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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 409282, 27 pages
Random Attractors for Stochastic Retarded Lattice Dynamical Systems
1School of Mathematics and Statistics, Henan University of Science and Technology, Henan, Luoyang 471023, China
2School of Mathematics and Science, Shanghai Normal University, Shanghai 200234, China
Received 27 August 2012; Accepted 16 September 2012
Academic Editor: Jinhu Lü
Copyright © 2012 Xiaoquan Ding and Jifa Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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 , chemical reaction , material science , electrical engineering , laser systems , pattern recognition , complex network , 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.  established the first result on the existence of a global attractor for LDSs. Wang , Zhou and Shi  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 , Flandoli andSchmalfuss, with notable developments given in [20–25] and others. Bates et al.  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 whose inner product and norm are given by for all , . For , let denote the Banach space of all continuous functions endowed with the supremum norm . For any real numbers and any continuous function denotes the element of given by for .
In this paper, we investigate the long time behavior of the following stochastic retarded LDS: with initial data where , is a bounded positive constant sequence, is a nonlinear mapping satisfying local Lipschitz condition, , , and 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 . 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 . 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.
Let be a separable Banach space with Borel -algebra and be a probability space.
Definition 2.1. is called a metric dynamical system if is -measurable, is the identity on , for all , , and for all .
Definition 2.2. A set is called invariant with respect to , if for all , it holds
Definition 2.3. A continuous random dynamical system on over a metric dynamical system , is a mapping which is -measurable, and for all , (i) is continuous for all ; (ii) is the identity on ; (iii) for all , .
Definition 2.4. A random set is a multivalued mapping such that for every , the mapping is measurable, where is the distance between the element and the set . It is said that the random set is bounded (resp., closed or compact) if is bounded (resp., closed or compact) for -a.e. .
Definition 2.5. A random variable is called tempered with respect to , if for -a.e. A random set is called tempered if is contained in a ball with center zero and tempered radius for all .
Remark 2.6. If is tempered, then for any , and -a.e. Therefore, for any , is also tempered. Moreover, if for -a.e. , is continuous in , then for any , is measurable and for all and -a.e. Hence, for any , is also tempered.
Remark 2.7. If is tempered, then for any and -a.e. Moreover, is tempered, and if for -a.e. , is continuous in , then is also continuous in for such .
Hereafter, we always assume that is a continuous random dynamical system over , and is a collection of random subsets of .
Definition 2.8. A random set is called a random absorbing set in if for every and -a.e. , there exists such that
Definition 2.9. A random set is called a -random attractor (-pullback attractor) for if the following hold: (i) is a random compact set; (ii) is strictly invariant, that is, for -a.e. and all , (iii) attracts all sets in , that is, for all and -a.e. , where is the Hausdorff semimetric given by for any and .
Definition 2.10. is said to be -pullback asymptotically compact in if for all and -a.e. , has a convergent subsequence in whenever , and .
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 of random subsets of is called inclusion closed if whenever is an arbitrary random set, and is in with for all , then must belong to .
Proposition 2.11. Let be an inclusion-closed collection of random subsets of and a continuous random dynamical system on over . Suppose that is a closed random absorbing set for in and is -pullback asymptotically compact in . Then has a unique -random attractor which is given by
In this paper, we will take as the collection of all tempered random subsets of and prove the stochastic retarded LDS has a -random attractor.
3. Compactness Criterion in
In this section, we provide a necessary and sufficient condition for the relative compactness of sequences in , which will be used to establish the asymptotic compactness of the retarded LDS.
Lemma 3.1. Let . Then for every , there exists such that for all ,
Proof. For every , by virtue of the uniform continuity of , there exist such that Since for each , , there exists such that for all , Take . Then for each , there exists such that . Therefore, we get from (3.2) and (3.3) that for all , which completes the proof.
Theorem 3.2. Let . Then is relative compact in if and only if the following conditions are satisfied: (i) is bounded in ; (ii) is equicontinuous; (iii).
Proof. The proof is divided into two steps. We first show the necessity of the conditions and then prove the sufficiency.
(1) Assume that is relative compact in . Then we want to show conditions (i), (ii), and (iii) hold. Clearly, in this case, by the Ascoli-Arzelà theorem, must be bounded and equicontinuous. So we only need to prove condition (iii).
Given , since is relative compact, there exists a finite subset of such that the balls of radii centered at form a finite covering of , that is, for each , there exists such that By Lemma 3.1, there exists such that for all , By (3.5) and (3.6), we find that for each , there exists such that Therefore, for all , we have which implies condition (iii).
(2) Assume that conditions (i), (ii), and (iii) are valid. We want to prove that is relative compact in . That is, given , we want to show that has a finite covering of balls of radii . By condition (iii), we find that there exists such that for all , Consider the set in . By conditions (i) and (ii), we know that is bounded and equicontinuous in . Then, by the Ascoli-Arzelà theorem, we obtain that is relative compact in and hence there exists a finite subset of such that the balls of radii centered at form a finite covering of , that is, for each , there exists such that Now for each , we choose such that for and for . Then by (3.9) and (3.10), we find that for each , there exists such that which implies that the set has a finite covering of balls with radii . 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.
Theorem 3.3. Let . Then is relative compact in if and only if the following conditions are satisfied: (i) is bounded in ; (ii) is equicontinuous; (iii).
Proof. If is relative compact in , 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 , it follows from condition (iii) that there exists such that which implies that there exists such that By Lemma 3.1, we find that there exists such that Take . It follows from (3.13) and (3.14) that which implies that Therefore, which together with conditions (i) and (ii) shows that the conditions in Theorem 3.2 are satisfied with . The proof is complete.
