# Existence of Random Attractors for a Class of Second-Order Lattice Dynamical Systems with Brownian Motions

Authors: Yamin Wang; Ziqiao Huang; Fuad E. Alsaadi; Stanislao Lauria; Yurong Liu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper is concerned with the random attractors for a class of second-order stochastic lattice dynamical systems. We first prove the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we establish the existence of the global random attractor. The system under consideration is quite general, and many existing results can be regarded as the special case of our results.


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## Body

## 1. Introduction

We consider the following second-order stochastic lattice dynamical system:(1)u¨+ξAu˙+hu˙+Au+ηu+fu=g+W˙t,hhihhit>0,u0=ui,0i∈Zn=u0,u˙0=u1i,0i∈Zn=u10,
where u=(ui)i∈Zn∈l2, u˙=(u˙i)i∈Zn∈l2 are real-value functions on R+; ξ=(ξi)i∈Zn and η=(ηi)i∈Zn are given vectors satisfyingbounded conditions; g=(gi)i∈Zn∈l2; f(u)=(fi(ui))i∈Zn and h(u˙)=(hi(u˙i))i∈Zn are nonlinear terms satisfying some growth assumptions to be given later; A is the linear operator on l2. In (1), W(t)=W(t,ω)=∑i∈Znaiwi(t,ω)ei, where a=(ai)i∈Zn∈l2 and ei∈l2 denotes the element having 1 at position i and all the other components 0 and {wi,i∈Zn} are independent two-side Brownian motions.Lattice dynamical systems (LDSs) are infinite systems of ordinary differential equations, modeled on an underlying spatial lattice with some regular structure, for example, the integer lattice in the plane. Such systems arise as models in many applications, including image processing and pattern recognition, electrical engineering laser systems, biology, and material science; see [1–8] and the references therein. LDSs in one sense lie between ordinary and partial differential equations, but very often they exhibit new phenomena not found in either of these fields. LDSs raise a host of challenges to the researcher and are of broad interest to scientists and mathematicians.So far, various properties of solutions about LDSs have been studied by many authors, such as the traveling solutions, the chaotic properties of solutions, and the phenomena of synchronization (see, e.g., [4, 5, 9]). One of the most important problems in mathematical physics is understanding of the asymptotic behavior of dynamical systems. Global attractor theory is an important tool to study the asymptotic behavior of infinite dimensional systems. For dissipative infinite dimensional dynamical systems given by partial differential equations, global attractor theory has been well developed; see [10, 11] and references therein. Recently, the long-term behavior of LDS has gained the extensive attention. For the LDSs without noise, the first result on existence of global attractors was established by Bates et al. in [12]. Since then, much work has been done for either first-order or second-order deterministic LDS (see, e.g., [12, 13]).On the other hand, when modeling real world systems, stochastic disturbance is probably one of the main resources of the performance degradations of the dynamical systems, since the actual dynamic behavior is very often a noisy process brought on by random fluctuations from probabilistic causes. Stochastic systems have found successful applications in more and more branches of science and engineering. Random attraction as an interesting dynamic behavior has received increasing research attention. For stochastic partial differential equations, Ruelle has initiated the study of global random attractors in [14]. And the fundamental theory of global attractors for stochastic partial differential equations has been established and developed by Crauel, Debussche, Flandoli, Schmalfuss, and others; see, for example, [15–18] and the references therein. Very recently, much attention has been focused on lattice dynamical system with stochastic noises. Bates et al. [19] first studied the existence of global random attractor for a class of first-order dynamical systems driven by white noises on lattice Z. Then, Lv and Sun [20] have extended the result in [19] to generalized first-order stochastic systems on the lattice Zk. For some latest results on first-order random attractors, we refer readers to, for example, [21] and the references therein.For the second-order SLDS with stochastic noises on the latticeZ or Zk, the existence of the random attractor is receiving the attention from research community [22, 23]. For example, [22] investigated the asymptotic behavior for a class of second-order stochastic lattice dynamical systems and proved the existence of the random attractor for the concerned second-order SLDS. Paper [23] addressed the asymptotic behavior of solutions to second-order SLDS with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences and discussed the existence of a tempered random bounded absorbing set and a random attractor for the SLDS. However, the asymptotic behavior of second-order SLDS has not yet been fully investigated because of the technical complexity and remains open and challenging. In this paper, based on the idea of [13, 19], we aim to prove the existence of a global random attractor for a class of second-order SLDS (1). It is worth pointing out that the second-order SLDS considered in this paper is quite general, and many existing results can be viewed as the special cases of our results.This paper is organized as follows. In Section2, we introduce some basic concepts related to stochastic dynamical systems and the global random attractor. Meanwhile, we present some notations and give a simple description of our system. In Section 3, Some bounded conditions and assumptions of nonlinear terms are given, and the existence and uniqueness of solutions of system (1) are established. In Section 4, we prove the existence of an absorbing set. In Section 5, we establish the existence conditions for a global random attractor of system (1), and some concluding remarks are given in Section 6.


