Exponential attractors for random dynamical systems and applications
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Abstract
The paper is devoted to constructing a random exponential attractor for some classes of stochastic PDE’s. We first prove the existence of an exponential attractor for abstract random dynamical systems and study its dependence on a parameter and then apply these results to a nonlinear reaction–diffusion system with a random perturbation. We show, in particular, that the attractors can be constructed in such a way that the symmetric distance between the attractors for stochastic and deterministic problems goes to zero with the amplitude of the random perturbation.
Keywords
Random exponential attractors Stochastic PDE’s Reactiondiffusion equationAMS subject classifications
35B41 35K57 35R60 60H15Introduction
The theory of attractors for partial differential equations (PDE’s) has been developed intensively since late seventies of the last century. It is by now well known that many autonomous dissipative PDE’s possess an attractor, even if the Cauchy problem is not known to be well posed. Moreover, one can establish explicit upper and lower bounds for the dimension of an attractor. A comprehensive presentation of the theory of attractors can be found in [3, 13, 33].
The situation becomes more complicated when dealing with nonautonomous dissipative systems. In that case, there are at least two natural ways to extend the concept of an attractor. The first one is based on the reduction of the nonautonomous dynamical system (DS) in question to the autonomous one and leads to the attractor which is independent of time and attracts the images of bounded sets uniformly with respect to time shifts. It is usually called a uniform attractor. A drawback of this approach is that the attractor is often huge (infinitedimensional), even in the case when the DS considered has trivial dynamics, with a single exponentially stable trajectory; see [13] and the references therein for details.
An alternative approach treats the attractor for a nonautonomous system as a family of timedepending subsets obtained by the restriction of all bounded trajectories to all possible times. In that case, the resulting object is usually finitedimensional (as in the autonomous case), but the attraction becomes nonuniform with respect to time shifts. Moreover, as a rule, the attraction forward in time is no longer true, and one has only the attraction property pullback in time, so those objects are called pullback attractors (or kernel sections in the terminology of Vishik and Chepyzhov); see the books [9, 13] and the literature cited there.
The theory of attractors can also be extended to the case of random dynamical systems (RDS) mainly based on the concept of a pullback attractor. Various results similar to the deterministic case were obtained for many RDS generated by stochastic PDE’s, such as the Navier–Stokes system or reaction–diffusion equations with random perturbations. The situation is even slightly better here since, in contrast to general nonautonomous deterministic DS, in the case of RDS one usually has forward attraction property in probability; see [4, 5]. Moreover, if the random dynamics is Markovian and mixing, then a minimal random attractor in probability can be described as the support of the disintegration for the unique Markovian invariant measure of the extended DS corresponding to the problem in question; see [25].
However, there is an intrinsic drawback of the theory of attractors; namely, the rate of attraction to the (global, uniform, pullback) attractor can be arbitrarily slow and there is no way, in general, to express or to estimate this rate of convergence in terms of physical parameters of the system under study. As a consequence, the attractor is also very sensitive to perturbations which makes it, in a sense, unobservable in experiments and numerical simulations.
This drawback can be overcome using the concept of an inertial manifold (IM) instead. This is an invariant finitedimensional manifold of the phase space which contains the attractor and possesses the socalled exponential tracking property (i.e., every trajectory of the considered DS is attracted exponentially to a trajectory on the manifold). The rate of attraction can be estimated in terms of physical parameters, and the manifold itself is robust with respect to perturbations; see [22, 33] and references therein. Moreover, the construction can be extended to the case of nonautonomous and random DS and the resulting inertial manifold resolves also the problem with the lack of forward attraction: under some natural assumptions, the rate of exponential attraction to the nonautonomous/random inertial manifold is uniform with respect to time shifts; see [2, 7, 8, 10, 12] and the literature cited there.
Unfortunately, being a kind of center manifold, an IM requires a separation of the phase space to “fast” and “slow” variables. This leads, in turn, to very restrictive spectral gap conditions which are violated for many interesting applications, including the 2D Navier–Stokes system, reaction–diffusion equations in higher dimensions, damped wave equations, etc. In addition, when a stochastic dissipative PDE is considered, e.g., with an additive white noise, to guarantee the existence of the IM, one should impose an additional condition that all nonlinear terms are globally Lipschitz continuous.
