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International Journal of Differential Equations
Volume 2011 (2011), Article ID 628459, 23 pages
Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces
Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA
Received 22 June 2011; Accepted 3 September 2011
Academic Editor: I. Chueshov
Copyright © 2011 Xiaoying Han. 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.
We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.
Stochastic lattice differential equations (SLDE’s) arise naturally in a wide variety of applications where the spatial structure has a discrete character and random spatiotemporal forcing, called noise, is taken into account. These random perturbations are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. SLDE’s may also arise as spatial discretization of stochastic partial differential equations (SPDE’s); however, this need not to be the case, and many of the most interesting models are those which are far away from any SPDE’s.
The long term behavior of SLDE’s is usually studied via global random attractors. For SLDE’s on regular spaces of infinite sequences, Bates et al. initiated the study on existence of a global random attractor for a certain type of first-order SLDE’s with additive white noise on 1D lattice . Continuing studies have been made on various types of SLDS’s with multiplicative or additive noise, see [2–7].
Note that regular spaces of infinite sequences may exclude many important and interesting solutions whose components are just bounded, considering that a weighted space of infinite sequences can make the study of stochastic LDE’s more intensive. More importantly, all existing works on SLDE’s consider either a noncoupled additive noise or a multiplicative white noise term at each individual node whereas in a realistic system randomness appears at each node as well as the coupling mode between two nodes. Han et al. initiated the asymptotic study of such SLDE’s in a weighted space of infinite sequences, with not only additive/multiplicative noise but also coefficients which are randomly coupled .
In this work, following the idea of , we will investigate the existence of a global random attractor for the following stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in weighted spaces: where ; (, are positive constants; is the coupling operator, , , , are random variables, and , are two-sided Brownian motions on proper probability spaces.
For deterministic partly dissipative lattice systems without noise, the existence of the global attractor has been studied in [9–13]. For stochastic lattice system (1.2) with additive noises, when , , for all , Huang  and Wang et al.  proved the existence of a global random attractor for the associated RDS in the regular phase space . In this work we will consider the existence of a compact global random attractor in the weighted space , which attracts random tempered bounded sets in pullback sense, for stochastic lattice systems (1.1) and (1.2). Here we choose a positive weight function such that . If , then contains any infinite sequences whose components are just bounded and . Note that when , our results recover the results obtained in [4, 14] while is reduced to the standard . Moreover, the required conditions in this work for the existence of a random attractor for system (1.2)-(1.1) in weighted space are weaker than those in .
The rest of this paper is organized as follows. In Section 2, we present some preliminary results for global random attractors of continuous random dynamical systems in weighted spaces of infinite sequences. We then discuss the existence of random attractors for stochastic lattice systems (1.1) and (1.2) in Sections 3 and 4, respectively.
Let be a positive function from to , where is a finite positive constant. Define for any , and then is a separable Hilbert space with the inner product and norm for , . Moreover, define with inner product and norm then is also a separable Hilbert space.
Let be a probability space and be a family of measure-preserving transformations such that is -measurable, and for all . The space is called a metric dynamical system. In the following, “property (P) holds for a.e. with respect to " means that there is with and such that (P) holds for all .
Recall the following definitions from existing literature.(i)A stochastic process is said to be a continuous RDS over with state space , if is -measurable, and for each , the mapping is continuous for , and for all and .(ii)A set-valued mapping (may be written as for short) is said to be a random set if the mapping is measurable for any .(iii)A random set is called a closed (compact) random set if is closed (compact) for each .(iv)A random set is said to be bounded if there exist and a random variable such that for all .(v)A random bounded set is said to be tempered if for a.e. , Denote by the set of all tempered random sets of .(vi)A random set is said to be a random absorbing set in if for any and a.e. , there exists such that .(vii)A random set is said to be a random attracting set if for any , we have in which is the Hausdorff semidistance defined via for any .(viii)A random compact set is said to be a random global attractor if it is a compact random attracting set and for a.e. and.
Definition 2.1 (see ). is said to be random asymptotically null in , if for any , a.e. , and any , there exist and such that
Theorem 2.2 (see ). Let be a continuous RDS over with state space and suppose that(a)there exists a random bounded closed absorbing set such that for a.e. and any , there exists yielding for all ;(b) is random asymptotically null on ; that is, for a.e. and for any , there exist and such that
Then the RDS possesses a unique global random attractor given by
3. Stochastic Partly Dissipative Lattice Systems with Multiplicative Noise in Weighted Spaces
This section is devoted to the study of asymptotic behavior for system (1.1) in weighted space . We first transform the stochastic lattice system (1.1) to random lattice system in Section 3.1. We then show in Section 3.2 that (1.1) generates random dynamical system in . Finally we prove in Section 3.3 the existence of a global random attractor for system (1.1).
Throughout the rest of this paper, a positive weight function is chosen to satisfy(P0) and for some positive constants and .
3.1. Mathematical Setting
Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise: where , ; with (, , ; are positive constants; are random variables on the probability space ; is a linear operator on defined by is a Brownian motion (Wiener process) on the probability space ; denotes the Stratonovich sense of the stochastic term.
