Random attractors for stochastic lattice systems with non-Lipschitz non-linearity

January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Journal of Difference Equations and Applications Vol. 00, No. 00, January 2008, 1–25 Paper Random attractors for stochastic lattice systems with non-Lipschitz non-linearity T. Caraballoa∗, F. Morillasband J. Valeroc aDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla, Spain.; bUniversitat de Val`encia, Departament d’Economia Aplicada, Facultat d’Economia, Campus dels Tarongers s/n, 46022 Val`encia, Spain. cUniversidad Miguel Hern´andez, Centro de Investigaci´on Operativa, Avda. Universidad s/n, 03202 Elche (Alicante), Spain. (XXXXXX) In this paper we study the asymptotic behaviour of solutions of a first-order stochastic lattice dynamical system with an additive noise. We do not assume any Lipschitz condition on the nonlinear term, just a continuity assumption together with growth and dissipative conditions, so that uniqueness of the Cauchy problem fails to be true. Using the theory of multi-valued random dynamical systems we prove the existence of a random compact global attractor. Keywords: Stochastic lattice dynamical systems, set-valued dynamical system, random attractor AMS Subject Classification: 35B40, 35B41, 35K40, 35K57, 37B25, 37L55, 60H10 1. Introduction In this paper we study the asymptotic behaviour of solutions of the stochastic first order lattice dynamical system with an additive noise given by (dui dt =ν(ui−1−2ui+ui+1)−hi(ui)−fi(ui) + ai dwi(t) dt , i ∈Z, ui(0) = u0 i, under some growth and dissipative conditions on the nonlinear terms fand h. Here wi(t) are real independent two-sided Brownian motions. We prove the existence of a random global pullback attractor for such systems, extending in this way the results given in [3] (see [11] for the case of a multiplicative noise, and also [21] for a partial dissipative stochastic lattice dynamical system with additive noise). The main difference with [3] is the fact that we do not assume conditions ensuring the uniqueness of solutions for the Cauchy problem. Hence, we use the theory of multi-valued random dynamical systems [6] in order to prove the existence of the pullback attractor. Comparing our results with the single-valued ∗Corresponding author. Email: [email protected] ISSN: 1023-6198 print/ISSN 1563-5120 online c 2008 Taylor & Francis DOI: 10.1080/1023619YYxxxxxxxx http://www.informaworld.com January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 2T. Caraballo, F. Morillas & J. Valero case from [3], the main technical difficulty appears in proving the measurability of the attractor. Our stochastic lattice dynamical system can be obtained by the spatial discretization of a stochastic reaction-diffusion equation in an arbitrary domain for the spatial variable (which can be unbounded and even the whole space RN). So, such systems can be viewed as approximations for reaction-diffusion equations in unbounded domains. In the deterministic case the theory of attractors for lattice dynamical systems has been intensively studied over the last years (see e.g. [1], [4], [5], [14], [15], [17], [18], [20], [22], [23], [24] and the references therein). This paper is organized as follows. In Section 2 we recall the general theory of multi-valued random dynamical systems as developed in [6]. First, we establish some preliminaries concerning the definitions of multi-valued non-autonomous dynamical systems (MNDS) and multivalued random dynamical systems (MRDS) which turns to be an MNDS with an additional measurability property. Then, we recall two theorems ensuring the existence of the global pullback attractor for an MNDS and the random global pullback attractor for an MRDS. In Section 3 we describe the setting of our problem. By a standard change of variable (involving the Ornstein-Uhlenbeck process), we transform the stochastic equation into a random one depending on a parameter ω. We then