Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise✩ Tom´as Caraballo, Xiaoying Han, Bj¨orn Schmalfuss, Jos´e Valero aDepto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla Apdo. de Correos 1160, 41080-Sevilla, Spain bDepartment of Mathematics and Statistics 221 Parker Hall, Auburn University Auburn, AL 36849, USA cInstitut f¨ur Mathematik Institut f¨ur Stochastik, Ernst Abbe Platz 2, 07737 Jena,Germany dCentro de Investigaci´on Operativa, Universidad Miguel Hern´andez, Avda. de la Universidad, s/n, 03202-Elche, Spain Abstract In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems. Key words: stochastic lattice differential equations, random dynamical systems, random attractors, multiplicative noise 2010 MSC: 60H15, 34C35, 58F11, 58F15, 58F36 ✩This work has been partially supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grants MTM2011-22411 and MTM2012-31698, and by Junta de Andaluc´ıa under Proyecto de Excelencia P12-FQM-1492. Email addresses: [email protected] (Tom´as Caraballo), [email protected] (Xiaoying Han), [email protected] (Bj¨orn Schmalfuss), [email protected] (Jos´e Valero) Preprint submitted to Elsevier January 14, 2016 1. Introduction It is well known that stochastic lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character and uncertainties or random influences, called noises, are taken into account. The lattice differential equations have been used to model systems such as cellular neural networks with applications to image processing, pattern recognition, and brain science (see [1] and references therein). They are also used to model the propagation of pulses in myelinated axons where the membrane is excitable only at spatially discrete sites, in which case uirepresents the potential at the i-th active site (see e.g., [2, 3]). Lattice differential equations can also be found in chemical reaction theory (see e.g., [4, 5]). In the absence of noise, many works have been done on various aspects of solutions to deterministic lattice dynamical systems. We refer the readers to [6, 7] and references therein for traveling waves, and [8, 9] and references therein for the chaotic properties of solutions. In this paper we will investigate the long term behavior for the following stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and a different multiplicative white noise at each node: dui(t) = [ρ(ui−1−2ui+ui+1)−fi(ui) + gi]dt +σiui◦dwi(t), i ∈Z,(1.1) where Zdenotes the integer set, u= (ui)i∈Z∈ℓ2:= {(ui)i∈Z:Pi∈Zu2 i<∞}, g= (gi)i∈Z∈ℓ2, ρ and σiare positive constants, fiis a smooth function satisfying proper dissipative conditions, and wi’s are mutually independent Brownian motions. Here ◦denotes the Stratonovich sense of the stochastic term. The study of global random attractors was initiated by Ruelle [10]. The fundamental theory of global random attractors for stochastic partial differential equations was developed by Crauel and Flandoli [11], Flandoli and Schmalfuss [12], Imkeller and Schmalfuss [13], Schmalfuss [14] amongst others. Due to the unbounded fluctuations in the systems caused by the white noise, the concept of global pullback/random attractor was introduced to capture the essential dynamics with possibly extremely wide fluctuations. This is significantly different from the deterministic case. The existence of a global attractor for the deterministic counterpart of (1.1) was established in [15]. For stochastic lattice dynamical systems with additive or multiplicative noise, the existence of global random attractors has been intensively analyzed in the recent literature (see e.g., Bates et al. [16], Caraballo et al. [17, 18], Caraballo and Lu [19], Han [20], Han et al. [21], amongst others). We emphasize that in the studies of stochastic lattice systems with multiplicative noise up to date, only a finite number of Wiener process is considered in each equation, being the same in all the equations, while the multiplicative noise 2 considered here in Equation (1.1) is different at each node. This can be the result of an environmental effect on the whole domain of the system, either on each equation of the lattice (additive noise case) [16] or in some parameters of the model (multiplicative noise case). To be more precise, and focus on the multiplicative case, let us recall that in some previous lattice models analyzed in the existing literature, the term which is responsible for the dissipative character of the problem is sometimes split into two terms, one of which is linear. For that reason, the deterministic counterpart of our model (1.1) is given as (see, e.g. Caraballo and Lu [19]) dui(t) = [ρ(ui−1−2ui+ui+1)−λiui−˜ fi(ui) + gi]dt, i ∈Z. Assuming now that the environmental effect produces a perturbation in the parameter λiin such a way that it becomes λi−σi˙wiat each node i∈Z, then our stochastically perturbed lattice becomes dui(t) = [ρ(ui−1−2ui+ui+1)−λiui−˜ fi(ui) + gi]dt +σiui◦dwi(t), i ∈Z, which is precisely the model we are interested in, if we denote fi(ui) = λiui+ ˜ fi(ui). In the present paper we will prove the existence of a global random attractor for the infinite dimensional random dynamical system generated by the stochastic lattice differential equation (1.1). An interesting feature of this structure is that, even though the spatial domain is unbounded and the solution operator is not smooth or compact, unlike parabolic type of partial differential equations on bounded domains, bounded sets of initial data converge in the pullback sense, under the forward flow to a random compact invariant set. It is worth mentioning again that the noise involved in the system is multiplicative, and more importantly, different at each node. We put emphasis again on this comment because, to the best of our knowledge, such systems have not been considered in the literature although they are fully justified by physical intuitions. More precisely, all the papers published on this topic to date consider at most a finite sum of noise in each node which means that the noise term is essentially the same at all the nodes. In previous studies in the literature, the stochastic systems are first rewritten as a stochastic differential equation in the Hilbert space ℓ2, then a suitable change of variable, involving usually an Ornstein-Uhlenbeck process, is performed in order to transform the stochastic equation into a random differential equation in ℓ2. This has subsequently become a standard way of formalization (see e.g. [19, 18, 21]). However, due to the appearance of the infinitely many noise terms, this scheme cannot be applied to handle our problem and hence a new methodology is ought to be developed. Unlike the previously described technique, we will perform a change of variable first to transform the stochastic lattice system (1.1) into a random lattice system. We then prove that the random dynamical system generated by the 3 resulting random lattice system possesses a global random attractor. The existence of global random attractors for the original stochastic lattice system can be obtained once it can be shown that the random dynamical system generated by the transformed equation is conjugated with the original one in the same space of sequences ℓ2. After the transformation, we need to formulate the random lattice system as an abstract evolution equation in a Gelfand triple of Hilbert spaces formed by sequences. Instead of carrying out our analysis working only for our particular random lattice system, we will first develop an abstract theory for the existence of weak solutions1to general random differential equations defined in a Gelfand triple of Hilbert spaces. Then we will apply it to our lattice model as a particular example. For this reason the goal of this paper is two-fold: 1. proving a general theorem on the existence and uniqueness of weak solutions for random differential equations in Hilbert spaces; 2. proving, as a special application, that our stochastic