# Random Attractors for Stochastic Retarded Lattice Dynamical Systems

Authors: Xiaoquan Ding; Jifa Jiang
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

This paper is devoted to a stochastic retarded lattice dynamical system with additive white noise. We extend the method of tail estimates to stochastic retarded lattice dynamical systems and prove the existence of a compact global random attractor within the set of tempered random bounded sets.


---

## Body

## 1. Introduction

Lattice dynamical systems (LDSs) arise naturally in a wide variety of applications in science and engineering where the spatial structure has a discrete character. Among such examples are brain science [1], chemical reaction [2], material science [3], electrical engineering [4], laser systems [5], pattern recognition [6], complex network [7], and many others. On the other hand, LDSs also appear as spatial discretizations of partial differential equations on unbounded domains.There are many works concerning deterministic LDSs. For example, the traveling wave solutions were studied in [8, 9], the chaotic properties of solutions were examined by [6, 10], the long-time behavior of LDSs was investigated by [11–17]. In particular, Bates et al. [11] established the first result on the existence of a global attractor for LDSs. Wang [13], Zhou and Shi [14] used the idea of tail estimates on solutions and obtained, respectively, some sufficient and necessary conditions for the existence of a global attractor for autonomous LDSs. Later, the method of tail estimates is extended to nonautonomous LDSs [15–17].It is noted that an evolutionary system in reality is usually affected by external perturbations which in many cases are of great uncertainty or random influence. These random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. Therefore, it is of prime importance to take into account these random effects in some models, and this has led to stochastic differential equations. Random attractors for stochastic partial differential equations were first introduced by Crauel and Flandoli [18], Flandoli andSchmalfuss[19], with notable developments given in [20–25] and others. Bates et al. [26] initiated the study of random attractors for stochastic LDSs. Since then, many works have been done for the existence of random attractors for stochastic LDSs, see, for example, [27–34] and the references therein. Similarly to deterministic LDSs, the method of tail estimates also plays a key role in the study of the existence of random attractors for stochastic LDSs.On the other hand, in the natural world, the current rate of change of the state in an evolutionary system always depends on the historical status of the system. Then, it is more reasonable to describe the evolutionary systems by functional differential equations. Many papers are devoted to the study of the asymptotic behavior of deterministic functional differential equations, see, for example, [35–41] and the references therein. Especially, Zhao and Zhou [40, 41] considered the asymptotic behavior of some deterministic retarded LDSs and extended the method of tail estimates to deterministic retarded LDSs. More recently, Yan et al. [42, 43] discussed the asymptotic behavior of some stochastic retarded LDSs with global Lipschitz nonlinearities.Consider the Hilbert space(1.1)ℓ2={u=(ui)i∈ℤ:ui∈ℝ,∑i∈ℤ‍|ui|2<∞},
whose inner product and norm are given by
(1.2)(u,v)=∑i∈ℤ‍uivi,∥u∥2=∑i∈ℤ‍ui2,
for all u=(ui)i∈ℤ, v=(vi)i∈ℤ∈ℓ2. For ν>0, let 𝒞:=C([-ν,0];ℓ2) denote the Banach space of all continuous functions ξ:[-ν,0]→ℓ2 endowed with the supremum norm ∥ξ∥𝒞=sups∈[-ν,0]∥ξ(s)∥. For any real numbers a≤b,t∈[a,b] and any continuous function u:[a-ν,b]→ℓ2,ut denotes the element of 𝒞 given by ut(s)=u(t+s) for s∈[-ν,0].In this paper, we investigate the long time behavior of the following stochastic retarded LDS:(1.3)dui(t)=((ui-1-2ui+ui+1)-λiui+fi(uit)+gi)dt+aidwi(t),t>0,i∈ℤ,
with initial data
(1.4)ui(t)=ui0(t),t∈[-ν,0],i∈ℤ,
where u=(ui)i∈ℤ∈ℓ2, (λi)i∈ℤ is a bounded positive constant sequence, f=(fi)i∈ℤ:𝒞→ℓ2 is a nonlinear mapping satisfying local Lipschitz condition, g=(gi)i∈ℤ∈ℓ2, a=(ai)i∈ℤ∈ℓ2, and {wi:i∈ℤ} are independent two-sided real-valued Wiener processes on a probability space which will be specified later.It is worth mentioning that in the absence of the white noise, the existence of a global attractor for (1.3)-(1.4) was established in [40]. The main contribution of this paper is to extend the method of tail estimates to stochastic retarded LDSs and prove the existence of a random attractor for the infinite dimensional random dynamical system generated by stochastic retarded LDS (1.3)-(1.4). It is clear that our method can be used for a variety of other stochastic retarded LDSs, as it was for the nonretarded case.The paper is organized as follows. In the next section, we recall some fundamental results on the existence of a pullback random attractor for random dynamical systems. In Section3, we establish a necessary and sufficient condition for the relative compactness of sequences in C([-ν,0];ℓ2). In Section 4, we define a continuous random dynamical system for stochastic retarded LDS (1.3)-(1.4). The existence of the random attractor for (1.3)-(1.4) is given in Section 5.


