Random Attractors for Stochastic Retarded Lattice Dynamical Systems

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.


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  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][36][37][38][39][40][41] whose inner product and norm are given by For ν > 0, let C : C −ν, 0 ; 2 denote the Banach space of all continuous functions ξ : −ν, 0 → 2 endowed with the supremum norm ξ C sup s∈ −ν,0 ξ s . For any real numbers a ≤ b, t ∈ a, b and any continuous function u : a − ν, b → 2 , u t denotes the element of C given by u t s u t s for s ∈ −ν, 0 . In this paper, we investigate the long time behavior of the following stochastic retarded LDS: with initial data where u u i i∈Z ∈ 2 , λ i i∈Z is a bounded positive constant sequence, f f i i∈Z : C → 2 is a nonlinear mapping satisfying local Lipschitz condition, g g i i∈Z ∈ 2 , a a i i∈Z ∈ 2 , and {w i : 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 Section 3, we establish a necessary and sufficient condition for the relative compactness of sequences in C −ν, 0 ; 2 . 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.

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 2.3. A continuous random dynamical system on X over a metric dynamical system Ω, F, P, ϑ t t∈R is a mapping Abstract and Applied Analysis 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.

2.6
Moreover, R is tempered, and if for P-a.e. ω ∈ Ω, r ϑ t ω is continuous in t, then R ϑ t ω is also continuous in t for such ω.
Hereafter, we always assume that ϕ is a continuous random dynamical system over Ω, F, P, ϑ t t∈R , and D is a collection of random subsets of X.
Definition 2.8. A random set K is called a random absorbing set in D if for every B ∈ D and P-a.e. ω ∈ Ω, there exists t B ω > 0 such that Definition 2.9. A random set A 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, Abstract and Applied Analysis 5 iii A attracts all sets in D, that is, for all B ∈ D and P-a.e. ω ∈ Ω, where d is the Hausdorff semimetric given by d E, F sup x∈E inf y∈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 t n , ϑ −t n ω, x n } ∞ n 1 has a convergent subsequence in X whenever t n → ∞, and x n ∈ B ϑ −t n ω .
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. Let D be an inclusion-closed collection of random subsets of X and ϕ a continuous random dynamical system on X over Ω, F, P, ϑ t 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 In this paper, we will take D as the collection of all tempered random subsets of C and prove the stochastic retarded LDS has a D-random attractor.

Compactness Criterion in
In this section, we provide a necessary and sufficient condition for the relative compactness of sequences in C −ν, 0 ; 2 , which will be used to establish the asymptotic compactness of the retarded LDS.

6 Abstract and Applied Analysis
Since for each s j , u s j ∈ 2 , there exists N j ε > 0 such that for all k ≥ N j ε , Take N ε max 1≤j≤p N j ε . Then for each s ∈ −ν, 0 , there exists j ∈ {1, 2, . . . , p} such that s ∈ s j−1 , s j . Therefore, we get from 3.2 and 3.3 that for all k ≥ N ε , which completes the proof.
Proof. The proof is divided into two steps. We first show the necessity of the conditions and then prove the sufficiency. 1 Assume that S is relative compact in C −ν, 0 ; 2 . 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 .
Given ε > 0, since S is relative compact, there exists a finite subset E of S such that the balls of radii ε/2 centered at E form a finite covering of S, that is, for each u ∈ S, there exists v ∈ E such that By 3.5 and 3.6 , we find that for each u ∈ S, there exists v ∈ E such that

3.7
Abstract and Applied Analysis 7 Therefore, for all k ≥ K * ε , we have which implies condition iii . 2 Assume that conditions i , ii , and iii are valid. We want to prove that S is relative compact in C −ν, 0 ; 2 . That is, given ε > 0, we want to show that S has a finite covering of balls of radii ε. By condition iii , we find that there exists K ε > 0 such that for all u u i i∈Z ∈ S, By conditions i and ii , we know that S| K is bounded and equicontinuous in C −ν, 0 ; R 2K ε 1 . Then, by the Ascoli-Arzelà theorem, we obtain that S| K is relative compact in C −ν, 0 ; R 2K ε 1 and hence there exists a finite subset H of S| K such that the balls of radii ε/2 centered at H form a finite covering of S| K , that is, for each u| K Now for each v| K v i |i|≤K ε ∈ H, we choose v v i i∈Z such that v i v i for |i| ≤ K ε and v i 0 for |i| > K ε . Then by 3.9 and 3.10 , we find that for each u ∈ S, there exists which implies that the set S has a finite covering of balls with radii ε. The proof is complete.
The next result is a variant of Theorem 3.2 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.
and only if the following conditions are satisfied:

Abstract and Applied Analysis
Proof. If {u n } ∞ n 1 is relative compact in C −ν, 0 ; 2 , 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 ε > 0, it follows from condition iii that there exists K 1 ε > 0 such that By Lemma 3.1, we find that there exists K 2 ε > 0 such that 3.14 Take K ε max{K 1 ε , K 2 ε }. It follows from 3.13 and 3.14 that which together with conditions i and ii shows that the conditions in Theorem 3.2 are satisfied with S {u n } ∞ n 1 . The proof is complete.