lattice model (1.1) generates, after a suitable change of variable, a random dynamical system which possesses a global random attractor. It is worth mentioning that the existence of the global random attractor for (1.1) is done by proving that the random dynamical system generated by (1.1) possesses a random compact absorbing set and taking benefit of the compact injections of the sequence of Hilbert spaces. In this way we avoid the calculation of uniform estimates on the tails of the solutions, as it has been done in all previous published works on this topic in order to prove either asymptotic compactness [19] or asymptotic nullness [21]. The rest of the paper is organized as follows. In Section 2, we introduce basic concepts concerning random dynamical systems and global random attractors. In Section 3, we perform a transformation which allows us to rewrite our lattice system as a random lattice system without white noise, which can be eventually written as an evolution equation in some appropriate Hilbert spaces. Section 4 is devoted to establishing the existence and uniqueness of weak solutions for general abstract random evolution equations, which will be used later to prove that (1.1) generates an infinite dimensional random dynamical system. The existence of the global random attractor for (1.1) is finally established in Section 5. Some closing remarks will be given in Section 6. 2. Preliminaries on random dynamical systems We recall in this section some of the basic concepts related to random dynamical systems and the concept of random attractors (see [23], [24] and [12] for more details). 1The term “weak solution” throughout this paper refers to the weak solution in the PDE (partial differential equation) context, instead of the weak solution in the SDE (stochastic differential equation) context (see, e.g. [22]). 4 Let (H, k·kH) be a separable Banach space and (Ω,F,P) a probability space. Definition 2.1. Ω,F,P,(θt)t∈Ris called a metric dynamical system, if (i) θ:R×Ω→Ωis (B(R)⊗ F,F)-measurable, (ii) θ0is the identity on Ω, (iii) θs+t=θt◦θsfor all s, t ∈R, (iv) θtP=Pfor all t∈R. Consider the probability space (Ω,F,P) = (C0,B(C0),P), where C0={ω∈ C(R,R) : ω(0) = 0} endowed with the compact open topology (see [23], Appendix A.2 and A.3), P is the corresponding Wiener measure on the Borel σ-algebra B(C0). Let θtω(·) = ω(·+t)−ω(t), t ∈R, then (C0,B(C0),P,(θt)t∈R) is a metric dynamical system. Definition 2.2. A stochastic process ϕ(t)is called a continuous random dynamical system over Ω,F,P,(θt)t∈Rif ϕis (B[0,∞)×F×B(H),B(H))- measurable, and for all ω∈Ω (i) the mapping ϕ(t, ω, ·) : H→His continuous for (t, ω)∈R+×Ω, (ii) ϕ(0, ω, ·)is the identity on H, (iii) ϕ(s+t, ω, ·) = ϕ(t, θsω, ·)◦ϕ(s, ω, ·)for all s, t ≥0(cocycle property). It is known that finite dimensional stochastic differential equations generate random dynamical systems (see Arnold [23] Chapter 1), but this is not necessarily true in general for infinite dimensional equations. However, for particular types of noise, as in our case, this can be achieved applying a suitable change of variable. Definition 2.3. A set-valued map A: Ω →2H\ ∅,ω7→ A(ω), where A(ω) is closed for all ω∈Ω, is called a random set if for each x∈Hthe map ω7→dist(x, A(ω)) is measurable. Definition 2.4. A random bounded set B(ω)⊂His called tempered with respect to (θt)t∈Rif for ω∈Ω lim t→±∞ log+d(B(θ−tω)) |t|= 0 , where d(B) = sup x∈B kxkH. Now consider a continuous random dynamical system ϕover (Ω,F,P,(θt)t∈R) and let Ddenote the collection of random tempered sets in H. 5 Definition 2.5. A random set K∈ D is called an absorbing set in Dif for B∈ D and ω∈Ωthere exists tB(ω)>0such that ϕ(t, θ−tω, B(θ−tω)) ⊂K(ω)for all t≥tB(ω). Definition 2.6. A random set Ais called a global random Dattractor (pullback Dattractor)for ϕif the following hold: (A1) A ∈ D,A(ω)is compact set for ω∈Ω; (A2) Ais strictly invariant, i.e. for ω∈Ωand all t≥0it holds ϕ(t, ω, A(ω)) = A(θtω); (A3) Aattracts