## 2. Preliminaries

In this section, we recall some basic concepts related to random attractors for random dynamical systems. The reader is referred to [18–21, 26, 44, 45] for more details.Let(X,∥·∥X) be a separable Banach space with Borel σ-algebra ℬ(X) and (Ω,ℱ,ℙ) be a probability space.Definition 2.1.
(
Ω
,
ℱ
,
ℙ
,
(
ϑ
t
)
t
∈
ℝ
) is called a metric dynamical system if ϑ:ℝ×Ω→Ω is (ℬ(ℝ)⊗ℱ,ℱ)-measurable, ϑ0 is the identity on Ω, ϑs+t=ϑt∘ϑs for all s, t∈ℝ, and ϑtℙ=ℙ for all t∈ℝ.Definition 2.2.
A setA⊂Ω is called invariant with respect to (ϑt)t∈ℝ, if for all t∈ℝ, it holds
(2.1)ϑt-1A=A.Definition 2.3.
A continuous random dynamical system onX over a metric dynamical system (Ω,ℱ,ℙ, (ϑt)t∈ℝ) is a mapping
(2.2)φ:ℝ+×Ω×X→X,(t,ω,x)→φ(t,ω,x),
which is (ℬ(ℝ+)⊗ℱ⊗ℬ(X),ℬ(X))-measurable, and for all ω∈Ω, (i)
φ(t,ω,·):X→X is continuous for all t∈ℝ+;(ii)
φ(0,ω,·) is the identity on X;(iii)
φ(t+s,ω,·)=φ(t,ϑsω,·)∘φ(s,ω,·) for all s, t∈ℝ+.Definition 2.4.
A random setD is a multivalued mapping D:Ω→2X∖∅ such that for every x∈X, the mapping ω→d(x,D(ω)) is measurable, where d(x,B) is the distance between the element x and the set B⊂X. It is said that the random set is bounded (resp., closed or compact) if D(ω) is bounded (resp., closed or compact) for ℙ-a.e. ω∈Ω.Definition 2.5.
A random variabler:Ω→(0,∞) is called tempered with respect to (ϑt)t∈ℝ, if for ℙ-a.e. ω∈Ω(2.3)limt→∞e-βtr(ϑ-tω)=0∀β>0.
A random set D is called tempered if D(ω) is contained in a ball with center zero and tempered radius r(ω) for all ω∈Ω.Remark 2.6.
Ifr>0 is tempered, then for any τ∈ℝ, β>0 and ℙ-a.e. ω∈Ω(2.4)limt→∞e-βtr(ϑ-t+τω)=e-βτ·limt→∞e-β(t-τ)r(ϑ-t+τω)=0.
Therefore, for any τ∈ℝ, r(ϑτ·) is also tempered. Moreover, if for ℙ-a.e. ω∈Ω, r(ϑtω) is continuous in t, then for any ν>0, supσ∈[-ν,0]r(ϑσ·) is measurable and for all β>0 and ℙ-a.e. ω∈Ω(2.5)limt→∞e-βtsupσ∈[-ν,0]r(ϑ-t+σω)≤limt→∞e(β/2)(ν-t)·sups∈(-∞,0]{e(β/2)sr(ϑsω)}=0.
Hence, for any ν>0, supσ∈[-ν,0]r(ϑσ·) is also tempered.Remark 2.7.
Ifr>0 is tempered, then for any α>0 and ℙ-a.e. ω∈Ω(2.6)R(ω)=∫-∞0‍eαsr(ϑsω)ds<∞.
Moreover, R is tempered, and if for ℙ-a.e. ω∈Ω, r(ϑtω) is continuous in t, then R(ϑtω) is also continuous in t for such ω.Hereafter, we always assume thatφ is a continuous random dynamical system over (Ω,ℱ,ℙ,(ϑt)t∈ℝ), and 𝒟 is a collection of random subsets of X.Definition 2.8.
A random setK is called a random absorbing set in 𝒟 if for every B∈𝒟 and ℙ-a.e. ω∈Ω, there exists tB(ω)>0 such that
(2.7)φ(t,ϑ-tω,B(ϑ-tω))⊆K(ω)∀t≥tB(ω).Definition 2.9.
A random set𝒜 is called a 𝒟-random attractor (𝒟-pullback attractor) for φ if the following hold: (i)
𝒜 is a random compact set;(ii)
𝒜 is strictly invariant, that is, for ℙ-a.e. ω∈Ω and all t≥0,
(2.8)φ(t,ω,𝒜(ω))=𝒜(ϑtω);(iii)
𝒜 attracts all sets in 𝒟, that is, for all B∈𝒟 and ℙ-a.e. ω∈Ω,
(2.9)limt→∞d(φ(t,ϑ-tω,B(ϑ-tω)),𝒜(ω))=0,whered is the Hausdorff semimetric given by d(E,F)=supx∈Einfy∈F∥x-y∥X for any E⊆X and F⊆X.Definition 2.10.
φ is said to be 𝒟-pullback asymptotically compact in X if for all B∈𝒟 and ℙ-a.e. ω∈Ω, {φ(tn,ϑ-tnω,xn)}n=1∞ has a convergent subsequence in X whenever tn→∞, and xn∈B(ϑ-tnω).The following existence result on a random attractor for a continuous random dynamical system can be found in [19, 26]. First, recall that a collection 𝒟 of random subsets of X is called inclusion closed if whenever E is an arbitrary random set, and F is in 𝒟 with E(ω)⊂F(ω) for all ω∈Ω, then E must belong to 𝒟.Proposition 2.11.
Let𝒟 be an inclusion-closed collection of random subsets of X and φ a continuous random dynamical system on X over (Ω,ℱ,ℙ,(ϑt)t∈ℝ). Suppose that K∈𝒟 is a closed random absorbing set for φ in 𝒟 and φ is 𝒟-pullback asymptotically compact in X. Then φ has a unique 𝒟-random attractor 𝒜 which is given by
(2.10)𝒜(ω)=⋂τ≥0‍⋃t≥τ‍φ(t,ϑ-tω,K(ϑ-t))¯.In this paper, we will take𝒟 as the collection of all tempered random subsets of 𝒞 and prove the stochastic retarded LDS has a 𝒟-random attractor.