## 2. Preliminaries and Equivalent Norm

### 2.1. Preliminaries

In this subsection, we recall some basic concepts about random dynamical systems and the definition of random global attractor (see [17, 19, 24] for details).LetH,·H be a Hilbert space and (Λ,F,P) a probability space. Denote D as a collection of random subsets of H. A continuous random dynamical system (s(t,ω)) over (Λ,F, P, (θt)t∈R) is defined as follows.Definition 1.
A stochastic process(s(t))t≥0 is a continuous random dynamical system over (Λ, F, P, (θt)t∈R) if s(t) is (B[0,∞)×F×B(H),B(H))-measurable and, for all ω∈Λ, (S1)
the mappings(·,ω)(·):[0,∞]×H→H is continuous;(S2)
s(0,ω)(·) is the identity operator on H;(S3)
s(p+t,ω)(·)=s(t,θpω)∘s(p,ω)(·) for all p,t≥0 (cocycle property).Definition 2.
A random bounded setB(ω)⊂H is called tempered with respect to (θt)t∈R, if, for all ω∈Λ,
(2)lim⁡t→∞e-βtdBθtω=0,∀β>0,
where d(B)=sup⁡x∈B∥x∥H.Definition 3.
A random setK is called an absorbing set in D if, for all B∈D and a.e. ω∈Λ, there exists tB(ω)>0 such that
(3)st,θ-tωBθ-tω⊂Kω,∀t≥tBω.Definition 4.
A random setA is called a global random (D) attractor for s(t) if the following hold: (A1)
A is a random compact set; that is, ω↦d(x,A(ω)) is measurable for every x∈H, and A(ω) is compact for a.e. ω∈Λ;(A2)
A is a strictly invariant set;(A3)
A attracts all sets in D; that is, for all B∈D and a.e. ω∈Λ one has
(4)lim⁡t→∞dst,θ-tωBθ-tω,Aω=0,where d(A,B)=sup⁡x∈Ainf⁡y∈B∥x-y∥H is Hausdorff semimetric (for any A⊆H,B⊆H). Also, the collection D is called the domain of attraction of A.The following theorem is needed to prove the existence of global random attractor of system (1).Theorem 5.
LetK∈D be an absorbing set for the continuous random dynamical system (s(t))t≥0. Suppose the set K is closed and, for a.e. ω∈Λ, K satisfies the following asymptotic compactness condition: each sequence ϕn∈s(tn,θ-tn)(K(θ-tnω)) (when t→∞) has a convergent subsequence in H. Then, s(t,ω) has a unique global random attractor
(5)Aω=⋂τ≥tKω‍⋃t≥τ‍st,θ-tωKθ-tω¯.Proof.
The proof of this theorem is similar to [12], so it is omitted here.


### 2.2. Equivalent Norm

Let(6)l2=u=uii∈Zn∣∑i∈Zn‍ui2<+∞,hhhui∈R,i=i1,i2,…,in∈Znu=uii∈Zn∣∑i∈Zn‍ui2<+∞,.
For all u,v∈l2, we define the inner product (·,·) and norm ∥·∥ as follows:
(7)u,v=∑i∈Zn‍uivi,u2=u,u=∑i∈Zn‍ui2.In this paper, we assume that the linear operatorA in (1) can be decomposed as
(8)A=A1+A2+⋯+An,Aj=DjDj*,j=1,2,…,n,
where Dj and Dj*:l2→l2 are bounded linear operators defined by
(9)Djui=u(i1,…,ij+1,…,in)-u(i1,…,ij,…,in),Dj*ui=u(i1,…,ij-1,…,in)-u(i1,…,ij,…,in).
It follows readily that
(10)Ajui=2u(i1,…,ij,…,in)-u(i1,…,ij-1,…,in)-u(i1,…,ij+1,…,in).For allu=(ui)i∈Zn, v=(vi)i∈Zn∈l2, define the bilinear forms by
(11)u,vλ=γ∑j=1n‍Dju,Djv+λu,v,uλ2=γ∑j=1n‍Dju2+λu2,
where γ=1-εξ¯¯ (ε and ξ¯¯ are given constants).Now, we prove that the spacesl2=(l2,(·,·),∥·∥) and lλ2=(lλ2,(·,·)λ,∥·∥λ) are equivalent spaces.Lemma 6.
The two bilinear forms(·,·), (·,·)λ in (7) and (11) are both the inner products, and the resulting norms ∥·∥ in (7) and ∥·∥λ in (11) are equivalent.Proof.
It is easy to check that the two bilinear forms(·,·) and (·,·)λ are both the inner products. We now only need to show that the norms ∥·∥ and ∥·∥λ are equivalent. Noticing that
(12)λu2≤uλ2=γ∑j=1n‍Dju2+λu2,uλ2=γ∑j=1n‍Dju2+λu2≤γ∑j=1n‍∑i∈Zn‍ui1,…,ij+1,…,in-ui1,…,ij,…,in2+λu2≤4nγ+λu2,
it follows that the norms · and ·λ are equivalent. The proof is completed.LetF=lλ2×l2, where lλ2:=(lλ2,(·,·)λ,∥·∥λ). From Lemma 6, we know that F=lλ2×l2 is a Hilbert space.


## 2.1. Preliminaries

In this subsection, we recall some basic concepts about random dynamical systems and the definition of random global attractor (see [17, 19, 24] for details).LetH,·H be a Hilbert space and (Λ,F,P) a probability space. Denote D as a collection of random subsets of H. A continuous random dynamical system (s(t,ω)) over (Λ,F, P, (θt)t∈R) is defined as follows.Definition 1.
A stochastic process(s(t))t≥0 is a continuous random dynamical system over (Λ, F, P, (θt)t∈R) if s(t) is (B[0,∞)×F×B(H),B(H))-measurable and, for all ω∈Λ, (S1)
the mappings(·,ω)(·):[0,∞]×H→H is continuous;(S2)
s(0,ω)(·) is the identity operator on H;(S3)
s(p+t,ω)(·)=s(t,θpω)∘s(p,ω)(·) for all p,t≥0 (cocycle property).Definition 2.
A random bounded setB(ω)⊂H is called tempered with respect to (θt)t∈R, if, for all ω∈Λ,
(2)lim⁡t→∞e-βtdBθtω=0,∀β>0,
where d(B)=sup⁡x∈B∥x∥H.Definition 3.
A random setK is called an absorbing set in D if, for all B∈D and a.e. ω∈Λ, there exists tB(ω)>0 such that
(3)st,θ-tωBθ-tω⊂Kω,∀t≥tBω.Definition 4.
A random setA is called a global random (D) attractor for s(t) if the following hold: (A1)
A is a random compact set; that is, ω↦d(x,A(ω)) is measurable for every x∈H, and A(ω) is compact for a.e. ω∈Λ;(A2)
A is a strictly invariant set;(A3)
A attracts all sets in D; that is, for all B∈D and a.e. ω∈Λ one has
(4)lim⁡t→∞dst,θ-tωBθ-tω,Aω=0,where d(A,B)=sup⁡x∈Ainf⁡y∈B∥x-y∥H is Hausdorff semimetric (for any A⊆H,B⊆H). Also, the collection D is called the domain of attraction of A.The following theorem is needed to prove the existence of global random attractor of system (1).Theorem 5.
LetK∈D be an absorbing set for the continuous random dynamical system (s(t))t≥0. Suppose the set K is closed and, for a.e. ω∈Λ, K satisfies the following asymptotic compactness condition: each sequence ϕn∈s(tn,θ-tn)(K(θ-tnω)) (when t→∞) has a convergent subsequence in H. Then, s(t,ω) has a unique global random attractor
(5)Aω=⋂τ≥tKω‍⋃t≥τ‍st,θ-tωKθ-tω¯.Proof.
The proof of this theorem is similar to [12], so it is omitted here.