all sets in D, i.e., for all B∈ D and a.e. ω∈Ωit holds lim t→∞ d(ϕ(t, θ−tω, B(θ−tω)),A(ω)) = 0, where d(X, Y ) = sup x∈X inf y∈Ykx−ykHis the Hausdorff semi-metric (here X⊂H, Y ⊂H). The collection Dis usually called the domain of attraction of A. Often it is enough to assume that the items (A1) - (A3) only hold for ω∈˜ Ω, where ˜ Ω∈ F is of full Pmeasure and (θt)t∈R–invariant, i.e., θt˜ Ω = ˜ Ω for all t∈R. The next proposition is an abstract result on the existence of global random attractor for a continuous random dynamical system. Although there are different versions of sufficient conditions ensuring the existence of global random attractors, the following one fits our purpose in this work, see [12]. Proposition 2.7. Let ϕ(t)be a continuous random dynamical system over (Ω, F,P,(θt)t∈R). Suppose that ϕ(t)has a random absorbing set K∈ D,K(ω) compact for ω∈Ω, then ϕpossesses a D-random attractor Agiven by A(ω) = \ τ≥0[ t≥τ ϕ(t, θ−tω, K(θ−tω)). Moreover, Ais unique in D. 3. Mathematical preparation Our goal is to study the global random attractor for the random dynamical system generated by (1.1). To this end we will first transform the stochastic lattice equation (1.1) containing white noise terms into a random lattice equation without white noise terms but with random coefficients, which can be written as a random evolution equation eventually. We will then investigate the random attractor for the resulting random evolution equation. 6 A finite version of system (1.1) as follows dui(t) dt =ρ(ui−1−2ui+ui+1)−fi(ui) + gi+ N X j=1 σjui◦dwj(t) dt , i ∈Z(3.1) has been analyzed in [19], for a finite sum of noises in each node, which in principle is equivalent to the system with the same noise at all nodes. A similar version to (3.1) was later studied in [21] by using weighted spaces. Note that in [19], [21] and any other existing work of stochastic lattice systems with single or a finite sum of noise at all nodes, the system can be reformulated as a stochastic equation in the (regular or weighted) space of infinite sequences, before the change of variable process transforming the stochastic equation into a random equation. However, due to the infinite number of noises in this problem, we have to perform the change of variables to (1.1) before formulating it as an evolution equation in appropriate spaces of infinite sequences. In doing so, we will obtain some operators which depend now on the random variable ωand which are not defined on the whole ℓ2, their domains being a subspace of ℓ2. As a consequence, the techniques used in [19], [21] and other existing works are not applicable to our problem in this work. At this point, we will proceed with the transformation from a stochastic to a random system in a heuristic way without imposing any specific assumptions on the functions fi, giin (1.1). Later we will investigate appropriate conditions on fiand gito ensure the existence of a global random attractor. Let P= (C0,B(C0),P) be a Wiener space. We consider the product space (Ω,F,P) := ΠiPdefined in the usual way. In particular, because C0is a Polish space, Fis generated by the product topology of C0(see Kallenberg [25], Page 2). Note that Ω is a Fr´echet–space where the convergence in this space is the component–wise convergence (point–wise convergence). Let us extend the Wiener–shift from R× C0to R×Ω by θtω= (··· , θtωi,···),Ω∋ω= (ωi)i∈Z. Note that t7→ θtωiis continuous for any ωi∈ C0, ωi7→ θtωiis continuous for any t∈R, which gives the continuity (measurability) of the mappings θt·on Ω with respect to the metric of the Fr´echet–space and θ·ωon R. We obtain by Aliprantis and Border [26, Lemma 4.51, Page 153] the measurability of θ: (R×Ω,B(R×Ω)) = (R×Ω,B(R)⊗ F)→(Ω,F). The measures Pobtained by the projections of Pto B(C0) are still θ–ergodic. 7 For each i∈Zwe consider zi(θtω) = −σiZ0 −∞ eτθtωi(τ)dτ, t ∈R, ω ∈Ω. Note that each zi(θtω) is an Ornstein-Uhlenbeck process on (Ω,F,P,(θt)t∈R) and solves the following Langevin equation (see, e.g., Caraballo et al. [27]) dzi=−zidt +σidwi(t).