## 3. Compactness Criterion inC([-ν,0];ℓ2)

In this section, we provide a necessary and sufficient condition for the relative compactness of sequences inC([-ν,0];ℓ2), which will be used to establish the asymptotic compactness of the retarded LDS.Lemma 3.1.
Letu∈C([-ν,0];ℓ2). Then for every ɛ>0, there exists N(ɛ)>0 such that for all k≥N(ɛ),
(3.1)sups∈[-ν,0]∑|i|≥k‍|ui(s)|2<ɛ.Proof.
For everyɛ>0, by virtue of the uniform continuity of u, there exist -ν=s0<s1<s2<⋯<sp=0 such that
(3.2)∥u(s)-u(sj)∥<ɛ2,fors∈[sj-1,sj],j=1,2,…,p.
Since for each sj, u(sj)∈ℓ2, there exists Nj(ɛ)>0 such that for all k≥Nj(ɛ),
(3.3)∑|i|≥k‍|ui(sj)|2<ɛ4.
Take N(ɛ)=max1≤j≤pNj(ɛ). Then for each s∈[-ν,0], there exists j∈{1,2,…,p} such that s∈[sj-1,sj]. Therefore, we get from (3.2) and (3.3) that for all k≥N(ɛ),
(3.4)∑|i|≥k‍|ui(s)|2≤2∑|i|≥k‍|ui(sj)|2+2∑|i|≥k‍|ui(s)-ui(sj)|2≤2∑|i|≥k‍|ui(sj)|2+2∥u(s)-u(sj)∥2<ɛ,
which completes the proof.Theorem 3.2.
Let𝒮⊂C([-ν,0];ℓ2). Then 𝒮 is relative compact in C([-ν,0];ℓ2) if and only if the following conditions are satisfied: (i)
𝒮 is bounded in C([-ν,0];ℓ2);(ii)
𝒮 is equicontinuous;(iii)
limk→∞supu=(ui)i∈ℤ∈𝒮sups∈[-ν,0]∑|i|≥k‍|ui(s)|2=0.Proof.
The proof is divided into two steps. We first show the necessity of the conditions and then prove the sufficiency.
(1) Assume that𝒮 is relative compact in C([-ν,0];ℓ2). Then we want to show conditions (i), (ii), and (iii) hold. Clearly, in this case, by the Ascoli-Arzelà theorem, 𝒮 must be bounded and equicontinuous. So we only need to prove condition (iii).
Givenɛ>0, since 𝒮 is relative compact, there exists a finite subset ℰ of 𝒮 such that the balls of radii ɛ/2 centered at ℰ form a finite covering of 𝒮, that is, for each u∈𝒮, there exists v∈ℰ such that
(3.5)sups∈[-ν,0]∥u(s)-v(s)∥<ɛ2.
By Lemma 3.1, there exists K*(ɛ)>0 such that for all v∈ℰ,
(3.6)sups∈[-ν,0]∑|i|≥K*(ɛ)‍|vi(s)|2<ɛ24.
By (3.5) and (3.6), we find that for each u∈𝒮, there exists v∈ℰ such that
(3.7)sups∈[-ν,0]∑|i|≥K*(ɛ)‍|ui(s)|2≤2sups∈[-ν,0]∑|i|≥K*(ɛ)‍|ui(s)-vi(s)|2+2sups∈[-ν,0]∑|i|≥K*(ɛ)‍|vi(s)|2<ɛ2.
Therefore, for all k≥K*(ɛ), we have
(3.8)supu=(ui)i∈ℤ∈𝒮sups∈[-ν,0]∑|i|≥k‍|ui(s)|2≤ɛ2,
which implies condition (iii).
(2) Assume that conditions (i), (ii), and (iii) are valid. We want to prove that𝒮 is relative compact in C([-ν,0];ℓ2). That is, given ɛ>0, we want to show that 𝒮 has a finite covering of balls of radii ɛ. By condition (iii), we find that there exists K(ɛ)>0 such that for all u=(ui)i∈ℤ∈𝒮,
(3.9)sups∈[-ν,0]∑|i|≥K(ɛ)‍|ui(s)|2<ɛ24.
Consider the set 𝒮|K={u|K=(ui)|i|≤K(ɛ):u=(ui)i∈ℤ∈𝒮} in C([-ν,0];ℝ2K(ɛ)+1). By conditions (i) and (ii), we know that 𝒮|K is bounded and equicontinuous in C([-ν,0];ℝ2K(ɛ)+1). Then, by the Ascoli-Arzelà theorem, we obtain that 𝒮|K is relative compact in C([-ν,0];ℝ2K(ɛ)+1) and hence there exists a finite subset ℋ¯ of 𝒮|K such that the balls of radii ɛ/2 centered at ℋ¯ form a finite covering of 𝒮|K, that is, for each u|K∈𝒮|K, there exists v|K∈ℋ¯ such that
(3.10)sups∈[-ν,0]∑|i|≤K(ɛ)‍|ui(s)-vi(s)|2<ɛ24.
Now for each v|K=(vi)|i|≤K(ɛ)∈ℋ¯, we choose v~=(v~i)i∈ℤ such that v~i=vi for |i|≤K(ɛ) and v~i=0 for |i|>K(ɛ). Then by (3.9) and (3.10), we find that for each u∈𝒮, there exists v~∈ℋ={v~:v|K∈ℋ¯} such that
(3.11)sups∈[-ν,0]∥u(s)-v~(s)∥2≤sups∈[-ν,0]∑|i|≤K(ɛ)‍|ui(s)-vi(s)|2+sups∈[-ν,0]∑|i|>K(ɛ)‍|ui(s)|2<ɛ2,
which implies that the set 𝒮 has a finite covering of balls with radii ɛ. The proof is complete.The next result is a variant of Theorem3.2 which shows that condition (iii) in Theorem 3.2 has an equivalent form which is easier to verify for asymptotic compactness of dynamical systems associated with retarded LDSs.Theorem 3.3.
Let{un}n=1∞={(uin)i∈ℤ}n=1∞⊂C([-ν,0];ℓ2). Then {un}n=1∞ is relative compact in C([-ν,0];ℓ2) if and only if the following conditions are satisfied: (i)
{un}n=1∞ is bounded in C([-ν,0];ℓ2);(ii)
{un}n=1∞ is equicontinuous;(iii)
limk→∞limsupn→∞sups∈[-ν,0]∑|i|≥k‍|uin(s)|2=0.Proof.
If{un}n=1∞ is relative compact in C([-ν,0];ℓ2), then it follows from Theorem 3.2 that the above conditions (i), (ii), and (iii) are satisfied. So, to complete the proof, we only need to show that the above conditions (i), (ii), and (iii) imply the conditions in Theorem 3.2. Given ɛ>0, it follows from condition (iii) that there exists K1(ɛ)>0 such that
(3.12)limsupn→∞sups∈[-ν,0]∑|i|≥K1(ɛ)‍|uin(s)|2<ɛ22,
which implies that there exists N(ɛ)>0 such that
(3.13)sups∈[-ν,0]∑|i|≥K1(ɛ)‍|uin(s)|2<ɛ2,∀n>N(ɛ).
By Lemma 3.1, we find that there exists K2(ɛ)>0 such that
(3.14)sups∈[-ν,0]∑|i|≥K2(ɛ)‍|uin(s)|2<ɛ2,∀1≤n≤N(ɛ).
Take K(ɛ)=max{K1(ɛ),K2(ɛ)}. It follows from (3.13) and (3.14) that
(3.15)sups∈[-ν,0]∑|i|≥K(ɛ)‍|uin(s)|2<ɛ2,∀n≥1,
which implies that
(3.16)sup{un}n=1∞sups∈[-ν,0]∑|i|≥k‍|uin(s)|2≤ɛ2,∀k≥K(ɛ).
Therefore,
(3.17)limk→∞supn∈ℕsups∈[-ν,0]∑|i|≥k‍|uin(s)|2=0,
which together with conditions (i) and (ii) shows that the conditions in Theorem 3.2 are satisfied with 𝒮={un}n=1∞. The proof is complete.