To overcome these restrictive assumptions, an intermediate (between the IM and attractors) object, socalled exponential attractor (or inertial set), was introduced in [16] for the case of autonomous DS. This is a semiinvariant finitedimensional set (but not necessarily a manifold) which contains the attractor and possesses the exponential attraction property, like an IM. Moreover, the rate of attraction is controlled, which leads to some stability under perturbations. The initial construction of an exponential attractor given in [16] was restricted to the case of Hilbert phase spaces only and involved the Zorn lemma. A relatively simple and effective explicit construction of this object was suggested later in [17], and as believed nowadays, the exponential attractors are almost as general as the usual ones and no restrictive or artificial assumptions are required for their existence; see the survey [31] and the references therein.
An extension of the theory of exponential attractors theory to the case of nonautonomous DS (including the robustness) was given in [19] (see also [18, 30] for the socalled uniform exponential attractors and [28] for a slight extension of the result of [19]). As shown there, a nonautonomous exponential attractor remains finitedimensional as a pullback attractor, but attracts the images of bounded sets uniformly with respect to time shifts as a uniform attractor. Thus, like an IM, a nonautonomous exponential attractor contains the pullback one and possesses the forward attraction property, but in contrast to the IM, no restrictive spectral gap assumptions are required.
The aim of the present paper is to extend the theory of exponential attractors to the case of dissipative RDS. Although the theory of random attractors is often similar to the nonautonomous deterministic one and our study is also strongly based on the construction given in [17, 19], there is a fundamental difference between the two cases. Namely, in contrast to the deterministic case considered in [19], a typical trajectory of an RDS is unbounded in time. For instance, this is the case for a dissipative stochastic PDE with an additive white noise. Thus, if we do not impose the restrictive assumption on the global Lipschitz continuity of all nonlinear terms, then all the constants in appropriate squeezing/smoothing properties (which play a key role in the construction of an exponential attractor) will depend on time (in other words, will be random), and a straightforward extension does not work. However, some time averages of these quantities can be controlled, and this turns out to be sufficient for constructing an exponential attractor.
Theorem A
There is a random compact set \(\mathcal{M }_\omega \subset H\) and an event \(\Omega _*\subset \Omega \) of full measure such that the following properties hold for \(\omega \in \Omega _*\).
Semiinvariance: \(\varphi _t^\omega (\mathcal{M }_\omega )\subset \mathcal{M }_{\theta _t\omega }\) for all \(t\ge 0\).
Finitedimensionality: There is a number \(d>0\) such that \(\dim _f(\mathcal{M }_\omega )\le d\), where \(\dim _f\) stands for the fractal dimension of \(\mathcal{M }_\omega \).
Let us now assume that the random force \(\eta \) in Eq. (1.1) is replaced by \({\varepsilon }\eta \), where \({\varepsilon }\in [1,1]\) is a parameter. We denote by \(\mathcal{M }_\omega ^{\varepsilon }\) the corresponding exponential attractors. Since in the limit case \({\varepsilon }=0\) the equation is no longer stochastic, the corresponding attractor \(\mathcal{M }=\mathcal{M }^0\) is also independent of \(\omega \). A natural question is whether one can construct \(\mathcal{M }_\omega ^{\varepsilon }\) in such a way that the symmetric distance between the attractors of stochastic and deterministic equations goes to zero as \({\varepsilon }\rightarrow 0\). The following theorem gives a positive answer to that question.
Theorem B
We refer the reader to Sect. 4 for more precise statements of the results on the existence of exponential attractors and their dependence on a parameter. Let us note that various results similar to Theorem B were established earlier in the case of deterministic PDE’s; e.g., see the papers [19, 20], the first of which is devoted to studying the behaviour of exponential attractors under singular perturbations, while the second deals with nonautonomous DS and proves Hölder continuous dependence of the exponential attractor on a parameter.
In conclusion, let us mention that some results similar to those described above hold for other stochastic PDE’s, including the 2D Navier–Stokes system. They will be considered in a subsequent publication.
The paper is organised as follows. In Sect. 2, we present some preliminaries on RDS and a reaction–diffusion equation perturbed by a spatially regular white noise. Section 3 is devoted to some general results on the existence of exponential attractors and their dependence on a parameter. In Sect. 4, we apply our abstract construction to the stochastic reaction–diffusion system (1.1–1.3). Appendix gathers some results on coverings of random compact sets and their image under random mappings, as well as the timeregularity of stochastic processes.