(3.2) where wi(t)(ω) = wi(t, ω) = ωi(t) for any ω∈Ω and t∈R. Moreover, by a similar statement to that provided in [27] we have the following result. Lemma 3.1. There exists a {θt}t∈R-invariant subset ˜ Ω∈ F of Ωof full measure such that for any ω∈˜ Ω 1. For each i∈Z, lim t→±∞ |ωi(t)| t= 0; 2. For each i∈Zthe random variable given by zi(ω) := −σiZ0 −∞ eτωi(τ)dτ is well defined, and the mapping (t, ω)→zi(θtω) = −σiZ0 −∞ eτθtωi(τ)dτ =−σiZ0 −∞ eτωi(t+τ)dτ +σiωi(t) is a stationary solution of (3.2) with continuous trajectories; 3. For each i∈Z lim t→±∞ |zi(θtω)| |t|= 0, lim t→±∞ 1 tZt 0 zi(θτω)dτ = 0, lim t→±∞ 1 tZt 0 |zi(θτω)|dτ =E|zi|<∞. Remark 3.2. In the sequel we will consider θdefined as in (2) on ˜ Ωinstead of Ω. This mapping possesses the same properties as the original one if we choose for Fthe trace σ-algebra with respect to ˜ Ωdenoted also by F. For each ui(i∈Z) satisfying (1.1), we perform the following change of variables vi(t) = e−zi(θtω)ui(t), i ∈Z. 8 Then the equations for vi(i∈Z) read dvi dt =ρezi−1(θtω)−zi(θtω)vi−1−2vi+ezi+1(θtω)−zi(θtω)vi+1(3.3) +zi(θtω)vi+gie−zi(θtω)−e−zi(θtω)fi(ezi(θtω)vi), with initial condition vi(0) = v0 i∈Rsuch that v(0) := (v0 i)i∈Z=ν0∈ℓ2. Our goal is to justify that the lattice equations (3.3) can be reformulated as an abstract random evolution equation over an Gelfand evolution triplet V⊂H⊂V′, where His a separable Hilbert space, Va separable Banach space with dense and compact embedding V⊂H, and V′denotes the dual space of V. For this purpose we will formulate an abstract random evolution equation in the next section, such that the conditions we impose for the lattice-system (3.3) can be fit to the coefficients of this abstract system. We will then show the existence of weak solutions to the evolution equation, and prove in addition that this solution generates a random dynamical system under certain assumptions. Remark 3.3. It is nontrivial to prove that the transformation from (1.1) to (3.3) is a conjugation for random dynamical systems. According to the existing results in the literature, to prove that the transformation generates conjugated random dynamical systems, we should verify assumptions such as in Caraballo and Lu [19, Lemma 2.3]. In particular, we need to check that the mapping T(·,·) : Ω ×ℓ2→ℓ2defined as T(ω, u) = (e−zi(ω)ui)i∈Z is a homeomorphism in ℓ2for each fixed ω∈Ω. At this point, without proving the conjugation property, we will consider a new technique based on Gelfand triple, which allows us to prove the existence of random attractors for equation (3.3). 4. Weak solutions to general random evolution equations In this section, we will state and prove an existence theorem for weak solutions to general random evolution equations with particular type of operators and study the random dynamical systems generated by the weak solutions. Let Hbe a separable Hilbert space equipped with the inner product (·,·) and the norm k · k. Let Vbe a dense subspace of Hwith the inner product (·,·)Vand the norm k·kV, and assume that Vis given a topological vector space structure for which the inclusion map is continuous. Then the triplet (V, H, V ′) forms a rigged Hilbert space where V′is the dual space of Vequipped with the norm k · kV′. Let h·,·i denote the duality map between Vand V′. Then the duality map is compatible with the inner product on H, in the sense that hu, vi= (u, v),∀u∈V⊂H, v ∈H=H′⊂V′. 9 To fit the lattice equation (3.3) in the framework of the random evolution equation (4.3), let us define the operator A1by A1= 2ρIdH, where IdHis the identity operator of H. Define the operator Γ by (Γ(ω)v)i=−ρezi−1(ω)−zi(ω)vi−1+ezi+1(ω)−zi(ω)vi+1, v = (vi)i∈Z.