show that the Cauchy problem for the transformed lattice system possesses, at least, one global solution for each ω, although it can be non-unique. In Section 4 we first define an MNDS associated to our problem, and prove the existence of a global pullback attractor. In this part it is crucial to obtain suitable estimates of the tails of solutions leading to the pullback asymptotic compactness of the MNDS. Finally, we prove some measurability properties of the MNDS which imply that we have in fact a MRDS having a random pullback attractor. As remarked before, these last results are much more difficult to obtain than in the single-valued case. 2. Multi-valued random dynamical systems We recall now some basic definitions for set-valued non-autonomous and random dynamical systems and formulate sufficient conditions ensuring the existence of a pullback attractor for these systems. A pair (Ω, θ) where θ= (θt)t∈Ris a flow on Ω, that is, θ:R×Ω→Ω, θ0= idΩ, θt+τ=θt◦θτ=: θtθτfor t, τ ∈R, is called a non-autonomous perturbation. Let P:= (Ω,F,P) be a probability space. On this probability space we consider a measurable non–autonomous flow θ: θ: (R×Ω,B(R)⊗ F)→(Ω,F). In addition, Pis supposed to be ergodic with respect to θ, which means that every θt-invariant set has measure zero or one for t∈R. Hence Pis invariant with respect to θt. The quadruple (Ω,F,P, θ),which is the model for a noise, is called a metric dynamical system. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 3 If we replace in the definition of a metric dynamical system the probability space Pby its completion Pc:= (Ω,¯ F,¯ P) the above measurability property is not true in general (see Arnold [2, Appendix A]). But for fixed t∈Rwe have that the mapping θt: (Ω,¯ F)→(Ω,¯ F) is measurable. Let X= (X, dX) be a Polish space. Let D:ω→D(ω)∈2Xbe a multi– valued mapping. The set of multi–functions D:ω→D(ω)∈2Xwith closed and non–empty images is denoted by C(X).Let also denote by Pf(X) the set of all non-empty closed subsets of the space X. Thus, it is equivalent to write that Dis in C(X), or D: Ω →Pf(X). Let D:ω→D(ω) be a multi–valued mapping in Xover P. Such a mapping is called a random set if ω→inf y∈D(ω)dX(x, y) is a random variable for every x∈X. It is well known that a mapping is a random set if and only if for every open set Oin Xthe inverse image {ω:D(ω)∩O6=∅} is measurable, i.e., it belongs to F[13, Proposition 2.1.4]. Clearly, this is also valid if we replace Pby Pcand Fby F. It is straightforward to see that, if Dis a random set with respect to P, then it is also random with respect to Pc. We now introduce non-autonomous and random dynamical systems. Definition 2.1.A multi-valued map G:R+×Ω×X→Pf(X) is called a multi-valued non-autonomous dynamical system (MNDS) if i) G(0, ω, ·) = idX, ii) G(t+τ, ω, x)⊂G(t, θτω, G(τ, ω, x)) (cocycle property) for all t, τ ∈R+, x ∈ X, ω ∈Ω. It is called a strict MNDS if, moreover, iii) G(t+τ, ω, x) = G(t, θτω, G(τ, ω, x)) for all t, τ ∈R+, x ∈X, ω ∈Ω. An MNDS is called a multi–valued random dynamical system (MRDS) if the multi–valued mapping (t, ω, x)→G(t, ω, x) is B(R+)⊗ F ⊗ B(X) measurable, i.e. {(t, ω, x) : G(t, ω, x)∩O6=∅} ∈ B(R+)⊗ F ⊗ B(X) for every open set Oof the topological space X. For the above composition of multi-valued mappings we use that for any nonempty set V⊂X,G(t, ω, V ) is defined by G(t, ω, V ) = [ x0∈V G(t, ω, x0). We now formulate a general condition ensuring that an MNDS defines an MRDS. We need the particular assumption that Ω is a Polish space and Fthe associated Borel σ-algebra. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 4T. Caraballo, F. Morillas & J. Valero Lemma 2.2.