Notation
We shall use the following function spaces:
\(L^p=L^p(D)\) denotes the usual Lebesgue space in \(D\) endowed with the standard norm \(\Vert \cdot \Vert _{L^p}\). In the case \(p=2\), we omit the subscript from the notation of the norm. We shall write \(L^p(D,\mathbb{R }^k)\) if we need to emphasise the range of functions.
\(W^{s,p}=W^{s,p}(D)\) stands for the standard Sobolev space with a norm \(\Vert \cdot \Vert _{s,p}\). In the case \(p=2\), we write \(H^s=H^s(D)\) and \(\Vert \cdot \Vert _s\), respectively. We denote by \(H_0^s=H^s_0(D)\) the closure in \(H^s\) of the space of infinitely smooth functions with compact support.
\(C(J,X)\) stands for the space of continuous functions \(f:J\rightarrow X\).
When describing a property involving a random parameter \(\omega \), we shall assume that it holds almost surely, unless specified otherwise. Furthermore, when dealing with a property depending on \(\omega \) and an additional parameter \(y\in Y\), we say that it holds almost surely for \(y\in Y\) if there is a set of full measure \(\Omega _*\subset \Omega \) such that the property is true for \(\omega \in \Omega _*\) and \(y\in Y\).
We denote by \(c_i\) and \(C_i\) unessential positive constants not depending on other parameters.
Preliminaries
Random dynamical systems and their attractors

Measurability The mapping \((t,\omega ,u)\mapsto \varphi _t^\omega (u)\) from \(\mathbb{R }_+\times \Omega \times X\) to \(X\) is measurable with respect to the \(\sigma \)algebras \(\mathcal{B }_{\mathbb{R }_+}\otimes \mathcal{F }\otimes \mathcal{B }_X\) and \(\mathcal{B }_X\).
 Perfect cocycle property For almost every \(\omega \in \Omega \), we have the identity$$\begin{aligned} \varphi _{t+s}^\omega =\varphi _t^{\theta _s\omega }\circ \varphi _s^\omega , \quad t,s\ge 0. \end{aligned}$$(2.1)

Time regularity For almost every \(\omega \in \Omega \), the function \((t,\tau )\mapsto \varphi _t^{\theta _{\tau }\omega }(u)\), defined on \(\mathbb{R }_+\times \mathbb{R }\) with range in \(X\), is Höldercontinuous with some deterministic exponent \(\gamma >0\), uniformly with respect to \(u\in \mathcal{K }\) for any compact subset \(\mathcal{K }\subset X\).
Largetime asymptotics of trajectories for RDS is often described in terms of attractors. This paper deals with random exponential attractors, and we now define some basic concepts.
Definition 2.1

Semiinvariance For any \(t\ge 0\), we have \(\varphi _t^\omega (\mathcal{M }_\omega )\subset \mathcal{M }_{\theta _t\omega }\).
 Exponential attraction There is a constant \(\beta >0\) such thatwhere \(B\subset H\) is an arbitrary ball and \(C(B)\) is a constant that depends only on \(B\).$$\begin{aligned} d\bigl (\varphi _t^\omega (B),\mathcal{M }_{\theta _t\omega }\bigr ) \le C(B) e^{\beta t} \quad \text{ for} t\ge 0, \end{aligned}$$(2.2)
 Finitedimensionality There is random variable \(d_\omega \ge 0\) which is finite on \(\Omega _*\) such that$$\begin{aligned} \dim _f\bigl (\mathcal{M }_\omega \bigr )\le d_\omega . \end{aligned}$$(2.3)

Time continuity The function \(t\mapsto d^s\bigl (\mathcal{M }_{\theta _t\omega },\mathcal{M }_\omega \bigr )\) is Höldercontinuous on \(\mathbb{R }\) with some exponent \(\delta >0\).
Reaction–diffusion system perturbed by white noise
Let us denote \(H=L^2(D,\mathbb{R }^k)\) and \(V=H_0^1(D,\mathbb{R }^k)\). The following result on the wellposedness of problem (1.1–1.3) can be established by standard methods used in the theory of stochastic PDE’s (e.g., see [15, 21]).