(5.1) Remark 5.1. Notice that Γdoes not generate a bounded linear operator from H to H. Hence the theory of Banach-space valued ordinary differential equations does not apply here. We will use the theory of weak solutions to evolution equations containing unbounded operators to analyze system (3.3). The following lemma provides some properties of the linear operator Γ. Lemma 5.2. There exists a (θt)t∈R- invariant set ˜ Ω∈ F of full measure such that the operator Γ(ω)defined in (5.1) can be considered as a linear operator Γ(ω) : H→V′and satisfies EkΓ(ω)k2 L(H,V ′)<∞. Moreover, the mapping t7→ kΓ(θtω)kL(H,V ′)∈L2 loc(R)and is measurable. In addition ω7→ Γ(ω)vis measurable for all v∈V(or v∈H). Proof. For any v∈V, kΓ(ω)vk2 V′=ρ2X i∈Z λ−1 iezi−1−zivi−1+ezi+1−zivi+12 ≤2ρ2X i∈Z λ−1 ie2(zi−1−zi)v2 i−1+e2(zi+1−zi)v2 i+1 ≤2ρ2kvk2 H·sup i∈Z e2(zi−zi+1) λi + sup i∈Z e2(zi−zi−1) λi, which implies that kΓ(ω)kL(H,V ′)≤2ρ2·sup i∈Z e2(zi−zi+1) λi + sup i∈Z e2(zi−zi−1) λi. Observe that if ziis N(0, σ2) distributed then 2(zi−zi−1) and 2(zi−zi+1) are N(0,8σ2) distributed, and hence E[e2(zi−zi±1)]<∞. Also note that by Pi∈Zλ−1 i<∞, we have Esup i∈Z e2(zi−zi±1) λi ≤X i∈Z Ee2(zi−zi±1) λi <∞, which implies that EkΓ(ω)k2 L(H,V ′)<∞. 16 At the same time we obtain lim t→±∞ 1 tZt 0 sup i∈Z e2(zi(θsω)−zi±1(θsω)) λi ds =Esup i∈Z e2(zi(ω)−zi±1(ω)) λi <∞, i.e., t7→ kΓ(θtω)kL(H,V ′)∈L2 loc(R). It remains to show that Γ is measurable. In fact, for any v= (vi)i∈Z∈H, we have that ω7→ (Γ(ω)v)iǫiis measurable and in addition X i∈Z ((Γ(ω)v)iǫi)2 λi <∞. This implies the measurability of Γ. Let A2:V→V′be defined by (A2u)i=λiui, i ∈Z. Then equation (3.3) can be cast into the format of (4.3): dv(t) dt +Av(t) = F(θtω, v(t)) + G(θtω),(5.2) where A=A1+A2, A1= 2ρIdH,(A2v) = (λivi)i∈Z, F(ω, v) = −Γ(ω)v+λivi+zi(ω)vi−e−zi(ω)fi(ezi(ω)vi)i∈N ,(5.3) G(ω) = gie−zi(ω)i∈N .(5.4) We will next verify that the operator Fas defined in (5.3) satisfies assumptions (F1) - (F3) proposed in Section 4 with α= 1, and Gas defined in (5.4) satisfies assumption (F4). To this end, we make the following standing assumptions on function fiand gi: (f0) fi:R→Ris continuous for each i∈Z. (f1) There exists β= (βi)i∈Z∈Hsuch that fi(s)2≤λis2+β2 i,∀s∈R. (f2) There exists γ= (γi)i∈Z∈Vsuch that sfi(s)≥ −γ2 i+3λi 4s2,∀s∈R. (f3) For any ε > 0, there exists a constant l > 0 such that −(fi(s1)−fi(s2))(s1−s2) + 2ελi(s1−s2)2≤l(s1−s2)2,∀s1, s2∈R. 17 (f4) g= (gi)i∈Z∈H. First, for any v∈V, we have kF(ω, v)k2 V′=X i∈Z 1 λiρ(ezi−1−zivi−1+ezi+1−zivi+1) +λivi+zi(ω)vi−e−zi(ω)fi(ezi(ω)vi)2 ≤4kΓ(ω)vk2 V′+ 4 X i∈Z λiv2 i+X i∈Z 4 λi z2 i(ω)v2 i +X i∈Z 4 λi e−2zi(ω)f2 i(ezi(ω)vi), for which we have the following estimates of each term on the RHS: kΓ(ω)vk2 V′≤ kΓ(ω)k2 L(H,V ′)kvk2 H; (5.5) X i∈Z 4 λi z2 i(ω)v2 i≤4kvk2 H·sup i∈Z |zi(ω)|2 λi .(5.6) Due to assumption (f1) we have X i∈Z 4 λi e−2zi(ω)f2 i(ezi(ω)vi)≤4X i∈Z v2 i+X i∈Z 4 λi e−2zi(ω)β2 i ≤4kvk2 H+ 4 sup i∈Z e−2zi(ω) λi kβk2 H.(5.7) Let C1(ω) = 4 sup e−2zi(ω) λi kβk2 H C2(ω) = 4kΓ(ω)k2 L(H,V ′)+ 4 sup i∈Z |zi(ω)|2 λi + 4. Since zi(ω) and 2zi(ω) are Gauß random variables and the series of elements λ−1 iis convergent, we have that Pi∈Z e±2zi(ω) λi∈L1(Ω). Hence by the ergodic theorem and Lemma 5.2 we have t7→ Ci(θtω)∈L1 loc(R), i= 1,2, on a (θt)t∈R- invariant set of full measure. Collecting (5.5) - (5.7) we obtain the first inequality in (F2): kF(ω, v)k2 V′≤C1(ω) + C2(ω)kvk2 H+ 4kvk2 V. Let v∈L2(0, T ;V). Then by the continuity of fi, (t, ω)7→ e−zi(θtω)fi(ezi(θtω)vi(t)) ∈R 18 is measurable. Hence X i∈Z e−zi(θtωfi(ezi(θtω)vi(t, ω))ξi, ξ = (ξi)∈Z∈V, is measurable as a sum of measurable mappings. Similarly, the other terms forming hF(θtω, v(t)), ξiare also measurable. Thus (ω, t)7→ hF(θtω, v(t)), ξiis measurable for all v∈L2(0, T ;V) and ξ∈V. Also, kF(θtω, v)kV′is finite for almost all tand t7→ F(θtω, v(t)) ∈L2(0, T, V ′) for all ω∈Ω. In our particular case we can simply choose Rm,m∈N, as the finitedimensional spaces Hm. The complete orthonormal basis of Hgenerating Hm is given by ek=(ǫk 2, if kis even, ǫ−k−1 2, if kis odd. Let ξ∈ ∪m∈NHmand let (v(n))n∈Nbe a sequence such that v(n)→vstrongly in L2(0, T ;H) as n→ ∞. Then by the continuity of fiwe have that