For convenience, we first transform (3.1) into a random differential equation without white noise. Let then is an Ornstein-Uhlenbeck process on that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details) where for , , and possesses the following properties.
Lemma 3.1 (see [2, 15]). There exists a -invariant set of offull measure such that for , one has(i)the random variable is tempered;(ii)the mapping is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)
The mapping of on possesses same properties as the original one if we choose the trace -algebra with respect to to be denoted also by . Therefore we can change our metric dynamical system with respect to , still denoted by the symbols .
Let where is a solution of (3.1), then is a homomorphism in . System (3.1) can then be transformed to the following random system with random coefficients but without white noise: Letting , (3.8) are equivalent to where
We now make the following standing assumptions on , , , and and study in the following subsections asymptotic behavior of system (3.9).(H1) , . (H2)Let belongs to with respect to for each . and is tempered, that is, there exists a -invariant set of full measure such that for , In the following, we will consider and still write as .(H3), where .(H4)There exists a function such that(H5), , , , and there exists a constant such that , , .
3.2. Random Dynamical System Generated by Random Lattice System
In this subsection, we show that the random lattice system (3.9) generates a random dynamical system on .
Definition 3.2. We call a solution of the following random differential equation where , if satisfies
Theorem 3.3. Let and (P0), (H1), (H2), (H4), and (H5) hold. Then for any and any initial data , (3.9) admits a unique solution with .
Proof. (1) Denote , we first show that if and , then (3.9) admits a unique solution on with . Given , , and , note that is continuous in and measurable in from to .
By (3.2) and (H2), By (H4), and therefore For any , , and for some Also It then follows that For any compact set with , defining random variable via we have and for any , According to [15, 19, 20], problem (3.9) possesses a unique local solution () satisfying the integral equation
We will next show that . Since the set of continuous random process in is dense in the set (see [18, 21]), for each , there exists a sequence of continuous random process in such that
Consider the random differential equation with initial data where Follow the same procedure as above, (3.28) has a unique solution , that is, and by the continuity of in , there holds Note that multiplying (3.31) by and sum over results in Applying Gronwall’s inequality to (3.33) we obtain that where is independent of , which implies that the interval of existence of is , and .
By (3.34), Since for some and , then for any , By the Arzela-Acoli Theorem, there exists a convergent subsequence of such that and is continuous on . Moreover, for . By (3.27), (3.35), assumption (H2), and the Lebesgue Dominated Convergence Theorem we have Thus replacing by in (3.31) and letting give By the uniquness of the solutions of (3.9), we have for . By (3.34), for , which implies that the solution of (3.9) exists globally on .
(2) Next we prove that for any and , (3.9) has a solution on with . Let and , ,. Let be two solutions of (3.28) with initial data and replaced by , , , , respectively. Set . Take inner product of with and evaluate each term as follows. By (P0), (H1), (H2), and (H4), It then follows that For , applying Gronwall’s inequality to (3.41) on implies that for some constant depending on , and thus where is a constant depending on . Denote by , where with the norm . By (3.43), there exists a mapping such that , where is the solution of (3.28) on with . Since is dense in , the mapping can be extended uniquely to a continuous mapping .
For given and , for . There exist sequences , such that Let be the solution of (3.28), then it satisfies the integral equation By the continuity of , we have for , Thus for each , Moreover, is bounded in . Let , then we have and satisfies the differential equation (3.31).
Multiply equation (3.31) by and sum over , we obtain Similar to the process (3.35)–(3.39) in part (1), we obtain the existence of a unique solution of (3.9) with initial data , which is the limit function of a subsequence of in for . In the latter part of this paper, we may write as for simplicity.
Proof. By Theorem 3.3, the solution of (3.9) with exists globally on . It is then left to show that is measurable in .
In fact, for and , the solution of (3.9) for . In this case, function is continuous in ,, and measurable in , which implies that , is -measurable.
For and , the solution for . For any given , define , by and write Then is continuous and for any , and Thus is -measurable. Observe also that is -measurable. Hence is -measurable. It then follows from (3.54) that is -measurable. Therefore, fix we have that is measurable in . The other statements then follow directly.
Remark 3.5. If , system (3.1) defines a continuous RDS over in both state spaces and .
3.3. Existence of Tempered Random Bounded Absorbing Sets and Global Random Attractors in Weighted Space
In this subsection, we study the existence of a tempered random bounded absorbing set and a global random attractor for the random dynamical system generated by (3.1) in weighted space .
Theorem 3.6. Assume that (P0), (H1)–(H5) hold, then there exists a closed tempered random bounded absorbing set of such that for any and each , there exists yielding for all . In particular, there exists such that for all.
Proof. (1) For initial condition and , let be a solution of (3.28) with , where , then for all . Let be such that
By (H4) and (H5), we have
Applying Gronwall’s inequality to (3.57), we obtain that for ,
(2) For any and , let and be sequences such that Then as in , and (3.58) holds for . Therefore, where