[6, Lemma 2.5] Let Ωbe a Polish space and let Fbe the Borel σ–algebra. Suppose that (t, ω, x)7→ U(t, ω, x)is upper-semicontinuous. Then this mapping is measurable in the sense of Definition 2.1. A multi–valued mapping Dis said to be negatively, strictly, or positively invariant for the MNDS Gif D(θtω) ⊂ = ⊃ G(t, ω, D(ω)) for ω∈Ω, t ∈R+. Let Dbe the family of multi–valued mappings with values in C(X). We say that a family K∈ D is pullback D-attracting if for every D∈ D lim t→+∞distX(G(t, θ−tω, D (θ−tω)), K(ω)) = 0,for all ω∈Ω. B∈ D is said to be pullback D-absorbing if for every D∈ D there exists T= T(ω, D)>0 such that G(t, θ−tω, D (θ−tω)) ⊂B(ω),for all t≥T. (1) The following definition provides the main objective of this article. We have to introduce a particular set system (see Schmalfuß [19]). Let Dbe a set of multi– valued mappings in C(X) satisfying the inclusion closed property: if we suppose that D∈ D and D′is a multi–valued mapping in C(X) such that D′(ω)⊂D(ω) for ω∈Ω, then D′∈ D. Definition 2.3.A family A ∈ D is said to be a global pullback D-attractor for the MNDS Gif it satisfies: (1) A(ω) is compact for any ω∈Ω; (2) Ais pullback D-attracting; (3) Ais negatively invariant. Ais said to be a strict global pullback D-attractor if the invariance property in the third item is strict. A natural modification of this definition for MRDS is the following. Definition 2.4.Suppose that Gis an MRDS and suppose that the properties of Definition 2.3 are satisfied. In addition, we suppose that Ais a random set with respect to Pc.Then Ais called a random global pullback D-attractor. Now we recall two general results on the existence and uniqueness of pullback and random attractors associated to MNDS and MRDS respectively, which were proved in [6]. Theorem 2.5 .Suppose that the MNDS G(t, ω, ·)is upper-semicontinuous for t≥0and ω∈Ω. Let K∈ D be a multi-valued mapping such that the MNDS is pullback D-asymptotically compact with respect to K, i.e. for every sequence tn→+∞, ω ∈Ωevery sequence yn∈G(tn, θ−tnω, K(θ−tnω)) is pre–compact. In addition, suppose that Kis pullback D-absorbing. Then, the set Agiven by A(ω) := \ s≥0[ t≥s G(t, θ−tω, K (θ−tω)) (2) January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 5 is a pullback D-attractor. Furthermore, Ais the unique element from Dwith these properties. In addition, if Gis a strict MNDS, then Ais strictly invariant. Remark 1.When the property A ∈ D is not satisfied, some difficulties may appear, for example, in proving the strict invariance of the attractor. This is the case if D is just the set of all bounded subsets of X(see [9]-[10]). Theorem 2.6 .Let Gbe a MRDS. Under the assumptions in Theorem 2.5, let ω→G(t, ω, K(ω)) be a random set for t≥0with respect to Pc. Assume also that G(t, ω, K(ω)) is closed for all t≥0and ω∈Ω. Then the set Adefined by (2) is a random set with respect to Pc,so that it is a random global pullback D-attractor. 3. Setting of the problem Let us consider the lattice stochastic system      dui dt =ν(ui−1−2ui+ui+1)−hi(ui)−fi(ui) + ai dwi(t) dt , i ∈Z, ui(0) = u0 i, (3) where u= (ui)i∈Z∈l2,u0=u0 ii∈Z∈l2,ν > 0, a= (ai)i∈Z∈l2,wiare real independent two-sided Brownian motions (so they satisfy wi(0) = 0) and fi, hi:R→R, i ∈Z, are functions satisfying the following conditions: (H1) For all x∈R, hi(x)x≥α|x|2−c1,i, |hi(x)| ≤ β|x|+c2,i, where c1∈l1, c2∈l2, c2,i ≥0, α, β > 0. (H2) For all x∈R, fi(x)x≥γ|x|p−c3,i, |fi(x)| ≤ η|x|p−1+c4,i, where p≥2, c3∈l1,c4∈l2, c4,i ≥0, γ, η > 0. (H3) The maps hi, fi:R→Rare continuous. We define the operators A, B, B∗:l2→l2given by (Bu)i=ui+1 −ui,(B∗u)i=ui−1−ui, (Au)i=−ui−1+ 2ui−ui+1. It is easy to see that A=B∗B=BB∗and (B∗u, v) = (u, Bv), for all u, v ∈l2, where (·,·) denotes the scalar product in l2. Using conditions (H1)−(H2) it is not difficult to see that the Nemytski operators f, h :l2→l2given by f(u) = (fi(ui))i∈Z,h(u) = (hi(ui))i∈Zare well defined. On January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 6T. Caraballo, F. Morillas & J. Valero the other hand, we can obtain the following estimates for the norms of f(u), h (u) : kh(u)k2 l2≤X i∈Z (c2,i +β|ui|)2≤2kc2k2 l2+β2kuk2 l2,(4) kf(u)k2 l2≤X i∈Zc4,i +η|ui|p−12≤2kc4k2 l2+η2kuk2p−2 l2p−2 ≤Ckc4k2 l2+kuk2p−2 l2. Moreover, the maps f, h are continuous