1 for all x X, and then µ(A) = 1 by invariance (6.4), and µ is ergodic. ∈ To show that µ is unique, let ν be another invariant measure; then µ and ν are ergodic and equivalent, and so they coincide by lemma (122). A classical argument for proving existence of invariant measures is the so called Krylov- Bogoliubov method: 6.4. EXISTENCE AND UNIQUENESS OF INVARIANT MEASURES 101 Theorem 124 Let (Pt)t 0 be a Feller semigroup on a Polish space (X, ). Assume that ≥ B there exists a probability measure ν such that the family of measures (µT )T >0, defined by 1 T µT [f] = νPs[f]ds for each f Cb(X), T ∈ Z0 is tight. Then µ is an invariant measure for (Pt)t 0. ≥ Proof. By the Prokhorov theorem, there exists a sequence Tn and a probability 1 Tn → ∞ measure µ such that µT [f] = νPs[f]ds converges weakly to µ as n . Then if n Tn 0 → ∞ f Cb(X), as (Pt)t 0 is Feller we have ∈ ≥ R 1 Tn (µPt)[f] = µ[Ptf] = lim µT [Ptf] = lim νPs[Ptf]ds n n n T →∞ →∞ n Z0 1 Tn 1 Tn = lim (νPs)Pt[f]ds = lim (νPs+t)[f]ds n T n T →∞ n Z0 →∞ n Z0 1 Tn+t = lim (νPs)[f]ds n T →∞ n Zt 1 Tn = lim (νPs)[f]ds n T →∞ n Z0 1 t 1 Tn+t lim (νPs)[f]ds + lim (νPs)[f]ds. − n T n T →∞ n Z0 →∞ n ZTn The last two terms go to zero, and (µPt)[f] = µ[f]. The hypothesis of the previous theorem is verified in particular if the family (Pt(x0, ))t · is tight for some x0 X. ∈ Theorem 125 Let V : X [0, + ] be a Borel function such that the sets → ∞ KR = x X : V (x) R { ∈ ≤ } are compact for each R > 0. Suppose that there exists a point x0 X and a C(x0) > 0 such that ∈ V (y)Pt(x0, dy) C(x0) for all t 0. ≤ ≥ ZX Then there exists an invariant measures for (Pt)t. Proof. For all R > 0 we have C 1 C(x0) Pt(x0,KR ) = Pt(x0, dy) Pt(x0, dy)V (y) , V >R ≤ R X ≤ R Z{ } Z and the conclusion follows from the previous theorem. A function V as in the last theorem is called a Lyapunov function for the Markov semigroup. 102 CHAPTER 6. APPENDIX 6.5 Stationary Markov processes Let (Ω, ,P ) be a probability space. A Definition 126 Let (Pt)t R+ be a Markov transition function on a Polish space (X, ).A ∈ B Markov process associated to (Pt)t R+ , with respect to a filtration ( t)t R+ , is a stochastic ∈ F ∈ process (Xt)t R+ with values in (X, ), satisfying: ∈ B + E[IA(Xs+t) s] = (PtIA) Xs, for all s, t R ,A . |F ◦ ∈ ∈ B Observe that, if (Xt)t R+ is a Markov process with transition semigroup (Pt)t, and ν is ∈ the law of X0, then the law of Xt equals νPt. Moreover, we have that, for each ϕ Bb(X) ∈ E[ϕ(Xs+t) s] = Ptϕ(Xs). |F A Markov process (Xt)t R+ is said to be stationary if for every n-tuple 0 t1 ... tn the law of the random variable∈ ≤ ≤ ≤ (Xt1+h,...,Xtn+h) equals the law of (Xt1 ,...,Xtn ) for all h > 0. Proposition 127 Suppose that (Xt)t R+ is a Markov process, and µ is an invariant mea- ∈ sure for the associated transition semigroup (Pt)t. If µ is the law of X0, then (Xt)t R+ is stationary. ∈ Proof. We have to prove that, for each n-tuple 0 t1 ... tn, each h > 0 and each ≤ ≤ ≤ n-tuple ϕ1, ..., ϕn Bb(X) the expectation ∈ E[ϕ1(Xt +h) ϕn(Xt +h)] 1 ··· n does not depend on h. We proceed by induction on n. If n equals 1, then the thesis follows from the fact that the law of Xt+h equals µPt+h, which is independent of h by the invariance of µ. For the inductive step, it is suffi cient to observe that E[ϕ1(Xt1+h) ϕn(Xtn+h)] = E[E[ϕ1(Xt1+h) ϕn(Xtn+h) tn 1+h]] ··· ··· |F − = E[ϕ1(Xt1+h) ϕn 1(Xtn 1+h)E[ϕn(Xtn+h) tn 1+h]] ··· − − |F − = E[ϕ1(Xt1+h) ϕn 1(Xtn 1+h)Ptn tn 1 ϕn(Xtn 1+h)]. ··· − − − − − Next theorem is known as the law of large numbers for stationary processes. It can be found in [18] (theorem 3.3.1). Theorem 128 Suppose that (Xt)t R+ is a stationary Markov process, and the law µ of ∈ Xt is ergodic for the transition semigroup (Pt)t associated to the process. Then for each ϕ L2(X, µ) ∈ 1 t lim ϕ(Xs)ds = ϕdµ P -almost surely. t t →∞ Z0 ZX 6.5. STATIONARY MARKOV PROCESSES 103 The Ornstein-Uhlenbeck process Consider a stochastic differential equation in Rn of the form dZt + AZt = QdWt, (6.5) where A is symmetric n n matrix satisfyingpAx, x λ x 2, with λ > 0, and Q is a positive semidefinite symmetric× n n matrix. Theh solutioni ≥ | of| this equation with initial × condition Zt0 = x is given by t t0,x (t t0)A (t s)A Zt = e− − x + e− − QdW (s). t0 Z p It can be easily found by a stochastic version of variation of constant method. Moreover, (t t0)A the law of the solution is gaussian with mean e− − x and covariance operator Qt given by t (t s)A (t s)A Qt0,t = e− − Qe− − ds. Zt0 The associated Markov semigroup is given by P ϕ(x) = ϕ(y)N (t t )A (dy). t e 0 x,Qt ,t n − − 0 ZR tA λt The hypothesys on A imply that the norm of e− is bounded by e− . 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