Theorem 2.2
 Regularity: Almost every trajectory of \(u(t)\) belongs to the space$$\begin{aligned} \mathcal{X }=C(\mathbb{R }_+,H)\cap L_{\mathrm{loc}}^2(\mathbb{R }_+,V)\cap L_{\mathrm{loc}}^{p+1}(\mathbb{R }_+\times D). \end{aligned}$$
 Solution: With probability 1, we have the relationwhere the equality holds in the space \(H^{1}(D)\).$$\begin{aligned} u(t)=u_0+\int \limits _0^t\bigl (a\Delta uf(u)+h\bigr )\,ds+\zeta (t), \quad t\ge 0, \end{aligned}$$
The family of solutions for (1.1), (1.2) constructed in Theorem 2.2 form an RDS in the space \(H\). Let us describe in more detail a set of full measure on which the perfect cocycle property and the Höldercontinuity in time are true.
Abstract results on exponential attractors
Exponential attractor for discretetime RDS
Let \(H\) be a Hilbert space and let \({\varvec{\varPsi }}=\{\psi _k^\omega ,k\in \mathbb{Z }_+\}\) be a discretetime RDS in \(H\) over a group of measurepreserving transformations \(\{\sigma _k\}\) acting on a probability space \((\Omega ,\mathcal{F },\mathbb{P })\). We shall assume that \({\varvec{\varPsi }}\) satisfies the following condition.
Condition 3.1

Absorption The family \(\{\mathcal{A }_\omega \}\) is a random absorbing set for \({\varvec{\varPsi }}\).
 Stability With probability 1, we have$$\begin{aligned} \psi _1^\omega \bigl (\mathcal{O }_r(\mathcal{A }_\omega )\bigr )\subset \mathcal{A }_{\sigma _1\omega }. \end{aligned}$$(3.1)
 Lipschitz continuity There is an almost surely finite random variable \(K_\omega \ge 1\) such that \(K^m\in L^1(\Omega ,\mathbb{P })\) and$$\begin{aligned} \Vert \psi _1^\omega (u_1)\psi _1^\omega (u_2)\Vert _V\le K_\omega \Vert u_1u_2\Vert _H \quad \text{ for} u_1,u_2\in \mathcal{O }_r(\mathcal{A }_\omega ). \end{aligned}$$(3.2)
 Kolmogorov \({\varepsilon }\) entropy. There is a constant \(C\) and an almost surely finite random variable \(C_\omega \) such that \(C_\omega K_\omega ^m\in L^1(\Omega ,\mathbb{P })\),$$\begin{aligned} \mathcal{H }_{\varepsilon }(V,H)&\le C\,{\varepsilon }^{m},\end{aligned}$$(3.3)$$\begin{aligned} \mathcal{H }_{\varepsilon }(\mathcal{A }_\omega ,H)&\le C_\omega {\varepsilon }^{m}. \end{aligned}$$(3.4)
The following theorem is an analogue for RDS of a wellknown result on the existence of an exponential attractor for deterministic dynamical systems; e.g., see Sect. 3 of the paper [31] and the references therein.
Theorem 3.2
Proof
We repeat the scheme used in the case of deterministic dynamical systems. However, an essential difference is that we have a random parameter and need to follow the dependence on it. In addition, the constants entering various inequalities are now (unbounded) random variables, and we shall need to apply the Birkhoff ergodic theorem to bound some key quantities.
Step 3: Estimation of the fractal dimension. We shall need the following lemma, whose proof is given at the end of this subsection.
Lemma 3.3
Remark 3.4
It follows from (3.26) that if the random variable \(\xi _\omega \) entering the Birkhoff theorem is bounded [see (3.22)], then the fractal dimension of \(\mathcal{M }_\omega \) can be bounded by a deterministic constant. For instance, if the group of shift operators \(\{\sigma _k\}\) is ergodic, then \(\xi _\omega \) is constant, and the conclusion holds. This observation will be important in applications of Theorem 3.2.
Proof of Lemma 3.3
The cocycle property (2.1) and inclusion (3.1) imply that \(\psi _m^{\sigma _{km}\omega }(\mathcal{A }_{\sigma _{km}\omega })\supset \psi _l^{\sigma _{kl}\omega }(\mathcal{A }_{\sigma _{kl}\omega })\) for \(m\le l\). Hence, it suffices to establish (3.17) for \(m=l\).