## 4. Stochastic Retarded Lattice Differential Equations

In this section, we show that there is a continuous random dynamical system generated by stochastic retarded LDS (1.3)-(1.4).For convenience, we now formulate (1.3)-(1.4) as a stochastic functional differential equation in ℓ2. Define the linear operators A, B, B*, λ from ℓ2 to ℓ2 as follows. For u=(ui)i∈ℤ∈ℓ2,
(4.1)(Au)i=-ui-1+2ui-ui+1,(λu)i=λiui,(Bu)i=ui+1-ui,(B*u)i=ui-1-ui,
for each i∈ℤ. Then A=BB*=B*B and (B*u,v)=(u,Bv) for all u, v∈ℓ2. Therefore, (Au,u)≥0 for all u∈ℓ2. Let ei∈ℓ2 denote the element having 1 at position i and all the other components 0. Then
(4.2)w(t)=∑i∈ℤ‍aiwi(t)eiwith(ai)i∈ℤ∈ℓ2,
is an ℓ2-valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator Q such that Qei=aiei. For ξ∈𝒞, let f(ξ)=(fi(ξi))i∈ℤ. Then stochastic retarded LDS (1.3)-(1.4) can be rewritten as a stochastic functional equation in ℓ2(4.3)du=[-(A+λ)u+f(ut)+g]dt+dw,t>0,
with the initial data
(4.4)u(t)=u0(t),t∈[-ν,0].In the sequel, we consider the probability space(Ω,ℱ,ℙ) where
(4.5)Ω={ω∈C(ℝ,ℓ2):ω(0)=0},ℱ is the Borel σ-algebra induced by the compact-open topology of Ω, and ℙ the corresponding Wiener measure on (Ω,ℱ) with respect to the covariance operator Q. Let
(4.6)ϑtω(·)=ω(·+t)-ω(t),t∈ℝ.
Then (Ω,ℱ,ℙ,(ϑt)t∈ℝ) is an ergodic metric dynamical system. Since the above probability space is canonical, we have
(4.7)w(t,ω)=ω(t),w(t,ϑsω)=w(t+s,ω)-w(s,ω).
By Proposition A.1 in [26], there exists a {ϑt}t∈ℝ-invariant set Ω~∈ℱ of full ℙ-measure such that
(4.8)limt→±∞∥ω(t)∥t=0∀ω∈Ω~.Letℱ¯ be the ℙ-completion of ℱ and let
(4.9)ℱt=⋁s≤tℱst,t∈ℝ,
with
(4.10)ℱst=σ{w(τ2)-w(τ1):s≤τ1≤τ2≤t}∨𝒩,
where σ{w(τ2)-w(τ1):s≤τ1≤τ2≤t} is the smallest σ-algebra generated by the random variable w(τ2)-w(τ1) for all τ1, τ2 such that s≤τ1≤τ2≤t and 𝒩 is the collection of ℙ-null sets of ℱ¯. Note that
(4.11)ϑτ-1ℱst=ℱs+τt+τ,
so (Ω,ℱ,ℙ,(ϑt)t∈ℝ,(ℱst)s≤t) is a filtered metric dynamical system.Note that problem (4.3)-(4.4) is interpreted as an integral equation as follows:
(4.12)u(t)=u0(0)+∫0t‍(-(A+λ)u+f(us)+g)ds+w(t),t>0,u(t)=u0(t),t∈[-ν,0].ℙ-a.s. for any u0∈𝒞. By the theory in [46], we deal with (4.12) on the complete probability space (Ω,ℱ¯,ℙ). For λ and f, we make the following assumptions. (A1)
There exist positive constantsλl and λu such that
(4.13)0<λl≤λi≤λu<∞,i∈ℤ.(A2)
f(0)=0.(A3)
For anyr>0, there exists a constant l(r)>0 such that
(4.14)∥f(ξ)-f(η)∥≤l(r)∥ξ-η∥𝒞,for allξ, η∈C([-ν,0];ℓ2) with ∥ξ∥𝒞,∥η∥𝒞≤r.(A4)
There exist positive constantsα0 and cf such that
(4.15)∫0t‍eαs|fi(uis)|2ds≤cf2∫-νt‍eαs|ui(s)|2ds,for allα∈(0,α0), t>0, u∈C([-ν,t];ℓ2),i∈ℤ.(A5)
λl>cf.We now associate a continuous random dynamical system with the stochastic retarded lattice differential equations over(Ω,ℱ,ℙ,(ϑt)t∈ℝ). To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on (Ω,ℱ,ℙ,(ϑt)t∈ℝ) and transform the stochastic retarded lattice differential equations into a random one. Let
(4.16)z(t,ω)={∫-∞t‍(A+λ)e-(A+λ)(t-s)(w(t,ω)-w(s,ω))ds,ω∈Ω~,0,ω∉Ω~,
where e-(A+λ)t is the uniformly continuous semigroup on ℓ2 generated by bounded linear operator -A-λ. Then by (4.8), (4.16) is well defined. The process z(t),t∈ℝ is a stationary, Gaussian process. Moreover, the random variable ∥z(0,ω)∥ is tempered and for each ω∈Ω, the mapping t→z(t,ω) is continuous. Furthermore, by Lemma 5.13 in [46], we find that for all t∈ℝ and ℙ-a.s.,
