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Zhang, Yijin; Lin, Zongbing

doi:https://doi.org/10.1155/2020/5405947

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 5405947 | https://doi.org/10.1155/2020/5405947

Long Term Behavior for a Class of Stochastic Delay Lattice Systems in Space

Yijin Zhang^1,2and Zongbing Lin³

Academic Editor: Douglas R. Anderson

Received30 May 2019

Accepted05 Sept 2019

Published05 Jan 2020

Abstract

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We first show the existence of a continuous random dynamical system for the equations. Then we investigate the pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractor in space. Finally, ergodicity of the systems is achieved.

1. Introduction

We explore the asymptotic behavior of a class of stochastic lattice systems with time delay driven by additive white noise:

with initial data

where , denotes the integer set, . is a sequence in space (defined later), , , and are positive constants, is the intensity of noise, is a superlinear source term, is a nonlinear function satisfying certain structural conditions and capturing the time delay , (defined later), , is a two-side real valued Wiener process on a probability space.

Lattice differential equations are widely adopted in physics, biology, and engineering such as pattern formation, propagation of nerve pulses, electric circuits, and so on, see, e.g., ([1–7]). Stochastic lattice dynamical systems (SLDS) arise naturally, while random influences or uncertainties (called noises) are taken into account. These noises may play an important role as intrinsic phenomena rather than just compensation of defects in deterministic models [8].

The theory of attractors is a powerful tool to depict the asymptotic dynamics of an infinite-dimensional system. Random attractor is an important concept to describe asymptotic behavior for a random dynamical system and to capture the essential dynamics with possibly extremely wide fluctuations. Until now, random attractors have been investigated by many researchers, e.g., in [9–13] for autonomous stochastic equations, and in [14–21] for nonautonomous stochastic ones.

Lots of work have been done regarding the existence of global random attractors for SLDS with white noises of infinite sequences, see e.g., [22–29] and the references therein. Note that the stochastic equations considered in these papers do not contain nonlinearity with time delay. The differential equations with delays arise, for instance, from population dynamics where a time lag or after-effect is involved.

As far as we are aware, it seems that there are very few works in the literature dealing with random attractors of stochastic lattice equations containing nonlinearity with time delay except [30–33].

In the present paper, we consider the stochastic delay lattice equations with superlinear nonlinearity (delay terms). More precisely, we first prove the existence and uniqueness of tempered random attractors of equation (1). Then we show the ergodicity of the systems.

Denote

Let . It is known that is a Hilbert space with the inner product and the norm , where Then we define a Banach space by with the norm

In the sequel, we use and to denote the norm and inner product of , respectively. The norm of is written as .

The letters and are general positive constants, taking different values from line to line. Since their values are not significant, we do not care about their values and relationship between one and another.

This paper is organized as follows. In Section 2, we recall some basic concepts and already known results related to random dynamical systems and random attractors. In Section 3, we show that the stochastic delay lattice differential equation (1) generates an infinite dimensional random dynamical system. The existence of the global random attractor is given in Section 4. Finally, the proof of ergodicity of the systems is finished in Section 5.

2. Preliminaries

In the following, we recall some basic concepts on random dynamical systems and pullback attractors which are mentioned in [8, 28].

Let be a Banach space and be a metric dynamical system, a complete separable metric space with Borel -algebra . Suppose is a collection of some families of nonempty subsets of .

Definition 1. A mapping is called a continuous random dynamical system (RDS) on over if for all and ,(i) is -measurable;(ii) is the identity on ;(iii);(iv) is continuous.

Definition 2. A family is called a -pullback absorbing set for if for all and for every , there exists such thatIf, in addition, for all , is a closed nonempty subset of and is measurable in with respect to , then we say is a closed measurable -pullback absorbing set for .

Definition 3. The random dynamical system is said to be -pullback asymptotically compact in if for , the sequencewhenever , and with .

Definition 4. A family is called a -pullback attractor for if for every ,(i) is measurable in with respect to and is compact in (ii) is invariant: (iii) attracts every member of : for every ,where is the Hausdorff semi-distance in .
We borrow the following result for random dynamical systems from [28, 34] and omit its proof.