and weakly continuous. Lemma 3.1.The maps f, h :l2→l2are continuous. Proof . Let un→u0in l2.It is clear that for any ǫ > 0 there exist k0(ǫ) such that X |i|>k0u0 i 2≤ǫ,X |i|>k0 |c4,i|2≤ǫ, X |i|>k0 |un i|2≤ǫ,∀n. Also, by (H3) we can choose n0(ǫ, k0) such that if n≥n0, then X |i|≤k0fi(un i)−fiu0 i 2≤ǫ. Therefore, using (H2) we obtain  f(un)−fu0  2=X |i|≤k0fi(un i)−fiu0 i 2+X |i|>k0fi(un i)−fiu0 i 2 ≤ǫ+ 4 X |i|>k0|c4,i|2+η2|un i|2(p−1)+ 4 X |i|>k0|c4,i|2+η2u0 i 2(p−1) ≤9ǫ+ 4C0η2 X |i|>k0 |un i|2+X |i|>k0u0 i 2 ≤C1ǫ, where C0= supi∈Z,n∈N∪{0}|un i|2p−4. Thus, f(un)→fu0in l2, which implies the continuity. For hthe proof is similar.  Lemma 3.2.The maps f, h :l2→l2are weakly continuous. Proof . We note that as the space l2endowed with the weak topology satisfies the first axiom of countability, it is not difficult to show that fis weakly continuous if and only if un→u0weakly in l2implies f(un)→f(u0). Take an arbitrary ξ∈l2. Then for any ǫ > 0 there exists K1(ǫ, ξ) such that P|i|≥K1|ξi|2≤ǫ. Due to (4), if kukl2≤R, then there exists M=M(R) such that January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 7 kf(u)kl2≤M. Thus if un→uweakly in l2we have |(f(un)−f(u), ξ)| ≤ X |i|≤K1 |(fi(un i)−fi(ui)) ξi| + (kf(u)kl2+kf(un)kl2) X |i|≥K1 |ξi|2  1 2 ≤ kξkl2 X |i|≤K1 |(fi(un i)−fi(ui))|2  1 2 + 2Mǫ, for some M. For any ε > 0 we choose ǫ=ε 4M. Since un i→uifor all iand fiare continuous, we obtain the existence of N1(ε, ξ) such that P|i|≤K1|(fi(un i)−fi(ui))|21 2≤ε 2kξkl2 (for ξ6= 0). Hence |(f(un)−f(u), ξ)| ≤ εif n≥N1. For hthe proof is similar.  We rewrite (3) in the abstract form      du dt =−νAu −h(u)−f(u) + dW dt , u(0) = u0∈l2. (5) We will transform equation (5) into a random one by using a suitable change of variable. First let us describe the probability space associated to the Wiener processes wi. Let W(t, ω) (W(t) for short when no confusion is possible) be the white noise with values in the space l2given by W(t, ω) = (aiwi(t))i∈Z∈l2 with the probability space (Ω,F,P), where Ω = {ω∈ C R, l2:ω(0) = 0}=C0R, l2. Fis the Borel σ-algebra on Ω with respect to the compact open topology and Pis the Wiener measure on F. If we define the shift operator θtω(·) = ω(·+t)−ω(t) , for t∈R, then we obtain the metric dynamical system Ω,F,P,(θt)t∈R. We consider the filtration on Ω given by Ft s=σ{W(τ2)−W(τ1) : s≤τ1≤τ2≤t}, January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 8T. Caraballo, F. Morillas & J. Valero where σ{W(τ2)−W(τ1) : s≤τ1≤τ2≤t}is the smallest σ-algebra generated by W(τ2)−W(τ1) for all τ1, τ2satisfying s≤τ1≤τ2≤t. We have the property θ−1 rFt s=Ft+r s+r. The completion of (Ω,F,P) will be denoted by Ω,F,P. We introduce now an Ornstein-Uhlenbeck process in l2on the metric dynamical system Ω,F,P,(θt)t∈Rgenerated by the Wiener process. Namely, put z(θtω) := −αZ0 −∞ eαsθtω(s)ds, where α > 0 is taken from condition (H1). This integral makes sense for every path ωhaving sub-exponential growth. It is well known that zsolves the stochastic differential equation dz +αz =dW (t), as t→z(θtω) is a θtinvariant stationary solution known as the Ornstein-Uhlenbeck process. This process is adapted with respect to Ft −∞, where Ft −∞ =_ s≤t Ft s. Moreover, there exists a θt-invariant set Ω′⊂Ω of full measure such that the mapping s→z(θsω) is continuous for each ω∈Ω′(so that the process has continuous trajectories) and lim t→+∞ kW(t)k t= 0 for all ω∈Ω′.(6) Also, z(θtω) has sub-exponential growth, which means that lim t→±∞ log+kz(θtω)k t= 0, for ω∈Ω′. (7) See [3] for more details. Remark 2.In the sequel, all the statements are understood to hold on a θtinvariant set Ω′⊂Ω of full measure, although we shall write for simplicity that they are true for any ω∈Ω. That is, we keep the notation Ω for the set Ω′. We now perform the change of variable v(t) = u(t)−z(θtω), and problem (5) turns (formally) into      dv dt =−νAv −h(v+z(θtω)) −f(v+z(θtω)) + αz (θtω)−νAz (θtω) , v(0) = v0=u0−z(ω). (8) Let gω:R+×l2→l2be the Nemytski operator given by gω i(t, vi) = hi(vi+zi(θtω)) + fi(vi+zi(θtω)) −αzi(θtω) + ν(Az (θtω))i,i∈Z. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 9 From (H1) −(H2) we obtain that, for every x∈R, gω i(t, x)x=hi(x+zi(θtω)) x+fi(x+zi(θtω)) x−αzi(θtω)x+ν(Az (θtω))ix ≥α|x+zi(θtω)|2−hi(x+zi(θtω)) zi(θtω)−c1,i +γ|x+zi(θtω)|p−fi(x+zi(θtω)) zi(θtω)−c3,i −αzi(θtω)x+ν(Az (θtω))ix ≥α|x+zi(θtω)|2−β(|x+zi(θtω)|+c2,i)|zi(θtω)| − c1,i +γ|x+zi(θtω)|p−η|x+zi(θtω)|p−1+c4,izi(θtω)−c3,i −αzi(θtω)x+ν(Az (θtω))ix. Using Young’s inequality and α|x+zi(θtω)|2≥2α 3|x|2−2α|zi(θtω)|2, we have gω i(t, x)x≥α 2|x|2−K|zi(θtω)|2+|zi(θtω)|p+|(Az (θtω))i|2(9) −c1,i −c3,i −c2 2,i −c2 4,i. On the other hand, and thanks again to (H1) −(H2), we obtain |gω i(t, x)| ≤ β|x|+η|x|p−1+c2,i +c4,i +α|zi(θtω)|+ν|(Az (θtω))i|.(10) Since s→z(θsω) is continuous for every ω∈Ω′, it follows from (9)-(10) and the definition of the operator A, that the following properties hold: (G1) For all x∈R, gω i(x)x≥λ|x|2−d1,i (t), where λ > 0, d1∈C[0, T], l1for any T > 0. (G2) For all x∈R, |gω i(t, x)| ≤ C(|x|)|x|+d2,i (t), where d2∈C[0, T], l2for any T > 0, d2,i ≥0, and C(·)≥0 is a continuous increasing function. (G3) gω i: [0, T]×R→Rare continuous. Remark 3.Of course, the elements d1, d2depend on ωbut we omit this for simplicity of notation. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 16 T. Caraballo, F. Morillas & J. Valero Then, if we take T∗≥16 αlog 8r(ω)K7 αǫ we have that K6Z−T∗ −t eα 8shkz(θsω)k2 l2+kzi(θsω)kp lpids ≤ǫ 8. As for the other integral we use Lebegue’s theorem to find N2(ǫ, ω, T∗) such that, if N≥N2, K6Z0 −T∗ eα 8s X |i|≥N−1 |zi(θsω)|2+X |i|≥N−1 |zi(θsω)|p ds ≤ǫ 8. Then, if we take ˆ N > max{N0, N1, N2},N≥ˆ Nand T(ǫ, ω)>max{T1, T∗}, we deduce from (22) that X |i|>2N |vi(t, θ−tω, v0(θ−tω))|2≤X i∈Z ρN,i |vi(t, θ−tω, v0(θ−tω))|2≤ǫ, if t≥T. This implies that X |i|≥2N(ǫ,ω) |ui(t, θ−tω, u0(θ−tω))|2 ≤2X |i|≥2N(ǫ,ω) |vi(t, θ−tω, v0(θ−tω))|2+ 2 X |i|≥2N(ǫ,ω) |zi(ω)|2 ≤4ǫ, for some N≥ˆ N, if t≥T. Lemma 4.4. G is pullback D-asymptotically compact with respect to K∈ D. Proof . Let ω∈Ω. Consider {tn}n∈Nwith limn→∞ tn= +∞and ξn∈ G(tn, θ−tnω, K (θ−tnω)). We can find xn∈K(θ−tnω) such that ξn∈G(tn, θ−tnω, xn) . We will show that {ξn}n∈Npossesses a convergent subsequence. Let T(ω, K) be the time such that G(t, θ−tω, K (θ−tω)) ⊂K(ω) if t≥T. There exists n0verifying that tn≥T(ω), for n≥n0. Then {ξn} ⊂ K(ω), so it is weak convergent, that is, there exists ξ∈l2and a subsequence such that ξn→ξweakly in l2. (23) We shall check that this convergence is, in fact, strong. Let ǫ > 0 be arbitrary. Using Lemma 4.3 we obtain the existence of n1(ǫ, ω), M0(ǫ, ω) such that X |i|≥M0 |ξn i|2≤ǫ, if n≥n1. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 17 Also, as ξ∈l2, one can find M1(ǫ)≥M0such that X |i|≥M |ξi|2≤ǫ, if M≥M1. We take M(ǫ, ω) = max {M0, M1}. The weak convergence (23) ensures that for each |i| ≤ M, ξn i→ξias n→ ∞. From this convergence, there exists n2(M, ǫ, ω)≥n1such that X |i|≤M |ξn i−ξi|2≤ǫ, if n≥n2. Then, using these estimates for n≥n2we have that X i∈Z |ξn i−ξi|2≤X |i|≤M |ξn i−ξi|2+X |i|>M |ξn i−ξi|2 ≤ǫ+ 2 X |i|≥M |ξi|2+ 2 X |i|≥M |ξn i|2≤5ǫ. The proof is therefore finished.  Let us now prove other properties of the cocycle G. Lemma 4.5.Let v0nbe a sequence converging to v0in l2and fix T > 0. Then, for any ω∈Ωand ǫ > 0, there exists K(ǫ, ω)such that for any solution vn(·)∈ Sv0n, ωit follows X |i|≥2K(ǫ,ω) |vn i(t)|2≤ǫ, ∀t∈[0, T].(24) Moreover, there exists v(·)∈ S v0, ωand a subsequence vnksatisfying vnk→vin C[0, T], l2.(25) Proof . For any ǫ > 0 there exist K1(ǫ), N1(ǫ) such that X i∈Zv0n i−v0 i 2<ǫ 4,∀n≥N1, X i∈Z ρK,i v0 i 2<ǫ 4,∀K≥K1. Hence, X i∈Z ρK,i v0n i 2≤2 X i∈Z ρK,i v0n i−v0 i 2+X i∈Z ρK,i v0 i 2!