## 2.2. Equivalent Norm

Let(6)l2=u=uii∈Zn∣∑i∈Zn‍ui2<+∞,hhhui∈R,i=i1,i2,…,in∈Znu=uii∈Zn∣∑i∈Zn‍ui2<+∞,.
For all u,v∈l2, we define the inner product (·,·) and norm ∥·∥ as follows:
(7)u,v=∑i∈Zn‍uivi,u2=u,u=∑i∈Zn‍ui2.In this paper, we assume that the linear operatorA in (1) can be decomposed as
(8)A=A1+A2+⋯+An,Aj=DjDj*,j=1,2,…,n,
where Dj and Dj*:l2→l2 are bounded linear operators defined by
(9)Djui=u(i1,…,ij+1,…,in)-u(i1,…,ij,…,in),Dj*ui=u(i1,…,ij-1,…,in)-u(i1,…,ij,…,in).
It follows readily that
(10)Ajui=2u(i1,…,ij,…,in)-u(i1,…,ij-1,…,in)-u(i1,…,ij+1,…,in).For allu=(ui)i∈Zn, v=(vi)i∈Zn∈l2, define the bilinear forms by
(11)u,vλ=γ∑j=1n‍Dju,Djv+λu,v,uλ2=γ∑j=1n‍Dju2+λu2,
where γ=1-εξ¯¯ (ε and ξ¯¯ are given constants).Now, we prove that the spacesl2=(l2,(·,·),∥·∥) and lλ2=(lλ2,(·,·)λ,∥·∥λ) are equivalent spaces.Lemma 6.
The two bilinear forms(·,·), (·,·)λ in (7) and (11) are both the inner products, and the resulting norms ∥·∥ in (7) and ∥·∥λ in (11) are equivalent.Proof.
It is easy to check that the two bilinear forms(·,·) and (·,·)λ are both the inner products. We now only need to show that the norms ∥·∥ and ∥·∥λ are equivalent. Noticing that
(12)λu2≤uλ2=γ∑j=1n‍Dju2+λu2,uλ2=γ∑j=1n‍Dju2+λu2≤γ∑j=1n‍∑i∈Zn‍ui1,…,ij+1,…,in-ui1,…,ij,…,in2+λu2≤4nγ+λu2,
it follows that the norms · and ·λ are equivalent. The proof is completed.LetF=lλ2×l2, where lλ2:=(lλ2,(·,·)λ,∥·∥λ). From Lemma 6, we know that F=lλ2×l2 is a Hilbert space.