(4.17)z(t)=∫-∞t‍e-(A+λ)(t-s)dw(s).
Noticing that
(4.18)∫-∞t‍e-(A+λ)(t-s)dw(s)=e-(A+λ)tz(0)+∫0t‍e-(A+λ)(t-s)dw(s),
and using the Itô formula, we get from (4.17) that for all t>0 and ℙ-a.s.,
(4.19)z(t)=z(0)-∫0t‍(A+λ)z(s)ds+w(t).
Setting v(t)=u(t)-z(t) for t≥-ν in (4.12), then by (4.19), we obtain a deterministic equation, ℙ-a.s. 						(4.20)v(t)=v0(0)+∫0t‍(-(A+λ)v+f(vs+zs)+g)ds,t>0,v(t)=v0(t),t∈[-ν,0],
which is equivalent to the functional differential equation
(4.21)dvdt=-(A+λ)v+f(vt+zt)+g,t>0,
with initial condition
(4.22)v(t)=v0(t),t∈[-ν,0].
Here v0(t)=u0(t)-z0(t,ω), t∈[-ν,0].Problem (4.21)-(4.22) is a deterministic functional differential equation with random coefficients, which can be solved pathwise. We now establish the following result for problem (4.21)-(4.22).Theorem 4.1.
LetT>0 and ω∈Ω be fixed. Then the following properties hold.(1)
For eachv0∈𝒞, problem (4.21)-(4.22) has a unique solution v(·,ω,v0)∈C([-ν,T];ℓ2).(2)
Letv1(·,ω,v10) and v2(·,ω,v20) be the solutions of problem (4.21)-(4.22) for the initial data v10 and v20, respectively. Then there exists a constant c(T)>0 such that for all t∈[0,T](4.23)∥v1t(·,ω,v10)-v2t(·,ω,v20)∥𝒞≤∥v10-v20∥𝒞ec(T)t.Proof.
(1) Denote(4.24)F(t,ξ,ω)=-Aξ(0)-λξ(0)+f(ξ+zt(·,ω))+g,
for all t≥0, ξ∈𝒞 and ω∈Ω. Then by (A1)–(A3), we have that
(4.25)∥F(t,ξ,ω)-F(t,ξ,ω)∥≤[4+λu+lf(r)]∥ξ-η∥𝒞,
for any ξ, η∈𝒞 with ∥ξ∥𝒞∥≤r, ∥η∥𝒞∥≤r. Therefore, F satisfies local Lipschitz condition and maps the bounded sets of 𝒞 into the bounded sets of ℓ2. Then by using a standard argument, one can show that for each v0∈𝒞, there exists a Tmax≤∞ such that problem (4.21)-(4.22) has a unique solution v on [0,Tmax). Moreover, if Tmax<∞ then
(4.26)limsupt↑Tmax∥vt∥𝒞=∞.
We prove now that this local solution is a global one. Let T∈(0,Tmax). By (A5), we can choose β>0 small enough such that 2λl>2cf+β. Taking the inner product of (4.21) with v in ℓ2, we get
(4.27)12ddt∥v∥2+(Av,v)+(λv,v)=(f(vt+zt),v)+(g,v).
Clearly,
(4.28)(Av,v)=(Bv,Bv)≥0,(λv,v)=∑i∈ℤ‍λivi2≥λl∥v∥2.
Using the Young inequality, we find that
(4.29)(f(vt+zt),v)≤∥f(vt+zt)∥∥v∥≤cf2∥v∥2+12cf∥f(vt+zt)∥2,(g,v)≤∥g∥∥v∥≤β2∥v∥2+12β∥g∥2.
Then it follows from (4.27), (4.28), (4.29) that
(4.30)ddt∥v∥2≤-(2λl-cf-β)∥v∥2+1cf∥f(vt+zt(ϑ·ω))∥2+1β∥g∥2.
Choose α∈(0,α0) small enough such that 2λl>2cf+α+β. Then by (4.30), we obtain
(4.31)ddt(eαt∥v∥2)≤-(2λl-cf-α-β)eαt∥v∥2+eαtcf∥f(vt+zt)∥2+eαtβ∥g∥2.
Now, we can also choose γ>0 small enough such that 2λl>(2+γ)cf+α+β. Integrating (4.31) over [0,t](t∈[0,T]) leads to
(4.32)eαt∥v(t)∥2≤∥v(0)∥2-(2λl-cf-α-β)∫0t‍eαs∥v(s)∥2ds+1cf∫0t‍eαs∥f(vs+zs)∥2ds+∥g∥2β∫0t‍eαsds.
Using the Young inequality and (A4), we find that
(4.33)1cf∫0t‍eαs∥f(vs+zs)∥2ds≤cf∫-νt‍eαs∥v(s)+z(s)∥2ds≤cf∫0t‍eαs[(1+γ)∥v(s)∥2+(1+γ-1)∥z(s)∥2]ds+cf∫-ν0‍eαs∥v(s)+z(s)∥2ds.
Then by (4.32) and (4.33), we obtain
(4.34)eαt∥v(t)∥2≤-(2λl-(2+γ)cf-α-β)∫0t‍eαs∥v(s)∥2ds+∥v(0)∥2+∥g∥2αβeαt+cf∫-ν0‍eαs∥v(s)+z(s)∥2ds+c1∫0t‍eαs∥z(ϑsω)∥2ds≤(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2+c1∫0t‍eαs∥z(s)∥2ds+∥g∥2αβeαt,
where c1=cf(1+γ-1). Consequently,
(4.35)∥v(t)∥2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-αt+c1∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ.