Dependence of attractors on a parameter
We now turn to the case in which the RDS in question depends on a parameter. Namely, let \(Y\subset \mathbb{R }\) and \(\mathcal{T }\subset \mathbb{R }\) be bounded closed intervals. We consider a discretetime RDS \({\varvec{\varPsi }}^y=\{\psi _k^{y,\omega }:H\rightarrow H, k\ge 0\}\) depending on the parameter \(y\in Y\) and a family^{2} of measurable isomorphisms \(\{\theta _\tau :\Omega \rightarrow \Omega ,\tau \in \mathcal{T }\}\). We assume that \(\theta _\tau \) commutes with \(\sigma _1\) for any \(\tau \in \mathcal{T }\), and the following uniform version of Condition 3.1 is satisfied.
Condition 3.5
 Absorption and continuity For any ball \(B\subset H\) there is a time \(T(B)\ge 0\) such thatwhere we set \(\mathcal{A }_\omega ^y=B_V(R_\omega ^y)\). Moreover, there is an integrable random variable \(L_\omega \ge 1\) such that$$\begin{aligned} \psi _k^{y,\theta _\tau \omega }(B)\subset \mathcal{A }_\omega ^y\quad \text{ for}\, k\ge T(B), y\in Y, \tau \in \mathcal{T }, \omega \in \Omega , \end{aligned}$$(3.30)for \(y_1,y_2\in Y, \tau _1,\tau _2\in \mathcal{T }\), and \(\omega \in \Omega \).$$\begin{aligned} R_{\theta _{\tau _1}\omega }^{y_1}R_{\theta _{\tau _2}\omega }^{y_2} \le L_\omega \bigl (y_1y_2^\alpha +\tau _1\tau _2^\alpha \bigr ) \end{aligned}$$(3.31)
 Stability With probability 1, we have$$\begin{aligned} \psi _1^{y,\omega }\bigl (\mathcal{O }_r(\mathcal{A }_\omega ^y)\bigr ) \subset \mathcal{A }_{\sigma _1\omega }^y\quad \text{ for} y\in Y. \end{aligned}$$(3.32)
 Hölder continuity There are almost surely finite random variables \(K_\omega ^y,K_\omega \ge 1\) such that \(K_\omega ^y\le K_\omega \) for all \(y\in Y, (RK)^m\in L^1(\Omega ,\mathbb{P })\), andfor \(y_1,y_2\in Y, \tau _1,\tau _2\in \mathcal{T }, u_1,u_2\in \mathcal{O }_r(\mathcal{A }_\omega ^{y_1}\cup \mathcal{A }_\omega ^{y_2})\), and \(\omega \in \Omega \), where we set \(K_\omega ^{y_1,y_2}=\max (K_\omega ^{y_1},K_\omega ^{y_2})\).$$\begin{aligned} \Vert \psi _1^{y_1,\theta _{\tau _1}\omega }(u_1)\!\!\psi _1^{y_2, \theta _{\tau _2}\omega }(u_2)\Vert _V \le K_\omega ^{y_1,y_2} \bigl (y_1\!\!y_2^\alpha \!+\!\tau _1\!\!\tau _2^\alpha \!+\!\Vert u_1u_2\Vert _H\bigr )\nonumber \\ \end{aligned}$$(3.33)

Kolmogorov \({\varepsilon }\) entropy Inequalities (3.3) holds with some \(C\) not depending on \({\varepsilon }\).
In particular, for any fixed \(y\in Y\), the RDS \({\varvec{\varPsi }}^y\) satisfies Condition 3.1 and, hence, possesses an exponential attractor \(\mathcal{M }_\omega ^y\). The following result is a refinement of Theorem 3.2.
Theorem 3.6
Let \({\varvec{\varPsi }}^y\) be a family of RDS satisfying Condition 3.5. Then there is a random compact set \((y,\omega )\mapsto \mathcal{M }_\omega ^y\) with the underlying space \(Y\times \Omega \) and a set of full measure \(\Omega _*\in \mathcal{F }\) such that the following properties hold.
In addition, it can be shown that all the moments of the random variables \(P_\omega \) and \(Q_\omega \) are finite. The proof of this property requires some estimates for the rate of convergence in the Birkhoff ergodic theorem. Those estimates can be derived from exponential bounds for the time averages of some norms of solutions. Since the corresponding argument is technically rather complicated, we shall confine ourselves to the proof of the result stated above.
Proof of Theorem 3.6
We now turn to the property of Hölder continuity for \(\mathcal{M }_\omega ^y\). Inequalities (3.35) and (3.36) are proved by similar arguments, and therefore we give a detailed proof for the first of them and confine ourselves to the scheme of the proof for the other. Inequality (3.35) is established in four steps.