< ǫ, (26) January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 18 T. Caraballo, F. Morillas & J. Valero if n≥N1and K≥K1.Obviously, modifying K1appropriately, the result holds true for all n. Also, in view of (14), there exists R0(ω)>0 such that kvn(t)kl2≤R0(ω) , ∀t∈[0, T],∀n. (27) Using inequality (21) and the continuity of t→z(θtω), one can find K2(ǫ, ω) such that d dt X i∈Z ρK,i |vn i|2≤ǫ, if K≥K2. Integrating over (0, t) and using (26) we obtain X |i|≥2K(ǫ) |vn i(t)|2≤X i∈Z ρK,i |vn i(t)|2≤ǫ+Tǫ, (28) if K≥max {K1, K2}, so that (24) holds. Fix now t∈[0, T]. In view of (27), passing to a subsequence, we can state that vn(t)→wweakly in l2. Then, for any σ > 0, there exist N2(σ) and K3(σ) such that kvn(t)−wk2 l2≤X |i|≤K3(σ) |vn i(t)−wi|2+X |i|>K3(σ) |vn i(t)−wi|2 ≤X |i|≤K3(σ) |vn i(t)−wi|2+ 2 X |i|>K3(σ) |vn i(t)|2+ 2 X |i|>K3(σ) |wi(t)|2 < σ, if n≥N2. Hence, vn(t)→wstrongly in l2. It follows that the sequence vn(t) is pre-compact for any t. By (12) and (27), kFω(t, vn(t))k2 l2≤C(ω),∀n∈N,∀t∈[0, T], and from the equality d dtvn i=Fω i(vn i) we obtain that     d dtvn(t)   l2 ≤C1(ω), proving that the sequence vnis equi-continuous. The Ascoli-Arzel`a theorem implies the existence of a subsequence vnkconverging in C[0, T], l2to some function v(·). It is then easy to show that vis a solution of (11). Also, it is clear that v(0) = v0.  Lemma 4.5 implies several consequences. Corollary 4.6.For any ω∈Ωand t≥0the graph of the map u0→Gt, ω, u0 is closed. Hence, Ghas closed values. Proof . For ξn∈Gt, ω, u0 nthere are vn(·)∈ S u0 n−z(ω), ωsuch that ξn=vn(t) + z(θtω). Assume that ξn→ξand u0 n→u0. Applying Lemma 4.5 we obtain, passing to a subsequence, that vn→vin C[0, t], l2, where v(·)∈ S u0−z(ω), ω.Therefore, ξ=v(t) + z(θtω)∈Gt, ω, u0. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 19 Corollary 4.7. G is a strict MNDS. Proof . It follows from Lemma 4.1 and Corollary 4.6.  Corollary 4.8.For any ω∈Ωand t≥0, the map G(t, ω, ·)has compact values. Proof . Let ξn∈Gt, ω, u0be an arbitrary sequence. Take vn(·)∈ Su0−z(ω), ωsuch that ξn=vn(t) + z(θtω). By Lemma 4.5 there exists v(·)∈ S u0−z(ω), ωsatisfying vnk→vin C[0, t], l2for some subsequence. Hence, ξnk=vnk(t) + z(θtω)→v(t) + z(θtω)∈Gt, ω, u0in l2. Proposition 4.9.For any ω∈Ωand t≥0, the map u0→Gt, ω, u0is upper semi-continuous. Proof . Suppose the opposite. Then there exist u0,t > 0, a neighborhood Oof Gt, ω, u0and sequences u0 n→u0,ξn∈Gt, ω, u0 nsuch that ξn6∈ O. Let ξn=vn(t) + z(θtω), where vn(·)∈ S u0 n−z(ω), ω. By Lemma 4.5 we obtain that, up to a subsequence, vn→vin C[0, t], l2, where v(·)∈ S u0−z(ω), ω. Thus, ξn→v(t) + z(θtω)∈Gt, ω, u0in l2, which is a contradiction.  Finally, we have the following result. Theorem 4.10 .The MNDS Gpossesses a unique D-pullback global strictly invariant attractor A(ω), defined by (2), where K(ω)is the set given in (13). Proof . This theorem follows from Theorem 2.5 using Corollary 4.7, Lemmas 4.2, 4.4 and Proposition 4.9.  4.2 The random attractor In order to obtain that A(ω) is a random pullback D-attractor we need to check that it is a random set. And, to prove the measurability of the global attractor A(ω) we need to obtain some properties concerning the map ω→G(t, ω, K (ω)), where K(ω)∈ D is the pullback D-absorbing set given in Lemma 4.2. Also, we have to prove that Gis an MRDS. For N∈N, we consider the sets ΩN:= {ω∈Ω : kω(t)kl2≤Neζ|t|,for t∈R},(29) where 0 < pζ < α. Then it is not difficult to check that Ω = ∪NΩNand ΩN∈ F. It is shown in [7] that ΩNis a polish space for any N. We need the continuity of the map R×ΩN∋(t, ω)→z(θtω). Lemma 4.11.For any N∈Nthere exists M(N)such that kz(θtω)kl2≤Meζ|t|, for all ω∈ΩN. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 20 T. Caraballo, F. Morillas & J. Valero Proof . We have kz(θtω)kl2≤αZ0 −∞ eαs kθtω(s)kds ≤αZ0 −∞ eαs kω(t+s)kds +kω(t)k ≤αN Z0 −∞ eαseζ|t+s|ds +Neζ|t|≤Neζ|t|1 + αZ0 −∞ e(α−ζ)s ≤N1 + α α−ζeζ|t|.  