Hence, for fixed σ∈[-ν,0], we get that for t∈(-σ,T],
(4.36)∥v(t+σ)∥2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-α(t+σ)+c1∫0t+σ‍eα(s-t-σ)∥z(s)∥2ds+∥g∥2αβ≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ,
and for t∈[0,-σ],
(4.37)∥v(t+σ)∥2≤∥v0∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t).
In view of (4.36) and (4.37), we find that for all t∈[0,T],
(4.38)∥vt∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s)∥2ds+∥g∥2αβ.
Therefore, for all t∈[0,T],
(4.39)∥vt∥𝒞2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eαν+c1eαν∫0T‍∥z(s)∥2ds+∥g∥2αβ,
which, together with (4.26), implies that Tmax=∞. This proves the property (1).
(2) Letv~(t,ω)=v1(t,ω,v10)-v1(t,ω,v20). By (4.38), there exists a constant r(T)>0 such that
(4.40)∥v1t∥𝒞≤r(T),∥v2t∥𝒞≤r(T).
Then from (4.20) and (4.25), we have that for t∈[0,T](4.41)∥v~(t)∥≤∥v~(0)∥+[4+λu+lf(r(T))]∫0t‍∥v~s∥𝒞ds.
Hence, for fixed σ∈[-ν,0], we get that for t∈(-σ,T],
(4.42)∥v~(t+σ)∥≤∥v~(0)∥+[4+λu+lf(r(T))]∫0t+σ‍∥v~s∥𝒞ds≤∥v~0∥𝒞+[4+λu+lf(r(T))]∫0t‍∥v~s∥𝒞ds,
and for t∈[0,-σ],
(4.43)∥v~(t+σ)∥≤∥v~0∥𝒞.
In view of (4.42) and (4.43), we find that for all t∈[0,T],
(4.44)∥v~t∥𝒞≤∥v~0∥𝒞+[4+λu+lf(r(T))]∫0t‍∥v~s∥𝒞ds.
The Gronwall inequality implies that for all t∈[0,T],
(4.45)∥v~t∥𝒞≤∥v~0∥𝒞e[4+λu+lf(r(T))]t.
This proves the property (2). The proof is complete.Conversely, if for eachω∈Ω, v(t,ω,v0) is a solution of problem (4.21)-(4.22) with v0(·)=u0(·)-z0(·,ω), then the process
(4.46)u(t,ω,u0)=v(t,ω,v0)+z(t,ω)
is a solution of problem (4.3)-(4.4). And if u0 is a 𝒞-valued ℱ0-measurable random variable, then u(t,ω,u0) is an ℱt-adapted process.Theorem 4.2.
Problem (4.21)-(4.22) generates a continuous random dynamical system ϕ over (Ω,ℱ,ℙ,(ϑ)t∈ℝ), where
(4.47)ϕ(t,ω,v0)=vt(·,ω,v0),fort≥0,ω∈Ω,v0∈𝒞.
Moreover, if one defines ψ by
(4.48)ψ(t,ω,u0)=ut(·,ω,u0),fort≥0,ω∈Ω,u0∈𝒞,
then ψ is another continuous random dynamical system associated to problem (4.3)-(4.4).Proof.
From property (2) of Theorem4.1, it follows that ϕ(·,ω,·):[0,∞)×𝒞→𝒞 is continuous for all ω∈Ω. By (4.20), we have that for s,t≥0 and σ∈[-ν,0],
(4.49)ϕ(t,ϑsω,ϕ(s,ω,v0))(σ)=ϕ(s,ω,v0)(0)+∫0t+σ‍F(τ,ϕ(τ,ϑsω,ϕ(s,ω,v0)),ϑsω)dτ.
Then again by (4.20) and noticing that
(4.50)F(t,ξ,ϑsω)=F(t+s,ξ,ω),∀s,t≥0,ξ∈𝒞,
we get that
(4.51)ϕ(t,ϑsω,ϕ(s,ω,v0))(σ)=v0(0)+∫0s‍F(τ,ϕ(τ,ω,v0),ω)dτ+∫st+s+σ‍F(τ,ϕ(τ-s,ϑsω,ϕ(s,ω,v0)),ω)dτ.
For each ω∈Ω consider
(4.52)Φ(τ,ω,v0)={ϕ(τ,ω,v0),if0≤τ≤s,ϕ(τ-s,ϑsω,ϕ(s,ω,v0)),ifs<τ≤t+s.
Then for τ=t+s, we have
(4.53)Φ(t+s,ω,v0)=ϕ(t,ϑsω,ϕ(s,ω,v0))fors,t≥0.
It follows from (4.51) that
(4.54)Φ(t+s,ω,v0)(σ)=v0(0)+∫0t+s+σ‍F(τ,Φ(τ,ω,v0),ω)dτ,
for all σ∈[-ν,0]. By the uniqueness of the solution of (4.20), we find that
(4.55)Φ(t+s,ω,v0)=ϕ(t+s,ω,v0),
while (4.53) implies
(4.56)ϕ(t+s,ω,v0)=ϕ(t,ϑsω,ϕ(s,ω,v0))fors,t≥0.
Hence, ϕ is a continuous random dynamical system.
As forψ, noticing that
(4.57)ψ(t,ω,u0)=ϕ(t,ω,u0-z0(ω))+zt(ω),fort≥0,ω∈Ωandu0∈𝒞,
we get from (4.56) that for s, t≥0,
(4.58)ψ(t,ϑsω,ψ(s,ω,u0))=ϕ(t,ϑsω,ϕ(s,ω,u0-z0(ω)))+zt(ϑsω)=ϕ(t+s,ω,u0-z0(ω))+zt+s(ω)=ψ(t+s,ω,u0).
Therefore, ψ is also a continuous random dynamical system. Furthermore, ϕ and ψ are conjugated random dynamical systems, that is
(4.59)ψ(t,ω,T(ω,ξ))=T(ϑtω,ϕ(t,ω,ξ)),foranyξ∈𝒞,
where for every ω∈Ω, T(ω,ξ)=ξ+z0(ω) is a homeomorphism of 𝒞. The proof is complete.