As in the case of Theorem 3.2, inequality (3.34) holds for \(B=\mathcal{A }_\omega \) with \(C(B)=r\). Furthermore, if the group of shift operators \(\{\sigma _k\}\) is ergodic, then the Hölder exponent in (3.35) is a deterministic constant (cf. Remark 3.4). Finally, if \(\psi _k^{y,\omega }, R_\omega ^y\), and \(K_\omega ^y\) do not depend on \(\omega \) for some \(y=y_0\in Y\), then the exponential attractor \(\mathcal{M }_\omega ^{y_0}\) constructed in the above theorem is also independent of \(\omega \). Indeed, \(\mathcal{M }_\omega ^{y}\) was defined in terms of \(\psi _k^{y,\omega }, R_\omega ^y, K_\omega ^y\) and the random finite sets \(U_k^y(\omega )\) that form \(\delta \)nets for the random compact sets \(\mathcal{C }_k^y(\omega )\). As is mentioned after the proof of Lemma 5.5, these \(\delta \)nets are independent of \(\omega \) if so are the random compact sets to be covered. Using this observation, it is easy to prove by recurrence that \(\mathcal{C }_k^{y_0}(\omega )\) and \(U_k^{y_0}(\omega )\) do not depend on \(\omega \), and therefore the same property is true for the attractor \(\mathcal{M }_\omega ^{y_0}\).
Exponential attractor for continuoustime RDS
We now turn to a construction of an exponential attractor for continuoustime RDS. Let us fix a bounded closed interval \(Y\subset \mathbb{R }\) and consider a family of RDS \({\varvec{\varPhi }}^y=\{\varphi _t^{y,\omega }:H\rightarrow H, t\ge 0\}, y\in Y\). We shall always assume that the associated group of shift operators \(\theta _t:\Omega \rightarrow \Omega \) satisfies the following condition.
Condition 3.7
The discretetime dynamical system \(\{\theta _{k \tau _0}:\Omega \rightarrow \Omega , k\in \mathbb{Z }\}\) is ergodic for any \(\tau _0>0\).
Theorem 3.8
Proof
Step 5: Time continuity. Since mapping (3.51) is Hölder continuous, the required inequality (3.53) will be established if we prove that (3.53) is true for \(\widetilde{\mathcal{M }}_\omega ^y\). However, this is an immediate consequence inequalities (3.35), (3.36) and the ergodicity of the group of shift operators \(\{\sigma _k\}\). The proof of the theorem is complete.
As in the case of discretetime RDS, if \(\varphi _t^{y,\omega }, R_\omega ^y\), and the random objects entering Condition 3.5 do not depend on \(\omega \) for some \(y_0\), then the exponential attractor \(\mathcal{M }_\omega ^{y_0}\) is also independent of \(\omega \). This fact follows immediately from representation (3.54), because \(\varphi _{\tau }^{y_0,\theta _{\tau }\omega }\) and \(\widetilde{\mathcal{M }}_{\theta _{\tau }\omega }^{y_0}\) do not depend on \(\omega \).
Application to a reaction–diffusion system
Formulation of the main result
Theorem 4.1
To prove this result, we shall apply Theorem 3.8. For the reader’s convenience, let us describe briefly the conditions we need to check, postponing their verification to the next subsection.
Recall that \(H=L^2, V=H_0^1\), and the probability space \((\Omega ,\mathcal{F },\mathbb{P })\) and the corresponding group of shits operators \(\theta _t\) were defined in Sect. 2.2. The ergodicity of the restriction of \(\{\theta _t\}\) to any lattice \(\tau _0\mathbb{Z }\) is well known (see Condition 3.7), and the Kolmogorov \({\varepsilon }\)entropy of a unit ball in \(V\) regarded as a subset in \(H\) can be estimated by \(C{\varepsilon }^{n}\), where \(n\) is the space dimension (see the fourth item of Condition 3.5). We shall prove that the following properties are true for a sufficiently large \(\tau _0>0\).
We shall also prove that the random variables \(R_\omega ^0\) and \(K_\omega ^0\) are constants. If these properties are established, then all the hypotheses of Theorem 3.8 are fulfilled, and its application to the RDS associated with problem (4.1), (1.2) gives the conclusions of Theorem 4.1.