4. Stochastic Retarded Lattice Differential Equations
For convenience, we now formulate (1.3)-(1.4) as a stochastic functional differential equation in . Define the linear operators , , , from to as follows. For , for each . Then and for all , . Therefore, for all . Let denote the element having 1 at position and all the other components 0. Then is an -valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator such that . For , let . Then stochastic retarded LDS (1.3)-(1.4) can be rewritten as a stochastic functional equation in with the initial data
In the sequel, we consider the probability space where is the Borel -algebra induced by the compact-open topology of , and the corresponding Wiener measure on with respect to the covariance operator . Let Then is an ergodic metric dynamical system. Since the above probability space is canonical, we have By Proposition in , there exists a -invariant set of full -measure such that
Let be the -completion of and let with where is the smallest -algebra generated by the random variable for all , such that and is the collection of -null sets of . Note that so is a filtered metric dynamical system.
Note that problem (4.3)-(4.4) is interpreted as an integral equation as follows: -a.s. for any . By the theory in , we deal with (4.12) on the complete probability space . For and , we make the following assumptions. There exist positive constants and such that . For any , there exists a constant such that for all , with .There exist positive constants and such that for all , , ..
We now associate a continuous random dynamical system with the stochastic retarded lattice differential equations over . To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on and transform the stochastic retarded lattice differential equations into a random one. Let where is the uniformly continuous semigroup on generated by bounded linear operator . Then by (4.8), (4.16) is well defined. The process is a stationary, Gaussian process. Moreover, the random variable is tempered and for each , the mapping is continuous. Furthermore, by Lemma 5.13 in , we find that for all and -a.s., Noticing that and using the Itô formula, we get from (4.17) that for all and -a.s., Setting for in (4.12), then by (4.19), we obtain a deterministic equation, -a.s. which is equivalent to the functional differential equation with initial condition Here , .
Theorem 4.1. Let and be fixed. Then the following properties hold.(1)For each , problem (4.21)-(4.22) has a unique solution .(2)Let and be the solutions of problem (4.21)-(4.22) for the initial data and , respectively. Then there exists a constant such that for all
Proof. (1) Denote
for all , and . Then by ()–(), we have that
for any , with , . Therefore, satisfies local Lipschitz condition and maps the bounded sets of into the bounded sets of . Then by using a standard argument, one can show that for each , there exists a such that problem (4.21)-(4.22) has a unique solution on . Moreover, if then
We prove now that this local solution is a global one. Let . By (), we can choose small enough such that . Taking the inner product of (4.21) with in , we get
Using the Young inequality, we find that
Then it follows from (4.27), (4.28), (4.29) that
Choose small enough such that . Then by (4.30), we obtain
Now, we can also choose small enough such that . Integrating (4.31) over leads to
Using the Young inequality and (), we find that
Then by (4.32) and (4.33), we obtain
where . Consequently,
Hence, for fixed , we get that for ,
and for ,
In view of (4.36) and (4.37), we find that for all ,
Therefore, for all ,
which, together with (4.26), implies that . This proves the property (1).
(2) Let . By (4.38), there exists a constant such that Then from (4.20) and (4.25), we have that for Hence, for fixed , we get that for , and for , In view of (4.42) and (4.43), we find that for all , The Gronwall inequality implies that for all , This proves the property (2). The proof is complete.
Conversely, if for each , is a solution of problem (4.21)-(4.22) with , then the process is a solution of problem (4.3)-(4.4). And if is a -valued -measurable random variable, then is an -adapted process.
Theorem 4.2. Problem (4.21)-(4.22) generates a continuous random dynamical system over , where Moreover, if one defines by then is another continuous random dynamical system associated to problem (4.3)-(4.4).
Proof. From property (2) of Theorem 4.1, it follows that is continuous for all . By (4.20), we have that for and ,
Then again by (4.20) and noticing that
we get that
For each consider
Then for , we have
It follows from (4.51) that
for all . By the uniqueness of the solution of (4.20), we find that
while (4.53) implies
Hence, is a continuous random dynamical system.
As for , noticing that we get from (4.56) that for , , Therefore, is also a continuous random dynamical system. Furthermore, and are conjugated random dynamical systems, that is where for every , is a homeomorphism of . The proof is complete.
5. Existence of Random Attractors
In this section, we prove the existence of a -random attractor for the random dynamical system associated with (4.3)-(4.4). We first establish the existence of a -random attractor for its conjugated random dynamical system , then the existence of a -random attractor for follows from the conjugation relation between and . To this end, we will derive uniform estimates on the solutions of problem (4.21)-(4.22) when with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for .
From now on, we always assume that is the collection of all tempered subsets of with respect to . The next lemma shows that has a random absorbing set in .
Lemma 5.1. There exists such that is a random absorbing set for in , that is, for any and -a.e. , there exists such that
Proof. Replacing by in (4.38), we get that for all , By assumption, is tempered. On the other hand, by Remark 2.6, is also tempered. Therefore, if , then there exists such that for all , where is tempered by Remark 2.7. Then it follows from (5.2) and (5.3) that for all , Given , denote by where is tempered. Then . Further, (5.5) indicates that is a random absorbing set for in , which completes the proof.
Proof. Let be a smooth function defined on such that for all , and
Then there exists a positive deterministic constant such that for all . Taking the inner product of (4.21) with in , we obtain that
We now estimate terms in (5.10) as follows. First, we get from () that
Secondly, by the property of the cutoff function , we estimate
Thirdly, using the Young inequality and (), we find that
Finally, using the Young inequality again, we obtain that
Taking into account (5.10), (5.11), (5.12), (5.13), and (5.14), we obtain that
Using the Young inequality and (), we get that
Integrating (5.16) over