## 3. Existence and Uniqueness of Solutions

In this section, we will deal with the existence and uniqueness of solutions of system (1). For system (1), we make the following assumption (bounded conditions).(C1):(13)0<ξ¯≤ξi≤ξ¯¯<+∞,0<λ≤ηi≤η¯¯<+∞,hhhhhhhhhhhhhhhhhhhhhi=i1,i2,…,in∈Zn,
where λ, η¯¯, ξ¯, and ξ¯¯ are known positive constants;(C2): For alli∈Zn, fi(s)∈C1(R) and for each of bounded sets B, sup⁡s∈B|fi′(s)|<+∞; moreover,
(14)fiss≥c1Vis≥c2s2p+2≥0,∀s∈R,i∈Zn,(15)fis≤c2s2p+1+s,∀s∈R,i∈Zn,
where c1, c2, and p are positive constants and Vi(s)=∫0s‍fi(t)dt;(C3): Assumehi∈C1((R,Λ),R). For all ω∈Λ, hi(0,ω)=0. Furthermore, there exist constants α and β such that
(16)0<α≤hi′x,ω≤β<+∞,∀x∈R,ω∈Λ,i∈Zn.
Also, hi satisfies cocycle property
(17)hit+τ,ω=hit,θτω+hiτ,ω,hhhhhhhh∀τ,t≥0,ω∈Λ,i∈Zn;(C4):(18)g=gii∈Zn∈l2.(C5):W(t) is a Brownian motion with values in l2 defined on the probability space (Λ,F,P), where Λ={ω∈C(R,l2):ω(0)=0}, F is a complete σ-algebra, and P is the corresponding Wiener measure on F. To be specific, W(t)=W(t,ω)=∑i∈Znaiwi(t,ω)ei, where a=(ai)i∈Zn∈l2 and ei∈l2 denotes the element being 1 at position i and all the other components being 0 and {wi,i∈Zn} are independent of the two-side Brownian motions.Remark 7.
As pointed out in [19], the function f(s)=|u|2pu satisfies condition (C2).Next, we show the existence and uniqueness of the solution to system (1) under assumptions (C1)–(C5).Letφ=(u,v)T where v=u˙+ϵu-W, u∈l2 and ϵ=αλ/(β2+4λ), and (4p+2)/(4p+3)≤ϵ≤1. Equation (1) can be rewritten as the following equation with initial condition:
(19)φ˙=Cφ+Eφ+Gφ,φ0=u0,u10+ϵu0T,
where
(20)Cφ=-ϵ1ϵξA-A-η-ϵ2-ξA+ϵφ,Gφ=0-fu+g,Eφ=W-ξA+ϵW-hv-ϵu+W.Lemma 8.
For allω∈Λ, if conditions (C1)–(C5) hold, then the operators E and G in (19) map F=lλ2×l2 into themselves, and they are locally Lipschitz on F.Proof.
From the above assumptions, it follows that(21)f(u)2=fu-f02=∑i∈Zn‍fiui-fi02=∑i∈Zn‍fi′θiui2ui2≤sup⁡r∈-u,uf′r2u2.
By the definition of W(t), for all ω∈Λ, we have
(22)hv-ϵu+W2=∑i∈Zn‍hivi-ϵui+aiwit,ω2=∑i∈Zn‍hi′θivi-ϵui+aiwit,ω2hhhh×vi-ϵui+aiwit,ω2≤β2∑i∈Zn‍vi-ϵui+aiwit,ω2≤3β2v2+ϵ2u2+Wt,ω2,
where θi∈(0,1), i=(i1,i2,…,in)∈Zn. Hence, we can infer that f(u), h(v-ϵu+W)∈l2 for all u=(ui)i∈Zn, v=(vi)i∈Zn∈l2,ω∈Λ.
LetB be a bounded set in F, φj=(uj,vj)=((ui(j)),(vi(j)))i∈Zn∈B, j=1,2. Similar to the derivation of (21) and (22), there exists a constant L(B) dependent on the bounded set B such that
(23)Gφ1-Gφ2F2=fu1-fu22=∑i∈Zn‍fiui1-fiui22≤LBu1-u22≤LBφ1-φ2F2,Eφ1-Eφ2F2=hv1-ϵu1+Wfff-hv2-ϵu2+W2≤2β2v1-v22+ϵ2u1-u22≤2β21+ϵ2λφ1-φ2F2.
The above two inequalities imply that E and G are locally Lipschitz on F, and the proof is then complete.Theorem 9.
If (C1)–(C5) hold, then, for any initial dataφ(0)=(u0,u10+ϵu0)T∈F, there exists a unique local solution φ(t)=(u(t),v(t))T of (19) such that φ∈L2(Λ,C([0,T],F)), where T is a positive constant. In addition, for all ω∈Λ, we have the following estimate:
(24)sup⁡t∈0,TφtF2≤M2φ0F2+M1∫0T‍Wt,ω2+Wt,ω2p+2hhhhhhhhhh+g2dt,
where
(25)M1=max⁡12σ+α,c2p+1,4nγλ+λ2+c22σλ,M2=1+2c1λmax⁡s∈-u0,u0f′s.
Moreover, the solution of (19) depends continuously on the initial data φ(0); that is, for each ω∈Λ, the mapping φ(0)∈F↦φ(·,ω,φ(0))∈C([0,T],F) is continuous.Proof.
Taking the inner product(·,·)F of (19) with φ(t)=(u(t),v(t))T=(u(t),u˙(t)+ϵu+W)T∈F, we have
(26)φ˙,φF=Cφ,φF+Eφ,φF+Gφ,φF.
DenoteC*(φ)=C(φ)+0-h(v-ϵu+W) and E*(φ)=W-(ξA+ϵ)W.
It is easy to check that(27)φ˙,φF=Cφ,φF+Eφ,φF+Gφ,φF=C*φ,φF+E*φ,φF+Gφ,φF.
Now, let us estimate the terms of (27). First, we get