Lemma 4.12.The map R×ΩN∋(t, ω)→z(θtω)is continuous. Proof . Let tn→t0,ωn→ω0. Then kz(θtnωn)−z(θt0ω0)kl2=    αZ0 −∞ eαsθtnωn(s)ds −αZ0 −∞ eαsθt0ω0(s)ds   l2 ≤αZ0 −∞ eαs kθtnωn(s)−θt0ω0(s)kl2ds ≤αZ0 −∞ eαs kωn(s+tn)−ω0(s+t0)kl2ds +kωn(tn)−ω0(t0)kl2. Due to the definition of ΩNwe have kωn(τ)−ω0(τ)kl2≤ kωn(τ)kl2+ kω0(τ)kl2≤Neζ|τ|+eζ|τ|.Thus, for any ε > 0, there exists T(ε)> t0(and then, without loss of generality we can assume that T(ε)> tnalso) such that αZ−T −∞ eαs kωn(s+tn)−ω0(s+t0)kl2ds ≤2αNeζ|t|Z−T −∞ e(α−ζ)sds < ε 3, where |t0|<t(and thus, again, we can assume |tn|<t). Thus, if we take n0(ε, T (ε)) such that kωn(s+tn)−ω0(s+t0)kl2< ε/3, for any s∈[−T, 0] and n≥n0, then we get kz(θtnωn)−z(θt0ω0)kl2< ε, if n≥n0.  Let FΩNbe the trace σ-algebra of Fwith respect to ΩNand let BΩN(a, r), a ∈ ΩN, r > 0 be a ball in ΩN. These balls can be generated by BΩ(a, r)∩ΩNwhere BΩ(a, r) is a ball in Ω. The same is true for all open sets in ΩN. Hence FΩNis just the Borel–σ–algebra of ΩN. Moreover, since ΩN∈ F we have FΩN⊂ F. Let us define PΩN(A) := P(A),for A∈ FΩN, January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 21 that is, PΩNis just the restriction of Pto FΩN. Also, let ¯ FΩNbe the completion of FΩNwith respect to PΩN. The following facts are proved in [7]: (1) PΩNis a finite measure on (ΩN,FΩN). (2) If A∈¯ FΩN, then A∈¯ F. Now we establish the continuity of the random radius R(ω) given in Lemma 4.2 over ΩN. Lemma 4.13.The map ω→R(ω)is continuous on ΩN. Proof . Let ωn→ω0in ΩN. By Lemma 4.12 we have eατ kθτz(ωn)k2 l2+kθτz(ωn)kp l2+ 1→eατ kθτz(ω0)k2 l2+kθτz(ω0)kp l2+ 1, as n→ ∞. On the other hand, by Lemma 4.11 we obtain the majorant eατ kθτz(ωn)k2 l2+kθτz(ωn)kp l2+ 1≤eατ +M2e(α−2ζ)τ+Mpe(α−pζ)τ, which is integrable due to the restriction pζ < α. The result follows from Lebesgue’s theorem.  Concerning ΩNwe can obtain stronger properties for the cocycle G. Lemma 4.14.The map R+×ΩN×l2∋t, ω, u0→Gt, ω, u0is upper semicontinuous. Proof . If this is not true, then there exist u0,t0>0, ω0∈ΩN,a neighborhood Oof Gt0, ω0, u0and sequences tn→t0, ωn→ω0in ΩN,u0 n→u0in l2,ξn∈ Gtn, ωn, u0 nsuch that ξn6∈ O. We will prove that, up to a subsequence, ξn→ ξ∈Gt0, ω0, u0, which is a contradiction. Let vn(·)∈ S v0n, ωnbe such that ξn=vn(tn) + z(θtnωn), where v0n= u0n−z(ωn). By Lemma 4.12 we have that z(θtnωn)→z(θt0ω0) and v0n→v0= u0−z(ω0). Due to these properties and Lemma 4.13, one can state that (26) and (27) hold in the same way as in Lemma 4.5, where T > tn,T > t0, and R0is a common constant for any ωn. Lemma 4.12 and the continuity of t→z(θtω0) imply that z(θtωn)→z(θtω0) in C[0, T], l2. Hence, arguing as in the proof of Lemma 4.5, one can prove that, for any ǫ > 0, there exist K2(ǫ), N2(ǫ) such that Pi∈ZρK,i |zi(θtωn)|2< ǫ, if K≥K2,n≥N2. Then, using inequalities (21) and (27), we can find K3(ǫ) such that d dt X i∈Z ρK,i |vn i|2≤ǫ, if K≥K3. Similarly to the proof of Lemma 4.5 we obtain that the sequence vn(t) is pre-compact for any t. By (12), (27) and the fact that z(θtωn)→z(θtω0) in C[0, T], l2,we obtain kFωn(t, vn(t))k2 l2≤C, ∀n∈N,∀t∈[0, T], January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 22 T. Caraballo, F. Morillas & J. Valero and the equality d dtvn i=Fωn i(t, vn i) implies     d dtvn(t)   l2 ≤C1, which proves that the sequence vnis equi-continuous. The Ascoli-Arzel`a theorem implies then the existence of a subsequence vnkconverging to some function v(·) in C[0, T], l2. From the continuity of the maps h, f :l2→l2and (t, ω)→z(θtω) it follows that the map (ω, t, v)→Fω(t, v) is continuous. Thus, it is easy to show that vis a solution of (11). Also, it is clear that v(0) = v0. It follows that ξnk→ξ=v(t0) + z(θt0ω0)∈Gt, ω0, u0. Now, the following result is a consequence of Lemma 2.2. Corollary 4.15.The map t, ω, u0→Gt, ω, u0is B(R+)⊗ FΩN⊗ B l2 measurable. Further, we need some properties of the map ΩN∋ω→G(t, ω, K (ω)). Lemma 4.16.Let v0n→v0weakly in l2,ωn→ω0in ΩNand fix T > 0. Then there exist v(·)∈ S v0, ω0and a subsequence vnk∈ S v0nk, ωnksuch that vnk→vweakly in L20, T;l2, vnk(t)→v(t)weakly in l2for all t∈[0, T]. Proof . At light of (14) and Lemma 4.13 there exists R0>0 such that kvn(t)kl2≤R0,∀t∈[0, T ],∀n. Thus, by (12) and the convergence z(θtωn)→z(θtω0) in C[0, T], l2, kFωn(t, vn(t))kl2≤C, ∀n∈N,∀t∈[0, T],(30) and