## 5. Existence of Random Attractors

In this section, we prove the existence of a𝒟-random attractor for the random dynamical system ψ associated with (4.3)-(4.4). We first establish the existence of a 𝒟-random attractor for its conjugated random dynamical system ϕ, then the existence of a 𝒟-random attractor for ψ follows from the conjugation relation between ϕ and ψ. To this end, we will derive uniform estimates on the solutions of problem (4.21)-(4.22) when t→∞ with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for ϕ.From now on, we always assume that𝒟 is the collection of all tempered subsets of 𝒞 with respect to (Ω,ℱ,ℙ,(ϑt)t∈ℝ). The next lemma shows that ϕ has a random absorbing set in 𝒟.Lemma 5.1.
There existsK∈𝒟 such that K is a random absorbing set for ϕ in 𝒟, that is, for any B∈𝒟 and ℙ-a.e. ω∈Ω, there exists TB(ω)>0 such that
(5.1)ϕ(t,ϑ-tω,B(ϑ-tω))⊆K(ω)∀t≥TB(ω).Proof.
Replacingω by ϑ-tω in (4.38), we get that for all t≥0,
(5.2)∥vt(ϑ-tω,v0(ϑ-tω))∥𝒞2≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)+c1eαν∫0t‍eα(s-t)∥z(s,ϑ-tω))∥2ds+∥g∥2αβ≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)+c1eαν∫-∞0‍eαs∥z(0,ϑsω))∥2ds+∥g∥2αβ.
By assumption, B∈𝒟 is tempered. On the other hand, by Remark 2.6, ∥z0(ω)∥𝒞2 is also tempered. Therefore, if v0(ϑ-tω)∈B(ϑ-tω), then there exists TB(ω)>0 such that for all t≥TB(ω),
(5.3)[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)≤1+r(ω),
where
(5.4)r(ω)=∫-∞0‍eαs∥z(0,ϑsω)∥2ds
is tempered by Remark 2.7. Then it follows from (5.2) and (5.3) that for all t≥TB(ω),
(5.5)∥vt(ϑ-tω,v0(ϑ-tω))∥𝒞2≤(c1eαν+1)r(ω)+∥g∥2αβ+1.
Given ω∈Ω, denote by
(5.6)K(ω)={ξ∈𝒞:∥ξ∥𝒞2≤r1(ω)},
where
(5.7)r1(ω)=(c1eαν+1)r(ω)+∥g∥2αβ+1
is tempered. Then K∈𝒟. Further, (5.5) indicates that K is a random absorbing set for ϕ in 𝒟, which completes the proof.Lemma 5.2.
LetB∈𝒟 and v0(ω)∈B(ω). Then for every ɛ>0 and ℙ-a.e. ω∈Ω, there exist T*=T*(B,ω,ɛ)>0 and N*=N*(ω,ɛ)>0 such that the solution v(t,ω,v0(ω)) of problem (4.21)-(4.22) satisfies, for all t≥T*,
(5.8)sups∈[-ν,0]∑|i|≥N*‍|vit(s,ϑ-tω,u0(ϑ-tω))|2≤ɛ.Proof.
Letρ be a smooth function defined on ℝ+ such that 0≤ρ(s)≤1 for all s≥0, and
(5.9)ρ(s)={0,0≤s≤1,1,s≥2.
Then there exists a positive deterministic constant c2 such that |ρ'(s)|≤c2 for all s≥0. Taking the inner product of (4.21) with x=(ρ(|i|/k)vi) in ℓ2, we obtain that
(5.10)12ddt∑i∈ℤ‍ρ(|i|k)|vi|2+(Av,x)+(λv,x)=(f(vt+zt),x)+(g,x).
We now estimate terms in (5.10) as follows. First, we get from (A1) that
(5.11)(λv,x)=∑i∈ℤ‍λiρ(|i|k)vi2≥λl∑i∈ℤ‍ρ(|i|k)|vi|2.
Secondly, by the property of the cutoff function ρ, we estimate
(5.12)(Av,x)=(Bv,Bx)=∑i∈ℤ‍(vi+1-vi)[ρ(|i+1|k)vi+1-ρ(|i|k)vi]=∑i∈ℤ‍(vi+1-vi)[(ρ(|i+1|k)-ρ(|i|k))vi+1+ρ(|i|k)(vi+1-vi)]=∑i∈ℤ‍[ρ(|i+1|k)-ρ(|i|k)](vi+1-vi)vi+1+ρ(|i|k)(vi+1-vi)2≥∑i∈ℤ‍[ρ(|i+1|k)-ρ(|i|k)](vi+1-vi)vi+1≥-∑i∈ℤ‍|ρ′(ξi)|k|vi+1-vi||vi+1|≥-c2k∑i∈ℤ‍(|vi+1|2+|vi||vi+1|)≥-2c2k∥v∥2.
Thirdly, using the Young inequality and (A3), we find that
(5.13)(f(vt+zt),x)=∑i∈ℤ‍ρ(|i|k)f(vit+zit(ϑ·ω))≤cf2∑i∈ℤ‍ρ(|i|k)|vi|2+12cf∑i∈ℤ‍ρ(|i|k)|f(vit+zit(ϑ·ω)|2.
Finally, using the Young inequality again, we obtain that
(5.14)(g,x)=∑i∈ℤ‍ρ(|i|k)givi≤β2∑i∈ℤ‍ρ(|i|k)vi2+12β∑i∈ℤ‍ρ(|i|k)gi2.
Taking into account (5.10), (5.11), (5.12), (5.13), and (5.14), we obtain that
(5.15)ddt∑i∈ℤ‍ρ(|i|k)|vi|2≤-(2λl-cf-β)∑i∈ℤ‍ρ(|i|k)|vi|2+1cf∑i∈ℤ‍ρ(|i|k)|f(vit+zit)|2+1β∑i∈ℤ‍ρ(|i|k)gi2+4c2k∥v∥2,
which implies
(5.16)ddt(eαt∑i∈ℤ‍ρ(|i|k)|vi|2)≤-(2λl-cf-α-β)eαt∑i∈ℤ‍ρ(|i|k)|vi|2+1cfeαt∑i∈ℤ‍ρ(|i|k)|f(vit+zit)|2+1βeαt∑i∈ℤ‍ρ(|i|k)gi2+4c2keαt∥v∥2.
Using the Young inequality and (A4), we get that
(5.17)1cf∫0t‍eαs∑i∈ℤ‍ρ(|i|k)|f(vis+zis)|2ds≤cf∫-νt‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)+zi(s)|2ds≤cf∫0t‍eαs∑i∈ℤ‍ρ(|i|k)[(1+γ)|vi(s)|2+(1+γ-1)|zi(s)|2]ds+cf∫-ν0‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)+zi(s)|2ds.
Integrating (5.16) over [0,t](t≥0) leads to
(5.18)eαt∑i∈ℤ‍ρ(|i|k)|vi(t)|2-∑i∈ℤ‍ρ(|i|k)|vi(0)|2≤-(2λl-cf-α-β)∫0t‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)|2ds+1cf∫0t‍eαs∑i∈ℤ‍ρ(|i|k)|f(vis+zis|2ds+1β∫0t‍eαs∑i∈ℤ‍ρ(|i|k)gi2ds+4c2k∫0t‍eαs∥v(s)∥2ds.
It follows from (5.17) and (5.18) that
(5.19)eαt∑i∈ℤ‍ρ(|i|k)|vi(t)|2-∑i∈ℤ‍ρ(|i|k)|vi(0)|2≤-(2λl-(2+γ)cf-α-β)∫0t‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)|2ds+cf∫-ν0‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)+zi(s)|2ds+eαtαβ∑i∈ℤ‍ρ(|i|k)gi2+c1∫0t‍eαs∑|i|>k-1‍|zi(s)|2ds+4c2k∫0t‍eαs∥v(s)∥2ds≤cf∫-ν0‍eαs∑i∈ℤ‍ρ(|i|k)|vi(s)+zi(s)|2ds+eαtαβ∑i∈ℤ‍ρ(|i|k)gi2+c1∫0t‍eαs∑|i|>k-1‍|zi(s)|2ds+4c2k∫0t‍eαs∥v(s)∥2ds,