e−zi(θtω)fi(ezi(θtω), v(n) i(t)) →e−zi(θtω)fi(ezi(θtω), vi(t)), for all i,ωand a.a. t. It is clear that ξ∈Hmfor some m∈N. Consider, for example, that mis even. Then  m 2 X i=−m 2+1 e−zi(θtω)fi(ezi(θtω), v(n) i(t))ξi 2 ≤ m 2 X i=−m 2+1 e−2zi(θtω)f2 i(ezi(θtω), v(n) i(t)) m 2 X i=−m 2+1 ξ2 i ≤  m 2 X i=−m 2+1 λiv(n) i(t)2+ m X i=1 e−2zi(θtω)β2 i  m 2 X i=−m 2+1 ξ2 i ≤C1,m  v(n)(t)   2 H+ 1, for a.a. t∈[0, T ] and all n, where we have used condition (f1) and the fact that t7→ zi(θtω) are continuous functions. Since v(m)is convergent in L2(0, T ;H), due to Lebesgue’s theorem we deduce for every φ∈ C∞ 0(0, T ) that lim m→∞ ZT 0  m 2 X i=−m 2+1 e−zi(θtω)fi(ezi(θtω), v(m) i)ξi φ(t)dt =ZT 0  m 2 X i=−m 2+1 e−zi(θtω)fi(ezi(θtω), vi)ξi φ(t)dt. 19 Further, let us consider the term Γ (ω)v. We have that ρezi−1(θtω)−zi(θtω)v(n) i−1(t)−ezi+1(θtω)−zi(θtω)v(n) i+1 (t) −→ ρezi−1(θtω)−zi(θtω)vi−1(t)−ezi+1(θtω)−zi(θtω)vi+1 (t), for all i, ω and a.a. t. On the other hand,  m 2 X i=−m 2+1 ρezi−1(θtω)−zi(θtω)v(n) i−1(t)−ezi+1(θtω)−zi(θtω)v(n) i+1 (t)ξi 2 ≤2ρ2  m 2 X i=−m 2+1 e2(zi−1(θtω)−zi(θtω)) v(n) i−1(t)2 + m 2 X i=−m 2+1 e2(zi+1(θtω)−zi(θtω)) v(n) i+1 (t)2  m 2 X i=−m 2+1 ξ2 i ≤C2,m  v(n)(t)   2 H, for a.a. t∈[0, T ] and all n. Then as in the previous case we have lim n→∞ ZT 0 hΓ (ω)v(n), ξiφ(t)dt =ZT 0 hΓ (ω)v, ξiφ(t)dt for every φ∈ C∞ 0(0, T ). In a similar way, we obtain that lim n→∞ ZT 0 m 2 X i=−m 2+1 zi(θtω)v(n) i(t)ξiφ(t)dt =ZT 0 m 2 X i=−m 2+1 zi(θtω)vi(t)ξiφ(t)dt. Finally, v(n)→vin L2(0, T ;H) implies that for any φ∈ C∞ 0(0, T ), lim n→∞ ZT 0 m 2 X i=−m 2+1 λiv(n) i(t)ξiφ(t)dt = lim n→∞ ZT 0v(n)(t), ξVφ(t)dt =ZT 0 (v(t), ξ)Vφ(t)dt. Collecting all the terms of Fwe reach lim n→∞ ZT 0 hF(θtω, v(n)(t)), ξiφ(t)dt =ZT 0 hF(θtω, v(t)), ξiφ(t)dt, (5.8) for any φ∈ C∞ 0(0, T ), ξ ∈ ∪m∈NHm. 20 Hence, (F1) follows. Next for any v∈V, we have − hΓ(ω)v, vi ≤ kΓ(ω)vkV′kvkV ≤ kΓ(ω)kL(H,V ′)kvkHkvkV ≤2kΓ(ω)k2 L(H,V ′)kvk2 H+1 8kvk2 V; (5.9) X i∈Z zi(ω)v2 i=X i∈Z λ−1 2 izi(ω)viλ 1 2 ivi ≤2 sup i∈Z |zi(ω)|2 λi kvk2 H+1 8kvk2 V; (5.10) −X i∈Z e−zi(ω)fi(ezi(ω)vi)vi=−X i∈Z e−2zi(ω)fi(ezi(ω)vi)ezi(ω)vi ≤X i∈Z e−2zi(ω)γ2 i−3λi 4e2zi(ω)v2 i ≤ kγk2 VX i∈Z e−2zi(ω) λi −3 4kvk2 V.(5.11) Define K1(ω) := kγk2 VX i∈Z e−2zi(ω) λi , K2(ω) := 2 kΓ(ω)k2 L(H,V ′)+ sup i∈Z |zi(ω)|2 λi. By Lemma 5.2 and the ergodic theorem again we have t7→ Kj(θtω)∈ L1 loc(R) for j= 1,2 on a (θt)t∈R-invariant set of full measure. Collecting (5.9)- (5.11) we obtain hF(ω, v), vi=−hΓ(ω)v, vi+X i∈Zλiv2 i−zi(ω)v2 i−(e−zi(ω)fi(ezi(ω)vi)vi ≤K1(ω) + K2(ω)kvk2 H+1 2kvk2 V,∀v∈V. Hence, (F2) is proved. It remains to show that Fsatisfies (F3). In fact, for any x, y ∈Vwe have − hΓ(ω)(x−y),(x−y)i ≤ 1 4εkΓ(ω)k2 L(H,V ′)kx−yk2 H+εkx−yk2 V,(5.12) X i∈Z zi(ω)(xi−yi)2≤1 4εsup i∈Z |zi(ω)|2 λi kx−yk2 H+εkx−yk2 V,(5.13) 21 and according to (f3) we have −X i∈Z e−zi(ω)fi(ezi(ω)xi)−fi(ezi(ω)yi)(xi−yi) ≤X i∈Z [l−2ελi](xi−yi)2 ≤lkx−yk2 H−2εkx−yk2 V.(5.14) Define M(ω) := l+1 4εkΓ(ω)k2 L(H,V ′)+ sup i∈Z |zi(ω)|2 λi. Then M(ω)∈L1(Ω) and, by the ergodic theorem and Lemma 5.2, we have t7→ M(θtω)∈L1 loc(R) on a (θt)t∈R-invariant set of full measure. Collecting (5.12) - (5.14) we obtain h−A(x−y) + F(ω, x)−F(ω, y), x −yi ≤ − X i∈Z λi(xi−yi)2− hΓ(ω)(x−y),(x−y)i +X i∈Zλi(xi−yi)2+zi(ω)(xi−yi)2 −e−zi(ω)fi(ezi(ω)xi)−fi(ezi(ω)yi)(xi−yi) ≤M(ω)kx−yk2 H,∀x, y ∈V. Finally, kG(ω)k2 V′=X i∈Z 1 λi g2 ie−2zi(ω)≤ kgk2 Hsup i∈Z e−2zi(ω) λi := K3(ω)∈L1(Ω),(5.15) i.e., Gsatisfies (F4). Therefore, from Theorem 4.2 we conclude immediately the following result. Theorem 5.3. There exists a θ–invariant set ˜ Ωof full measure such that for any ν0∈Hand any T > 0equation (5.2) has a unique weak solution v= (vi)i∈Z∈ C([0, T ]; H)∩L2(0, T ;V)on [0, T ], with initial condition v(0) = ν0, and a weak derivative in L2(0, T ;V′). The following lemma provides some qualitative and quantitative properties of the random variables K1and K3appearing above. Lemma 5.4. The random variables K1(ω)and K3(ω)are tempered. Proof. To obtain the temperedness of K1we have to show that Esup t∈[0,1] log+K1(θtω)≤Esup t∈[0,1] K1(θtω)<∞. 