(28)C*φ,φF=v-ϵuϵξAu-Au-ηu-ξAv+ϵv-ϵ2u,uvF+0-hv-ϵu+W,uvF=v-ϵu,uλ+ϵξAu-Au-ηu-ξAv+ϵv,v-ϵ2u,v-hv-ϵu+W,v.
By condition (C3), we have
(29)ϵ2u,v+hv-ϵu+W,v≥αv2-ϵβ-ϵuv+αW,v.
Noticing thatγ=1-ϵξ¯¯, it follows from (29) that
(30)C*φ,φF≤-σφF-α2v2-αW,v+σ-ϵuλ2+σ-ϵ-α2v2+βϵλuλv.
Lettingσ=αλ/(β2+4λ(β+β2+4λ)), we can see that 4(ϵ-σ)(α/2-ϵ-σ)=(β2ϵ2)/λ, and then
(31)C*φ,φF≤-σφF-α2v2-αW,v.
From Young’s inequality, it follows that(32)E*φ,φF=W-ξA+ϵW,uvF=W,uλ+-ξA+ϵW,v≤12σWλ2+σ2uλ2+-ξAW,v+ϵW,v,(33)Gφ,φF=0fu+g,uvF=-fu,v+g,v.
By using Young’s inequality again, we have(34)g,v≤122σ+αg2+2σ+α2v2.
Based on assumption (C2) and the definition ofϵ, it follows that
(35)-fu,v=-fu,u˙+ϵu-W=-fu,u˙-ϵfu,u+fu,W=-∑i∈Zn‍fiuiu˙i-ϵ∑i∈Zn‍fiuiui+fu,W≤-ddt∑i∈Zn‍Viui-c1ϵ∑i∈Zn‍Viui+c2∑i∈Zn‍ui2p+1aiwit+c2∑i∈Zn‍uiaiwit≤-ddt∑i∈Zn‍Viui-ϵc2∑i∈Zn‍ui2p+2+c22p+12p+2∑i∈Zn‍ui2p+2+c212p+2W2p+2+c2∑i∈Zn‍uiaiwitHo¨lder’sinequality≤-ddt∑i∈Zn‍Viui+c212p+2W2p+2+σλ2u2+c222σλW2Young’sinequality.
Combining (31)–(35) with (27), we can calculate that
(36)dφF2dt+2ddt∑i∈Zn‍Viui≤M1g2+Wt2+Wt2p+2+-ξAW,v+ϵW,v-αW,v.
Sinceϵ=αλ/(β+4λ), it is easy to check that ϵ<α. Also by condition (C1), one has
(37)-ξAW,v+ϵW,v-αW,v<0.
Substituting (37) into (36) yields
(38)dφF2dt+2ddt∑i∈Zn‍Viui≤M1g2+Wt2+Wt2p+2.
From (14) it follows that
(39)∑i∈Zn‍Viui0≤1c1∑i∈Zn‍fiui0ui0≤1c1max⁡s∈-u0,u0f1′su02,
and then a combination of the above inequality and (38) leads to
(40)φF2≤φ0F2+2c1max⁡s∈-u0,u0f′su02+M1∫0t‍g2+Wτ2+Wτ2p+2dτ.
Hence, one has(41)φF2≤M2φ0F2+M1∫0t‍g2+Wτ2+Wτ2p+2dτ,
which implies that, for all ω∈Λ, ∥φ(t)∥F is bounded. So, for any T∈R+, (19) has a global solution on any interval [0,T], and therefore for all ω∈Λ,T>0, we have
(42)sup⁡t∈0,TφtF2≤M2φ0F2+M1∫0T‍g2+Wτ2+Wτ2p+2dτ,
which indicates that (22) has a global solution φ∈L2(Λ,C[0,T],F).
Next, we show that the solutions of (19) are dependent continuously on initial conditions. Let φi(0)=(u0(i),u10(i)+ϵu0(i))∈F and ∥φi(0)∥F<r0 and assume φi(t)=(φ(t,φi(0)))=(u(i),u˙(i)+ϵu(i)-W(t))(i=1,2) are the solutions of (19), where r0 is a constant.
SetR0=M2r02+M1∫0T‍(∥g∥2+∥W(τ)∥2+∥W(τ)∥2p+2)dτ. Since φ1(t) and φ2(t) are the solutions of (19), we have
(43)dφ2t-φ1tdt=Cφ2-Cφ1+Eφ2-Eφ1+Gφ2-Gφ1.
Taking the inner product of (43) with (φ2-φ1) in F, we get
(44)12dφ2-φ1F2dt=Cφ2-Cφ1,φ2-φ1F+Eφ2-Eφ1,φ2-φ1F+Gφ2-Gφ1,φ2-φ1F.
By Lemma8, we have
(45)12dφ2-φ1F2dt≤Cφ2-Cφ1,φ2-φ1F+β21+ϵ2λφ2-φ1F2+LR0φ2-φ1F2.
It is easy to see the operator C(·) in (19) is a linear operator, and then it follows from assumption (C1) that there exists a positive constant C0 such that ∥C∥F≤C0, where C0 depends only on the constants ϵ, ξ¯¯, ∥A∥F, and η¯¯. Hence, (45) implies that
(46)dφ2-φ1F2dt≤2C0+β21+ϵ2λ+LR0φ2-φ1F2.
Furthermore, by Grownwall inequality, it is clear that
(47)φ2-φ1F2≤φ20-φ10F2×exp⁡2C0+β21+ϵ2λ+LR0t,hhhhhhhhhhhhhhhhhhhhhhhhhhhhfort∈0,T.
Hence, we have
(48)sup⁡t∈0,Tφ2-φ1F2≤φ20-φ10F2×exp⁡2C0+β21+ϵ2λ+LR0T,
which implies that the solutions of system (19) depend continuously on the initial data. The proof is now complete.From assumption (C5) and noticingθτωt=ωτ+t-ωτ(τ,t∈R), it is easy to see that (Λ,F,P,(θt)t∈R) is a metric dynamical system. Also, from the definition of (θt)t∈R, we have
(49)Wt+τ,ω=Wt,θτω+Wτ,ω,∀τ,t∈R.Now, for anyt≥0,ω∈Λ, we introduce the map from F into F as follows:
(50)st,ωφ0=φt,ω,φ0,
where φ(t,ω,φ0) is the solution of (19) with initial data φ0 and s(·,ω) is continuous for ω from [0,∞]×F to F since the solution of (19) is dependent on initial data continuously.By (49), (C3), and Theorem 9, it is easy to check that (50) defines a continuous random dynamical system {s(t,ω)}t≥0,ω∈Λ over (Λ,F,P,(θt)t∈R).