from the equality d dtvn i=Fωn i(vn i) we obtain that     d dtvn(t)   l2 ≤C. (31) Hence, there exist v, χ ∈L20, T;l2such that, up to a subsequence, vn→v, d dtvn→d dtv, Fωn(·, vn(·)) →χweakly in v∈L20, T;l2. Also, for any ξ∈l2we have (vn(t), ξ) = v0n, ξ+Zt 0dvn dτ , ξdτ →v0, ξ+Zt 0dv dτ , ξdτ = (v(t), ξ), January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised Random attractors for stochastic lattice systems 23 where the last equality follows from v, dv dt ∈L20, T ;l2. Hence, vn(t)→v(t) weakly in l2for all t∈[0, T]. It follows from the weakly continuity of the maps h, f :l2→l2and the continuity of (t, ω)→z(θtω) that Fωn(t, vn(t)) →Fω0(t, v (t)) weakly in l2for all t∈[0, T]. Also by (30) we get |(Fωn(t, vn(t)) , ξ)| ≤ Ckξkl2,∀n∈N,∀t∈[0, T], and then Lebesgue’s theorem gives (Fωn(t, vn(t)) , ξ)→(Fω0(t, v (t)) , ξ) in L1(0, T) for all ξ∈l2. Thus, Fω0(·, v (·)) = χ, and v(t) = v0+Zt 0 dv dτ dτ =v0+Zt 0 Fω0(τ, v (τ)) dτ, for all t∈[0, T], which implies that v(·)∈ S v0, ω0. Lemma 4.17.The map ΩN∋ω→G(t, ω, K (ω)) is FΩN-measurable for any t≥0.Also, G(t, ω, K (ω)) is closed for all t≥0,ω∈ΩN. Proof . It is well known [12, Chapter III] that D(ω) is a random set with respect to Pcif and only if the graph of D(ω),given by Gr(D) := {(ω, x)∈ΩN×l2:x∈D(ω)}, belongs to FΩN⊗ B(l2). Hence to prove the first statement it is sufficient to show that the graph of the map ω→G(t, ω, K (ω)) belongs to FΩN⊗ B l2, and this is true if the graph is closed. Let ω→ω0in ΩNand ξn→ξin l2, where ξn∈Gt, ωn, u0nand u0n∈K(ωn). We have to show that ξ∈G(t, ω0, K (ω0)) .Take vn(·)∈ S u0n−z(ωn), ωnsuch that ξn=vn(t) + z(θtωn). By the definition of K(ω) we have that vn0=u0n−z(ωn) satisfies  v0n l2≤ R(ωn). Then by Lemmas 4.12, 4.13 we obtain, passing to a subsequence, that vn0→v0weakly in l2, where  v0 l2≤R(ω0), and u0n→u0=v0+z(ω0)∈K(ω0) weakly in l2. In view of Lemma 4.16 there exists v(·)∈ S v0, ω0and a subsequence such that vn(t)→v(t) weakly in l2for all t∈[0, T]. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised 24 REFERENCES Thus, ξn=vn(t) + z(θtωn)→v(t) + z(θtω0) weakly in l2, so that ξ=v(t) + z(θtω0)∈G(t, ω0, u0)⊂G(t, ω0, K (ω0)) . Since the graph is closed, it is obvious that G(t, ω, K (ω)) is closed for all t≥0, ω∈ΩN. Theorem 4.18 .The MNDS Gis an MRDS. Also, the pullback D-attractor A(ω) given in Theorem 4.10 is a random set with respect to FΩ, and then it is the unique random global pullback D-attractor for G. Proof . Let us prove that Gis an MRDS. As Gis an MNDS, it remains to show that the map (t, ω, x)→G(t, ω, x) is B(R+)⊗ F ⊗ B(X) measurable. Let Obe an open set of l2. Then, by Corollary 4.15, we have that the set {(t, ω, x)∈R+×ΩN×l2:G(t, ω, x)∩ O 6=∅} := AN,O belongs to B(R+)⊗ FΩN⊗ B(l2), so that AN,O∈ B(R+)⊗ F ⊗ B(l2). Hence {(t, ω, x)∈R+×Ω×l2:G(t, ω, x)∩ O 6=∅} = ∞ [ N=1 {(t, ω, x)∈R+×ΩN×l2:G(t, ω, x)∩ O 6=∅} = ∞ [ N=1 AN,O∈ B(R+)⊗ F ⊗ B(l2), and then Gis an MRDS. Furthermore, in view of Lemma 4.17, the map ΩN∋ω→G(t, ω, K (ω)) is FΩN-measurable for any t≥0.Hence, for a fixed t≥0, the set {ω∈ΩN:G(t, ω, K (ω)) ∩ O 6=∅} := CN,O belongs to FΩN, so CN,O∈FΩand then {ω∈Ω : G(t, ω, K (ω)) ∩ O 6=∅} = ∞ [ N=1 CN,O∈FΩ. Due to Theorem 2.6 the pullback D-attractor A(ω) is a random set with respect to FΩ, and then it is the unique random global pullback D-attractor for G. Acknowledgements This work was partially supported by Spanish Ministerio de Ciencia e Innovaci´on, Projects MTM2008-00088 and MTM2009-11820, the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa), Proyecto de Excelencia P07- FQM-02468, and the Consejer´ıa de Cultura y Educaci´on (Comunidad Aut´onoma de Murcia), grant 08667/PI/08. References [1] A.Y. Abdallah, Exponential attractors for first-order lattice dynamical systems, J. Math. Anal. Appl., 339 (2008), 217-224. January 25, 2011 12:44 Journal of Difference Equations and Applications 102JDEArevised REFERENCES 25 [2] L. 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