22 Noting that zi(θtω) = zi(ω)−Zt 0 zi(θsω)ds +σiωi(t), we consider Eesupt∈[0,1] |zi(θtω)|≤Ee4|zi(ω)|1 4EeR1 04|zi(θsω)|ds1 4 ·Ee4σisupt∈[0,1] ωi(t)1 4Ee4σisupt∈[0,1](−ωi(t))1 4.(5.16) First, since each ziis a Gauß random variable, Ee4|zi(ω)|<∞.(5.17) Next, due to the Jensen inequality, the Fubini Theorem, and the fact that each ziis a Gauß random variable, we have EeR1 04|zi(θsω)|ds ≤EZ1 0 e4|zi(θsω)|ds < ∞.(5.18) Finally, by the properties of ωi(t) we have Ee4σisupt∈[0,1] ±ωi(t)≤Z∞ 0 e4σisr2 πe−s2 2ds < ∞.(5.19) Thanks to (5.16) - (5.19) and the facts that (λ−1 i)i∈Z∈ℓ2and (zi)i∈Zis an iid–sequence we deduce the temperedness of K1and K3. Further, we shall obtain some useful estimates for the solutions vto the lattice random differential equation (5.2). Lemma 5.5. For any ω∈Ω, there exist functions J1(K, T, ω)and J2(K, T, ω), which are bounded for any (K, T )in bounded sets, such that kvk2 C([0,T ],H)≤J1(kν0kH, T, ω),ZT 0 kv(t)k2 Vdt ≤J2(kν0kH, T, ω). Proof. First notice that hA1v, vi= 2ρkvk2 Hand hA2v, vi=kvk2 V. Then from the estimate of hF(ω, v), viand (5.15) we obtain 1 2kv(t)k2 H+ 2ρZt 0 kv(s)k2 Hds +Zt 0 kv(s)k2 V ≤1 2kν0k2 H+Zt 0 (K1(θsω) + K3(θsω))ds +Zt 0 K2(θsω)kv(s)k2 Hds +1 2Zt 0 kv(s)k2 V, 23 and consequently the so-called energy inequality: kv(t)k2 H+ 4ρZt 0 kv(s)k2 Hds +Zt 0 kv(s)k2 V ≤ kν0k2 H+ 2 Zt 0 (K1(θsω) + K3(θsω))ds + 2 Zt 0 K2(θsω)kv(s)k2 Hds.(5.20) Define Q(ω) := −λ0−4ρ+ 2K2(ω). Then for any ω∈Ω the mapping t7→ Q(θtω) is locally integrable. Moreover, by using the Gronwall Lemma we obtain kv(t)k2 H≤eRt 0Q(θrω)drkν0k2 H+ 2 Zt 0 eRt sQ(θrω)dr(K1(θsω) + K3(θsω))ds. (5.21) It follows immediately that kvk2 C([0,T ],H)≤J1(kν0kH, T, ω), where J1(kν0kH, T, ω) is defined by J1(kν0kH, T, ω) := eRT 0|Q(θrω)|drkν0k2 H+2 ZT 0 eRT s|Q(θrω)|dr(K1(θsω)+K3(θsω))ds. By inequality (5.20), we have that RT 0kv(t)k2 Vdt ≤J2(kν0kH, T, ω), where J2(kν0kH, T, ω) := kν0k2 H+ 2 ZT 0 K2(θsω)J1(kν0kH, T, ω)ds +2 ZT 0 (K1(θsω) + K3(θsω))ds. Up to here we have shown a global existence and uniqueness result for (5.2). Denote by v(t, ω, ν0) the solution to (5.2) at time t≥0 starting in ν0∈Hfor a noise–path ω∈Ω. Theorem 5.6. (i) Let ω∈Ωand assume that (f1) - (f4) hold. Then the solution to system (5.2) generates a continuous random dynamical system ϕ(t, ω, ν0) over (Ω,F,P,(θt)t∈R)with state space H: ϕ(t, ω, ν0) = v(t, ω, ν0), ν0∈H, t ≥0, ω ∈Ω. (ii) Suppose that EQ < 0. Then the random dynamical system ϕhas a closed positively invariant tempered pullback absorbing set K(ω)in Hgiven by the ball in H,BH(0, R(ω)), centered at 0 with radius R(ω) := 2 Z0 −∞ eR0 sQ(θrω)dr(K1(θsω) + K3(θsω))ds1 2 . 24 Proof. (i) This follows directly from Theorem 4.2. (ii) According to (5.21) we have kϕ(t, ω, ν0)k2 H≤eRt 0Q(θrω)drkν0k2 H+ 2 Zt 0 eRt sQ(θrω)dr(K1(θsω) + K3(θsω))ds. Replacing ωby θ−tωin ϕwe obtain kϕ(t, θ−tω, ν0)k2 H≤eRt 0Q(θr−tω)drkν0k2 H +2 Zt 0 eRt sQ(θr−tω)dr(K1(θs−tω) + K3(θs−tω))ds =eR0 −tQ(θrω)drkν0k2 H +2 Z0 −t eR0 sQ(θrω)dr(K1(θsω) + K3(θsω))ds. First notice that since EQ < 0, then for any ν0∈B(θ−tω), lim t→∞ eR0 −tQ(θrω)drkν0k2 H≤lim t→∞ eR0 −tQ(θrω)drd(B(θ−tω))2= 0. In addition by the temperedness of K1and K3we have Z0 −∞ eR0 sQ(θrω)dr(K1(θsω) + K3(θsω))ds < ∞. Letting R2(ω) = 4 Z0 −∞ eR0 sQ(θrω)dr(K1(θsω) + K3(θsω))ds, gives that the ball K(ω) := BH(0, R(ω)) is a pullback absorbing set in H. The temperedness of R(ω) follows immediately from EQ < 0 and the fact that K1 and K3are tempered. It is easy to show that K(ω) is positively invariant. Consider some κ > 0 satisfying X i∈Z λκ−1 i<∞, see the beginning of this section. We now introduce the space Vκ={u= (ui)i∈Z:X i∈Z λκ iu2 i:= kuk2 Vκ<∞}, equipped with the inner product (u, v)Vκ=X i∈Z λκ iuivi, u = (ui)i∈Z, v = (vi)i∈Z. Then we have the compact imbedding Vκ⊂H(see [31] page 94). It then remains to show the compactness of the random dynamical system ϕ(t, ω, ν0), which will be presented in the following lemma. 25