## 4. Existence of the Absorbing Set

In this section, we are concerned with the existence of an absorbing setK(ω) for random dynamical system s(t,ω) generated by the stochastic system (19). We first introduce an Ornstein-Uhlenbeck process in l2 on the metric dynamic system (Λ,F,P,(θt)t∈R) (see [19, 24] for details).Lettingy(θtω)=-ϵ∫-∞0‍eϵτθtω(τ)dτ (t∈R) where ϵ=αλ/(β2+4λ), then y solves the Itô equation
(51)dy+ϵydt=dWt,fort≥0.From the properties of the Ornstein-Uhlenbeck process, we know that there exists aθt-invariant set Λ0⊂Λ of full P measure, and the following properties hold:(Y1)
the mappings→y(θsω) is continuous for each ω∈Λ0;(Y2)
the random variable∥y(ω)∥ is tempered;(Y3)
there exists a tempered functionb(ω)>0 such that
(52)yθtω2p+2+yθtω2≤bθtω≤bωe(ϵ/2)|t|.Theorem 10.
There exist aθt-invariant set Λ0⊂Λ of full P measure and an absorbing set K(ω),ω∈Λ0 for s(t,ω)ψ0. That is, for all B∈D and all ω∈Λ0, there exists TB(ω)>0 such that
(53)st,θ-tωBθ-tω⊂Kω,∀t>TBω.
Moreover, K∈D; that is, for all ω∈Λ0, there exists TK(ω) such that
(54)st,θ-tωKθ-tω⊂Kω,∀t>TKω.Proof.
Lettingψ(t)=ψ(t)=(u(t),v*(t))T=(u(t),u˙(t)+ϵu(t)-y(θtω))T=φ(t)+(0,W(t,ω)-y(θtω))T, where φ(t) is a solution of (19), then, for any ω∈Λ, y(ω) has properties (Y1), (Y2), and (Y3). By the Itô equation (51), it can be inferred that ψ(t) satisfies
(55)dψtdt=Cψ+Gψ+yθtω-hv*-ϵu+y+-ξA+2ϵyθtω.
Taking the inner product of (55) with ψ in F, we obtain
(56)12dψtF2dt=Cψ,ψF+Gψ,ψF+yθtω-hv*-ϵu+y+-ξA+2ϵyθtω,uv*F=C*φ,φF+E*φ,φF+G*φ,φF.
Similar to the derivation of (31) and (32), we have
(57)C*ψ,ψF=Cψ,ψFhhh+0-hv*-ϵu+y,uv*F≤-σψF2-α2v*2-αyθtω,v*;(58)E*ψ,ψF≤1σy(θtω)λ2+σ4uλ2+-ξAyθtω,v*+ϵyθtω,v*,
where σ, E*(·), and C*(·) are defined in (29) and (26), respectively.
Notice that(59)Gψ,ψF=0fu+g,uv*F=-fu,v*+g,v*=-fu,u˙+ϵu-yθtω+g,v*.
Similar to the derivation of (35), by condition (C2), (17), and (4p+2)/(4p+3)≤ϵ≤1, one has
(60)-fu,u˙+ϵu-yθtω=-fu,u˙-ϵfu,u+fu,yθtω≤-ddt∑i∈Zn‍Viui-c1ϵ∑i∈Zn‍Viui+c2∑i∈Zn‍ui2p+1aiyiθtω+c2∑i∈Zn‍uiaiyiθtω≤-ddt∑i∈Zn‍Viui-c1ϵ4p+4∑i∈Zn‍Viui-c2ϵ4p+34p+4∑i∈Zn‍ui2p+2+c22p+12p+2∑i∈Zn‍ui2p+2+c22p+2y2p+2+σλ4u2+c22σλy2≤-ddt∑i∈Zn‍Viui-c1ϵ4p+4∑i∈Zn‍Viui+c22p+2y2p+2+σλ4u2+c22σλy2,
and, by Young’s inequality, we also have
(61)g,v*≤12σ+αg2+σ+α2v*2.
Substituting (57)–(61) into (56), we obtain
(62)ddtψF2+2∑i∈Zn‍Viui≤-σψF2-2c1ϵ∑i∈Zn‍Viui+1σ+αg2+M3yθtω2+yθtω2p+2+-ξAyθtω,v*+2ϵyθtω,v*-αyθtω,v*.
By the definition of constantϵ, we find that 2ϵ<α, and then
(63)-ξAyθtω,v*+2ϵyθtω,v*-αyθtω,v*<0.
Thus, we arrive at
(64)ddtψF2+2∑i∈Zn‍Viui≤-σψF2-2c1ϵ∑i∈Zn‍Viui+1σ+αg2+M3yθtω2+yθtω2p+2.
Furthermore, from Grownwall inequality, it follows that
(65)ψF2≤M2ψ0F2e-c3t+1c3σ+αg2+M3∫0t‍ec3s-tyθsω2+yθsω2p+2ds,
where
(66)M3=max⁡c2p+1,4nγλ+λ2+c22σλ,c3=min⁡σ,c1ϵ4p+4.
Therefore, it follows from (65) and property (Y3) that
(67)ψt,θ-tω,ψ0θ-tωF2≤M2ψ0θ-tωF2e-c3t+1c3σ+αg2+M3∫0t‍ec3s-tyθs-tω2+yθs-tω2p+2ds≤M2ψ0θ-tωF2e-c3t+1c3σ+αg2+M3∫-t0‍ec3τyθτω2+yθτω2p+2dτ≤M2ψ0θ-tωF2e-c3t+1c3σ+αg2+M3∫-t0‍ec3τbωe(ϵ/2)|τ|dτ≤M2ψ0(θ-tω)F2e-c3t+1c3(σ+α)g2+2M32c3+ϵb(ω)≤M2ψ0θ-tωF2e-c3t+B~ω,
where B~(ω)=(1/c3(σ+α))∥g∥2+(2M3/(2c3+ϵ))b(ω) and B~(ω) is tempered thanks to the tempered function b(ω). Then, K~(ω)={ψ∈F,∥ψ∥F≤B~(ω)} is an absorbing set for ψ(t,ω,ψ(0)); that is, for all B∈D and all ω∈Λ0, there exists TB(ω)>0 such that ψ(t,θ-t,B(θ-tω))⊂K~(ω) for all t≥TB(ω).
LetK(ω)={φ∈F,∥φ∥F≤B~(ω)+∥y(ω)∥}; then K(ω) is an absorbing set for φ(t,ω,φ0)=ψ(t,ω,φ0-(0,y(ω))T-(0,y(θtω)-W(t,ω))T). Moreover, K∈D. The proof is now complete.