Proof of Theorem 4.1
Appendix
Coverings for random compact sets
In this section, we have gathered three auxiliary results on coverings of random compact sets by balls centred at the points of random finite sets. The first of them establishes the existence of a “minimal” covering with an explicit bound of the number of balls in terms of the Kolmogorov \({\varepsilon }\)entropy of the random compact set in question.
Lemma 5.1
Proof
The second result shows that, if a random compact set depends on a parameter in a Lipschitz manner, then the random finite set constructed above can be chosen to have a similar dependence on the parameter. To prove it, we shall need the following auxiliary construction.
Proposition 5.2
The proof given below will imply that if \(\mathcal{A }_\omega ^y\) does not depend on \(\omega \) for some \(y=y_0\), then the random set \(U_{\delta ,y}(\omega )\) satisfying (5.10–5.12) can be chosen in such a way that \(U_{\delta ,y_0}(\omega )\) is also independent of \(\omega \). Furthermore, if \(\mathcal{A }_\omega ^y\) does not depend on \(\omega \) for all \(y\in Y\), then \(U_{\delta ,y}\) is also independent of \(\omega \). The latter observation implies the following corollary used in the main text.
Corollary 5.3
To prove this result, it suffices to apply Proposition 5.2 to the nonrandom compact set \(B_V(R)\) depending on the parameter \(R\in \mathbb{R }_+\).
Proof of Proposition 5.2
Without loss of generality, we assume that the random variable \(C\) is constant, since one can represent \(\Omega \) as the union of the subsets \(\Omega _{l}=\{\omega \in \Omega : l\le C<l+1\}\) and construct required random finite sets on each \(\Omega _{l}\).
Lemma 5.4
And, finally, our third result refines Proposition 5.2 in a particular case.
Lemma 5.5
Proof
As is clear from the proof, if \(V^y(\omega )\) does not depend on \(\omega \) for some \(y=y_0\), then the random set \(U_{\delta ,y_0}(\omega )\) constructed in Lemma 5.5 is also independent of \(\omega \).
Proof of Lemma 5.4
Image of random compact sets
Proposition 5.6
Let \(X\) and \(Y\) be Polish spaces, let \((\Omega ,\mathcal{F })\) be a measurable space, let \(\{\mathcal{K }_\omega ,\omega \in \Omega \}\) be a random compact set in \(X\), and let \(\psi _\omega :X\rightarrow Y\) be a family of continuous mappings such that, for any \(u\in X\), the mapping \(\omega \mapsto \psi _\omega (u)\) is measurable from \(\Omega \) to \(Y\). Then \(\{\psi _\omega (\mathcal{K }_\omega ), \omega \in \Omega \}\) is a random compact set in \(Y\).
Proof
Kolmogorov–Čentsov theorem
The Kolmogorov–Čentsov theorem provides a sufficient condition for Höldercontinuity of trajectories of a random process. We shall need the following qualitative version of that result, which is a particular case^{6} of Theorem 1.4.4 in [27].
Theorem 5.7
Let us emphasise that we assume from the very beginning the continuity of almost all trajectories of \(\xi _t\), so that we do not need to modify our process.
Sketch of the proof
Proposition 5.8
Proof
Footnotes
 1.
In the case \(n=1\), the lefthand side of this inequality is zero.
 2.
This family of isomorphisms is needed to describe the regularity of dependence of random objects on \(\omega \). In the next subsection, when dealing with continuoustime RDS, we shall take for \(\theta _\tau \) the underlying group of measurepreserving transformations.
 3.
For instance, see Proposition 2.4.10 in [26] for the more complicated case of the Navier–Stokes system.
 4.
To have an absorbing set, one could take for \(R_\omega ^{{\varepsilon },2}\) the integral of \(\Vert {\varepsilon }\,U^\omega (\sigma +\tau _0)\Vert _2^{p+1}\) in \(\sigma \in [3,0]\). However, in this case the stability condition may not hold, and therefore we define \(R_\omega ^{{\varepsilon },2}\) in a different way. Our choice ensures that (4.21) holds for the radius of the absorbing ball.
 5.
We thank A. Iftimovici for the simple geometric argument proving Lemma 5.4.
 6.
Notes
Acknowledgments
We thank the anonymous referees for pertinent critical remarks which helped to improve the presentation and to remove some inaccuracies. This work was supported by the Royal Society–CNRS grant Long time behavior of solutions for stochastic Navier–Stokes equations (No. YFDRN93583). The first author was supported by the ANR grant STOSYMAP (ANR 2011 BS01 015 01).
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