which implies
(5.20)∑i∈ℤ‍ρ(|i|k)|vi(t)|2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-αt+1αβ∑i∈ℤ‍ρ(|i|k)gi2+c1∫0t‍eα(s-t)∑|i|>k-1‍|zi(s)|2ds+4c2k∫0t‍eα(s-t)∥v(s)∥2ds.
If we take t≥ν, we have that, for all σ∈[-ν,0],
(5.21)∑i∈ℤ‍ρ(|i|k)|vi(t+σ)|2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]e-α(t+σ)+1αβ∑i∈ℤ‍ρ(|i|k)gi2+c1∫0t+σ‍eα(s-t-σ)∑|i|>k-1‍|zi(s)|2ds+4c2k∫0t+σ‍eα(s-t-σ)∥v(s)∥2ds≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+1αβ∑i∈ℤ‍ρ(|i|k)gi2+c1eαν∫0t‍eα(s-t)∑|i|>k-1‍|zi(s)|2ds+4c2eανk∫0t‍eα(s-t)∥v(s)∥2ds,
whence for all t≥ν,
(5.22)supσ∈[-ν,0]∑i∈ℤ‍ρ(|i|k)|vit(σ)|2≤[(1+2νcf)∥v0∥𝒞2+2νcf∥z0∥𝒞2]eα(ν-t)+1αβ∑i∈ℤ‍ρ(|i|k)gi2+c1eαν∫0t‍eα(s-t)∑|i|>k-1‍|zi(s)|2ds+4c2eανk∫0t‍eα(s-t)∥v(s,ω,v0(ω))∥2ds.
Replacing ω by ϑ-tω, we find that
(5.23)supσ∈[-ν,0]∑i∈ℤ‍ρ(|i|k)|vit(σ,ϑ-tω,v0(ϑ-tω))|2≤[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)+1αβ∑i∈ℤ‍ρ(|i|k)gi2+c1eαν∫0t‍eα(s-t)∑|i|>k-1‍|zi(s,ϑ-tω)|2ds+4c2eανk∫0t‍eα(s-t)∥v(s,ϑ-tω,v0(ϑ-tω))∥2ds.
We now estimate terms in (5.23) as follows. Since B∈𝒟 is tempered set, and ∥z0(ω)∥𝒞2 is tempered function, if v0(ϑ-tω)∈B(ϑ-tω), then for every ɛ>0, there exists T1=T1(B,ω,ɛ)>0 such that for all t≥T1,
(5.24)[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]eα(ν-t)≤ɛ4.
Secondly, since g∈ℓ2, there exists N1=N1(ω,ɛ)>0 such that, for all k≥N1,
(5.25)1αβ∑i∈ℤ‍ρ(|i|k)gi2≤1αβ∑|i|>k‍ρ(|i|k)gi2≤ɛ4.
Thirdly, note that
(5.26)∫-∞0‍eα(s)∥z(0,ϑsω)∥2ds<∞.
Then by the Lebesgue theorem of dominated convergence, there exists N2=N2(ω,ɛ)>0 such that for all k≥N2,
(5.27)∫-∞0‍eαs∑|i|>k-1‍|zi(0,ϑsω)|2ds≤ɛ4c1eαν.
Then it follows from (5.27) that for all t≥0 and k≥N2,
(5.28)c1eαν∫0t‍eα(s-t)∑|i|>k-1‍|zi(s,ϑ-tω)|2ds≤c1eαν∫-∞0‍eαs∑|i|>k-1‍|zi(ϑsω)|2ds≤ɛ4.
Next, we get from (4.35) that
(5.29)4c2eανk∫0t‍eα(s-t)∥v(s,ϑ-tω,v0(ϑ-tω))∥2ds≤4c2∥g∥2eανkα2β+4c1c2eανk∫0t‍∫0s‍eα(τ-t)∥z(τ,ϑ-tω)∥2dτds+4c2eανk[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]te-αt.
For the integral on the right side of (5.29), we have that for all t≥0,
(5.30)∫0t‍∫0s‍eα(τ-t)∥z(τ,ϑ-tω)∥2dτds=∫0t‍se-αs∥z(0,ϑ-sω)∥2ds≤∫0∞‍se-αs∥z(0,ϑ-sω)∥2ds<∞.
Since B∈𝒟 is tempered set, and ∥z0(ω)∥𝒞2 is tempered function, there exists T3=T3(B,ω,ɛ)>0 such that, for all t≥T3 and k∈ℕ,
(5.31)4c2eανk[(1+2νcf)∥v0(ϑ-tω)∥𝒞2+2νcf∥z0(ϑ-tω)∥𝒞2]te-αt≤ɛ8.
At the same time, there exists N3=N3(ω,ɛ)>0 such that, for all k≥N3,
(5.32)4c2∥g∥2eανkα2β+4c1c2eανk∫0∞‍se-αs∥z(0,ϑ-sω)∥2ds≤ɛ8.
It follows from (5.29), (5.30), (5.31), and (5.32) that, for all t≥T3(ω,ɛ) and k≥N3(ω,ɛ),
(5.33)4c2eανk∫0t‍eα(s-t)∥v(s,ϑ-tω,v0(ϑ-tω))∥2ds≤ɛ4.
Let T4=T4(B,ω,ɛ)=max{T1,T2,T3}, N4=N4(ω,ɛ)=max{N1,N2,N3}. Then it follows from (5.23), (5.24), (5.25), (5.28), and (5.33) that, for all t≥T4 and k≥N4,
(5.34)supσ∈[-ν,0]∑|i|≥2k‍|vit(σ,ϑ-tω,v0(ϑ-tω))|2<supσ∈[-ν,0]∑i∈ℤ‍ρ(|i|k)|vit(σ,ϑ-tω,v0(ϑ-tω))|2≤ɛ.
The proof is complete.Lemma 5.3.
The random dynamical systemϕ is 𝒟-pullback asymptotically compact in 𝒞, that is, for ℙ-a.e. ω∈Ω, the sequence {ϕ(tn,ϑ-tnω,vn0(ϑ-tnω))}n=1∞ has a convergent subsequence in 𝒞 provided tn→∞, B∈𝒟 and vn0(ϑ-tnω)∈B(ϑ-tnω).Proof.
By (4.21) and Lemma 5.1, we find that, for every t≥TB(ω)+ν, and σ1, σ2∈[-ν,0],
(5.35)∥ϕ(t,ϑ-tω,v0(ϑ-tω))(σ1)-ϕ(t,ϑ-tω,v0(ϑ-tω))(σ2)∥=∥v(t+σ1,ϑ-tω,v0(ϑ-tω))-v(t+σ2,ϑ-tω,v0(ϑ-tω))∥≤∥v′(t+ξ,ϑ-tω,v0(ϑ-tω))∥|σ1-σ2|≤r2(ω)|σ1-σ2|,
where
(5.36)r2(ω)=4+λu+lf(supσ∈[-ν,0]{r1(ϑσω)+∥z(0,ϑσω)∥})+∥g∥,
and ξ is between σ1 and σ2.
By Lemma5.1, (5.35), and Lemma 5.2, {ϕ(tn,ϑ-tnω,vn0(ϑ-tnω))}n=1∞ satisfies conditions (i)–(iii) in Theorem 3.3. Therefore, {ϕ(tn,ϑ-tnω,vn0(ϑ-tnω))}n=1∞ is relative compact in 𝒞 and hence has a convergent subsequence in 𝒞.We are now in a position to present our main result about the existence of a𝒟-random attractor for ψ in 𝒞.Theorem 5.4.
The random dynamical systemψ has a unique 𝒟-random attractor in 𝒞.Proof.
Notice thatϕ has a closed absorbing set K in 𝒟 by Lemma 5.1 and is 𝒟-pullback asymptotically compact in 𝒞 by Lemma 5.3. Hence, the existence of a unique 𝒟-random attractor {𝒜1(ω)}ω∈Ω for ϕ follows from Proposition 2.11 immediately.
Sinceψ and ϕ are conjugated by the random homeomorphism T(ω,ξ)=ξ+z0(ω), and z0(ω)∈𝒞 is tempered, then by Proposition 1.8.3 in [45], ψ has a unique 𝒟-random attractor {𝒜2(ω)}ω∈Ω in 𝒞 which is given by
(5.37)𝒜2(ω)={ξ(ω)+z0(ω):ξ(ω)∈𝒜1(ω)}.
The proof is complete.



---

*Source: 409282-2012-10-16.xml*