## 5. Existence of a Global Random Attractor

In this section, we will show the existence of global random attractor related to the random lattice dynamical systems(t,ω) generated by system (19). In order to apply the result of Theorem 5, we need to prove the following lemmas.Lemma 11.
Let (C1)–(C5) hold andφ(0)=(u0,u10+ϵu0)∈K(ω). Then, for any μ>0, there exist T(μ,ω)>0 and N(μ,ω)>0 such that the solution of (19) satisfies
(68)∑∥i∥m≥N(μ,ω)‍φit,θ-tω,φ0θ-tωF2≤μ,∀t≥Tμ,ω,
where ∥i∥m=max⁡1≤j≤n|ij|, for any i=(i1,i2,…,in)∈Zn.Proof.
Assumeϑ∈C1(R+,R) is a smooth function satisfying
(69)ϑt=0,0≤t≤1;0≤ϑt≤1,1≤t≤2;ϑt=1,t≥2,
and there exists a constant M0 such that ϑ′t≤M0fort∈R+.
Letψ(t)=(u(t),v*(t))T=(ui(t),vi*(t))i∈ZnT be a solution of (55), where v*(t)=u˙(t)+ϵu(t)-y(θtω),ω∈Λ0.
SupposeM is a suitable large constant to be defined later. Set ψ~=(w~,z~)=(w~i,z~i)i∈Zn, where
(70)w~i=ϑimMui,z~i=ϑimMvi*
for all i∈Zn.
Taking the inner product(·,·)F of (55) with ψ~, we have
(71)dψtdt,ψ~tF=C*ψ,ψ~F+Gψ,ψ~F+yθtω-ξA+2ϵyθtω,w~z~F.
It is easy to see
(72)dψtdt,ψ~tF=12ddt∑i∈Zn‍ϑimMψi(t)F2,(73)C*ψ,ψ~F=Cψ,ψ~F+0-hv*-ϵu+y,w~z~F=v*-ϵu,w~λ+ϵξAu-Au-ηu-ξAv*+ϵv*-ϵ2u,z~+-hv*-ϵu+y,z~≤γ∑j=1n‍Djv*,Djw~+λv*,w~-ϵγ∑j=1n‍Dju,Djw~-ϵλu,w~+ϵξ¯¯∑j=1n‍Dju,Djz~-∑j=1n‍Dju,Djz~-λu,z~-ξ¯∑j=1n‍Djv*,Djz~-ϵ2u,z~+-hv*-ϵu+y,z~≤γ∑j=1n‍Djv*,Djw~+ϵξ¯¯-1∑j=1n‍Dju,Djz~+λv*,w~-ϵγ∑j=1n‍Dju,Djw~-ϵλu,w~-λ(u,z~)-ξ¯∑j=1n‍Djv*,Djz~-ϵ2u,z~+-hv*-ϵu+y,z~,
where the last inequality follows from assumption (C1), (9), and the definition of operator A in (8).
For simplicity, we denoteij=(i1,…,ij+1,…,in) and i=(i1,…,ij,…,in). Since γ=1-ϵξ¯¯, by the definition of ϑ, we have
(74)γ∑j=1n‍Djv*,Djw~+ϵξ¯¯-1∑j=1n‍Dju,Djz~=γ∑j=1n‍∑i∈Zn‍ϑijmMuijvij*-vi*hhhhhhhhhhhh-ϑimMuivij*-vi*ϑijmMuijvij*-vi*hhhhhhhhhhh-ϑijmMvij*uij-uihhhhhhhhhhhhhh-ϑimMvij*uij-uiϑijmMvij*uij-uiϑijmMuijvij*-vi*≤γ∑j=1n‍∑i∈Zn‍ϑijmM-ϑimMuijvi-uivij≤γM0Mγ∑j=1n‍∑i∈Zn‍uijvi-uivij≤2nγM0Muv,∀t∈0,T,T>0,ω∈Λ0,(75)∑j=1n‍Dju,Djw~=∑j=1n‍∑i∈Zn‍ϑijmM-ϑimMuijui-uijhhhhhhhhhhhh+ϑimMuij-ui2ϑijmM-ϑimMuijui-uij≥-2nM0Mu2,T>0,ω∈Λ0.
Also, similar to the derivation of (75), we get
(76)∑j=1n‍Dju,Djz~=∑j=1n‍∑i∈Zn‍ϑijmM-ϑimMvij*vi*-vij*hhhhhhhhhhh+ϑimMvij*-vi*2ϑijmM-ϑimMvij*vi*-vij*≥-2nM0Mv*2,T>0,ω∈Λ0,
where ∥ij∥m=max⁡{ij+1,∥i∥m}.
It is not difficult to see(77)v*,w~=∑i∈Zn‍ϑimMuivi*,(78)u,w~=∑i∈Zn‍ϑimMui2,(79)u,z~=∑i∈Zn‍ϑimMuivi*,(80)v*,z~=∑i∈Zn‍ϑimMvi*2.
Substituting (74)–(80) into (73) yields
(81)C*ψ,ψF≤2nγM0Muv+2nM0ϵγMu2+2nM0ξ¯Mv*2-∑i∈Zn‍ϑimMσψiF2+α2vi*2-αy,z~.
Also, it is obvious that
(82)E*ψ,ψ~F=yθtω-ξA+2ϵyθtω,w~z~F=∑i∈Zn‍ϑimMui,yiλ+-ξA+2ϵyθtω,z~.
Next, we estimate the term (G(ψ),ψ~)F in (71) as follows:
(83)Gψ,ψ~F=0fu+g,w~z~F=fu,z~+g,z~=∑i∈Zn‍fiui,ϑimMu˙i+∑i∈Zn‍fiui,ϑimMui-∑i∈Zn‍fiui,ϑimMyi+∑i∈Zn‍gi,ϑimMvi*.
By assumption (C2), we have
(84)∑i∈Zn‍fiui,ϑimMu˙i=ddt∑i∈Zn‍ϑimMViui,(85)∑i∈Zn‍fiui,ϑimMui=∑i∈Zn‍ϑimMViui,(86)∑i∈Zn‍fiui,ϑimMyi≤∑i∈Zn‍ϑimMc2ui2p+2yi+uiyi.
Furthermore, by Young’s inequality, we know
(87)∑i∈Zn‍gi,ϑimMvi*≤σ+α2∑im≥M‍ϑimMvi*2+12σ+α∑im≥M‍gi2,(88)uλv≤12λuλ2+λ2v2.
Substituting (72), (81)–(88) into (71) leads to
(89)ddt∑i∈Zn‍ϑimMψiF2+2Viui≤-σ∑i∈Zn‍ϑ(imM)ψiF2-2c1ϵ∑i∈Zn‍ϑimMViui+4γM0nMλuλv+4M0nϵγMλuλ+2M0nξ¯Mv*2+M4∑im≥M‍yi2+∑im≥M‍yi2p+2+1σ+α∑im≥M‍gi2,
where M4 only depends on α, β, n, p, λ, and c2.
Now, settingM5=max⁡2rM0n1+ϵ/Mλ,M0n(2γ+¯ξ)/M, we estimate
(90)ddt∑i∈Zn‍ϑimMψiF2+2Viui≤-σ∑i∈Zn‍ϑ(imM)ψiF2-2c1ϵ∑i∈Zn‍ϑimMViui+M5ψiF2+1σ+α∑im≥M‍gi2+M4∑im≥M‍yi2+∑im≥M‍yi2p+2.
By Grownwall inequality, for all t≥TK=TK(ω), we can deduce that
(91)∑i∈Zn‍ϑimMψit,ω,ψ0ωF2≤M2e-c3t-TK∑i∈Zn‍ϑimMψiTK,ω,ψ0ωF2+M5∫TKt‍ec3τ-tψτ,ω,ψ0ωF2dτ+1c3σ+α∑im≥M‍gi2+M4∫TKt‍ec3τ-t×∑im≥M‍yiθτω2p+2+yiθτω2dτ,
where c3, M2, M4, and M5 are defined above.
Next, let us estimate each term on the right hand side of (91). From (65) in Theorem 10 and property (Y3), we find that
(92)e-c3t-TK∑i∈Zn‍ϑimMψiTk,θ-tω,ψ0θ-tωF2≤e-c3(t-TK)M2ψ0θ-tωF2e-c3TK+1c3σ+αg2hhhhhhhhhhhh+M3∫0TK‍ec3(s-TK)yθs-tω2p+2hhhhhhhhhhhhhhhhhhhhhhhhhhhhh+yθs-tω2dsM2ψ0θ-tωF2e-c3TK+1c3σ+αg2M2ψ0θ-tωF2e-c3TK+1c3σ+αg2≤M2ψ0θ-tωF2e-c3t+1c3σ+αg2e-c3(t-TK)+2M32c3+ϵbωe-(ϵ/2)(t-TK).