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 .
Abstract and Applied Analysis 9 For convenience, we now formulate 1.3 -1.4 as a stochastic functional differential equation in 2 . Define the linear operators A, B, B * , λ from 2 to 2 as follows. For u u i i∈Z ∈ 2 , In the sequel, we consider the probability space Ω, F, P where Ω ω ∈ C R, 2 : ω 0 0 , 4.5 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 Then Ω, F, P, ϑ t t∈R is an ergodic metric dynamical system. Since the above probability space is canonical, we have where σ{w τ 2 − w τ 1 : s ≤ τ 1 ≤ τ 2 ≤ t} is the smallest σ-algebra generated by the random variable w τ 2 − w τ 1 for all τ 1 , τ 2 such that s ≤ τ 1 ≤ τ 2 ≤ t and N is the collection of P-null sets of F. Note that so Ω, F, P, ϑ t t∈R , F t s 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
P-a.s. for any u 0 ∈ 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.
A 5 λ l > c f .

Abstract and Applied Analysis 11
We now associate a continuous random dynamical system with the stochastic retarded lattice differential equations over Ω, F, P, ϑ t t∈R . To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on Ω, F, P, ϑ t t∈R and transform the stochastic retarded lattice differential equations into a random one. Let

4.16
where e − A λ t is the uniformly continuous semigroup on 2 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., Noticing that Setting v t u t −z t for t ≥ −ν in 4.12 , then by 4.19 , we obtain a deterministic equation,

4.20
which is equivalent to the functional differential equation Here v 0 t u 0 t − z 0 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 .
for all t ≥ 0, ξ ∈ C and ω ∈ Ω. Then by A 1 -A 3 , we have that 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 2 . Then by using a standard argument, one can show that for each v 0 ∈ C, there exists a T max ≤ ∞ such that problem 4.21 -4.22 has a unique solution v on 0, T max . Moreover, if T max < ∞ then lim sup

4.26
We prove now that this local solution is a global one. Let T ∈ 0, T max . By A 5 , we can choose β > 0 small enough such that 2λ l > 2c f β. Taking the inner product of 4.21 with v in 2 , we get Clearly,

4.28
Abstract and Applied Analysis 13 Using the Young inequality, we find that

4.43
In view of 4.42 and 4.43 , we find that for all The Gronwall inequality implies that for all t ∈ 0, T , This proves the property 2 . The proof is complete.
Conversely, if for each ω ∈ Ω, v t, ω, v 0 is a solution of problem 4.21 -4.22 with v 0 · u 0 · − z 0 ·, ω , then the process is a solution of problem 4.3 -4.4 . And if u 0 is a C-valued F 0 -measurable random variable, then u t, ω, u 0 is an F t -adapted process.

4.51
For each ω ∈ Ω consider Then for τ t s, we have It follows from 4.51 that for all σ ∈ −ν, 0 . By the uniqueness of the solution of 4.20 , we find that Hence, φ is a continuous random dynamical system. As for ψ, noticing that ψ t, ω, u 0 φ t, ω, u 0 − z 0 ω z t ω , for t ≥ 0, ω ∈ Ω and u 0 ∈ C, 4.57 we get from 4.56 that for s, t ≥ 0,

Existence of Random Attractors
In this section, we prove the existence of a D-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 that D is the collection of all tempered subsets of C with respect to Ω, F, P, ϑ t t∈R . The next lemma shows that φ has a random absorbing set in D.
Lemma 5.1. There exists K ∈ 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 T B ω > 0 such that Proof.

5.2
By assumption, B ∈ D is tempered. On the other hand, by Remark 2.6, z 0 ω 2 C is also tempered. Therefore, if v 0 ϑ −t ω ∈ B ϑ −t ω , then there exists T B ω > 0 such that for all t ≥ T B ω , Given ω ∈ Ω, denote by where r 1 ω c 1 e αν 1 r ω g 2 αβ 1 5.7 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.
Let B ∈ D and v 0 ω ∈ B ω . Then for every ε > 0 and P-a.e. ω ∈ Ω, there exist T * T * B, ω, ε > 0 and N * N * ω, ε > 0 such that the solution v t, ω, v 0 ω of problem 4.21 -4.22 satisfies, for all t ≥ T * , Proof. Let ρ be a smooth function defined on R such that 0 ≤ ρ s ≤ 1 for all s ≥ 0, and Then there exists a positive deterministic constant c 2 such that |ρ s | ≤ c 2 for all s ≥ 0. Taking the inner product of 4.21 with x ρ |i|/k v i in 2 , we obtain that We now estimate terms in 5.10 as follows. First, we get from A 1 that

5.11
Secondly, by the property of the cutoff function ρ, we estimate Av, x Bv, Bx

20
Abstract and Applied Analysis Thirdly, using the Young inequality and A 3 , we find that

5.13
Finally, using the Young inequality again, we obtain that

5.18
It follows from 5.17 and 5.18 that

5.22
Abstract and Applied Analysis

5.23
We now estimate terms in 5.23 as follows. Since B ∈ D is tempered set, and z 0 ω 2 C is tempered function, if v 0 ϑ −t ω ∈ B ϑ −t ω , then for every ε > 0, there exists T 1 T 1 B, ω, ε > 0 such that for all t ≥ T 1 ,

5.26
Then by the Lebesgue theorem of dominated convergence, there exists N 2 N 2 ω, ε > 0 such that for all k ≥ N 2 ,

5.29
For the integral on the right side of 5.29 , we have that for all t ≥ 0,

5.30
Since B ∈ D is tempered set, and z 0 ω 2 C is tempered function, there exists T 3 T 3 B, ω, ε > 0 such that, for all t ≥ T 3 and k ∈ N,

5.34
The proof is complete.