Hence, for any given constant μ>0, there exists T1(μ,ω)>TK(ω) such that, for all t>T1(μ,ω),
(93)M2e-c3t-TK∑i∈Zn‍ϑimMψiTK,θ-tω,ψ0θ-tωF2≤14μ.
Using (65) in Theorem 10 and property (Y3) again, we have
(94)∫TKt‍ec3(τ-t)ψτ,ω,ψ0ωF2dτ=∫TKt‍ec3(τ-t)M2ψ0θ-tωF2e-c3τ+1c3σ+αg2hhhhhhhhhhhhh+M3∫0τ‍ec3(s-τ)yθs-tω2p+2hhhhhhhhhhhhhhhhhhhhhhhhhhh+yθs-tω2dsM2ψ0θ-tωF2e-c3τ+1c3σ+αg2dτ≤M2ψ0θ-tωF2t-TKec3t+1c3σ+αg2+4M3ϵ4c3+ϵbω.
Sinceψ0(θ-tω)∈K(θ-tω), from Theorem 10, we have ∥ψ0(θ-tω)∥≤B~(θ-tω). So, for any μ, there exist T2(μ,ω)>TK(ω), N1(μ,ω)>0, and M>N1(μ,ω) such that
(95)M5∫TKt‍ec3(τ-t)ψτ,ω,ψ0ωF2dτ≤14μ.
By assumption (C4), there existsN2(μ,ω)>0 such that, for all M>N2(μ,ω), the following inequality holds:
(96)1c3σ+α∑im≥M‍gi2≤14μ.
LetT0 be a positive constant to be determined later. When t>T0+TK, we have
(97)M4∫TKt‍ec3s-t∑im≥M‍yiθs-tω2p+2+yiθs-tω2ds=M4∫TK-t0‍ec3τ∑im≥M‍yiθτω2p+2+yiθτω2dτ=M4∫-T00‍ec3τ∑im≥M‍yiθτω2p+2+yiθτω2dτ+M4∫TK-t-T0‍ec3τ∑im≥M‍yiθτω2p+2+yiθτω2dτ.
ChoosingT0≥2/(2c3+ϵ)ln⁡16M4b(ω)/μ(2c3+ϵ) and by property Y3, we have
(98)M4∫TK-t-T0‍ec3τ∑im≥M‍yiθτω2p+2hhhhhhhhhhhhhhhhhh+yiθτω2dτ≤18μ.
For the fixed T0, from Lebesgue’s theorem, there exist N3(μ,ω)>0 and M>N3(μ,ω) such that
(99)M4∫-T00‍ec3τ∑im≥M‍yiθτω2p+2hhhhhhhhhhhhhhhh+yiθτω2dτ≤18μ.
By setting
(100)Tμ,ω=max⁡T1μ,ω,T2μ,ω,TKμ,ω+T0μ,ω,N*μ,ω=max⁡N1μ,ω,N2μ,ω,N3μ,ω,
then, for t>T(μ,ω) and M>N*(μ,ω), we obtain that
(101)∑im≥2M‍ψit,θ-t,ψ0θ-tF2≤∑i∈Zn‍ϑimMψit,θ-t,ψ0θ-tF2≤μ.
It means that if constant N(μ,ω) is large enough,
(102)∑im≥N(μ,ω)‍φit,θ-t,ψ0θ-tF2≤2∑im≥Nμ,ω‍ψit,θ-t,ψ0θ-tF2+yiω2≤4μ
holds. The proof of the lemma is now completed.Finally, we prove the asymptotic compactness of the absorbing setK(ω).Theorem 12.
If assumptions (C1)–(C5) hold, then for anyω∈Λ0 the set K(ω) is asymptotically compact.Proof.
For anyω∈Λ, consider the sequence dn=s(tn,θ-tnω)xn in s(tn,θ-tnω)K(θ-tnω), where xn∈K(θ-tnω) and {tn}n∈N is an increasing sequence in R+ with tn→+∞ as n→+∞.
First, let us show that(dn)n∈N has a convergent subsequence. Since K(ω) is a bounded absorbing set, s(tn,θ-tnω)xn∈K(ω) holds for large n. Then, there exists a subsequence of {s(tn,θ-tnω)xn} (still denoted by {s(tn,θ-tnω)xn}) such that
(103)stn,θ-tnωxn⟶x0weaklyinF.
In what follows, we prove that the weak convergence (103) is actually strong convergence. In other words, we will show that, for every μ>0, there exists N~(μ,ω)>0 such that when n>N0(μ,ω),
(104)stn,θ-tnωxn-x0F2≤μ.
By Lemma11, there exist N~1(μ,ω)>0 and K1(μ,ω)>0 such that, for n>N~1(μ,ω),
(105)∑im≥K1(μ,ω)‍stn,θ-tnωxniF2≤18μ2.
Also, sincex0∈F, there exists K2(μ) such that
(106)∑im≥K2(μ)‍x0iF2≤18μ2.
LetK(μ,ω)=max⁡{K1(μ,ω),K2(μ)}. By (103), we infer that
(107)stn,θ-tnxni∥i∥m≤Kμ,ω⟶x0i∥i∥m≤Kμ,ωinR2nKμ,ω+1,hhhhhhhhhhhhhhihhhhhasn⟶∞,
which implies that there exists N~2(μ,ω)>0 such that when n≥N~2(μ,ω),
(108)∑im≤K(μ,ω)‍stn,θ-tnωxni-x0iF2≤12μ2.
SettingN~(μ,ω)=max⁡{N~1(μ,ω),N~2(μ,ω)}, we get from (105)–(108) that for n≥N~(μ,ω)(109)stn,θ-tnωxn-x0F2=∑im≤K(μ,ω)‍stn,θ-tnωxni-x0iF2+∑im≥K(μ,ω)‍stn,θ-tnωxni-x0iF2≤12μ2+12μ2≤μ2.
Therefore, we arrive at
(110)stn,θtnωxn⟶x0stronglyinF.
The proof is therefore completed.Finally, the main conclusion follows from Theorem5, Theorem 10, and Theorem 12.Theorem 13.
Assume that (C1)–(C5) hold. Then, for anyω∈Λ, the continuous random lattice dynamical system s(t,ω) generated by the general second-order stochastic lattice dynamical system (1) has a unique global random attractor.


## 6. Conclusions

In this paper, we have investigated the random attractors in second-order stochastic lattice dynamical systems. First, we first proved the uniqueness and existence of the solutions of second-order stochastic lattice dynamical systems in the spaceF=lλ2×l2. Then, by proving the asymptotic compactness of the random dynamical systems, we have established the existence of the global random attractor with the set of tempered bounded sets. These results could be further extended to more general nonlinear systems with uncertainties as in [25–27].



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