Proposition 5. Let be an inclusion closed collection of some families of nonempty subsets of , and be a continuous RDS on over and . Then has a -pullback attractor in if is -pullback asymptotically compact in and has a closed measurable -pullback absorbing set in . The -pullback attractor is unique and is given by, for each ,

3. RDSs for Stochastic Delay Lattice Systems

In this section, we first state some assumptions that will be used throughout this paper. Then we illustrate the existence of RDSs for stochastic delay lattice systems.

From now on, the functions are assumed to satisfy the following conditions with positive constants ,,, and .(A₁) For all ,

and . also possesses the local Lipschitz condition, i.e., for any bounded interval , there exists a positive constant , such that for every ,(A₂) is continuous and for all ,

where , and is a constant satisfying .(A₃)We also need this assumption:

(A₄)For sufficiently large , there is a positive constant small enough, such that

(A₅) We can choose a positive constant such that

For convenience, we now formulate system (1) as stochastic differential equations in . Denote by , and the linear operators from into in the following way: for any ,

and

Then we have , , and for all .

In the sequel, we consider the probability space where

is the Borel -algebra induced by the compact-open topology of , and is the Borel -algebra on . is the corresponding Wiener measure on .

Let us recall a filtration over the parametric space

which is the smallest -algebra generated by random variable for every This -algebra has the property: . So is adapted to .

There is a classical group acting on , which is defined by

Then is a parametric dynamical system (see [9] for more details).

Let denote the element having at position and all the other components . We have

is the white noise taking valus in defined on the probability space .

Then problem (1) and (2) can be written as the following abstract form:

with the initial conditions

Next, we define a continuous RDS for lattice system (1) and (2) in . This can be achieved by transferring the stochastic lattice system into a deterministic one with random parameters in a standard manner. Let satisfies the one-dimensional stochastic differential equation

This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein–Uhlenbeck process (see [9] for more details)

In fact, we have that there exists a -invariant subset of full measure such that is continuous in for every , and the random variable is tempered. Let and be the restrictions of and , respectively. We will define a continuous RDS for lattice system (1) and (2) in over and . For convenience, from now on, we will abuse the notation slightly and write the space as .

Given a bounded nonempty subset of , the Hausdorff semidistance between and the origin in is denoted by . Let be a family of nonempty subsets of . Such a is said to be tempered in if for every ,

Throughout the rest of this paper, we always use to denote the collection of all families of tempered nonempty subsets of .

The system (23) may be rewritten as an integral equation in ,

Theorem 6. Let then the following three properties hold:(1)Equation (28) possesses a unique solution ;(2)We have the following estimate, for every :(3)The solution of (28) depends continuously on the initial data , that is to say, for all , the mapping is continuous.

Proof. (1)Set For every , equation (28) has a solution if and only ifhas a solution for every and every . For each fixed , (30) becomes a deterministic equation. As we know, it has a local solution where is the maximal interval of the existence of the solution. Next we prove this local solution is a global one.
Suppose , from (30) we haveBy the assumption (A₁) and Young’s inequality, we obtain It follows from assumption (A₂) and Young’s inequality thatUtilizing Young’s inequality, we gain the following three inequalities:By (31)–(36), we getwhereBy the similar way, we receiveIt follows from (37)–(39) and assumption (A₃) thatwhere are positive constants depending on . This tells us that is bounded by a continuous function, hence there exists a global solution on any . For every ,Taking expectation on both sides of the above inequality, we known that . It implies that (28) has a global solution .(2)By (41) and , for every , we can conclude (29).(3)Let for some and be the corresponding solutions of (28). Then it follows from (28) thatSetBy the assumption of local Lipschitz condition of , we know there exists a constant such that on the ball SoBy Schwarz inequality, Young’s inequality and the assumption (A₂), we getwhere . Therefore, we inferBy Gronwall’s inequality, we findHenceThis inequality implies the uniqueness and continuous dependence on the initial data of the solution of (30). This proof is completed.
Similar to the proof of the Theorem 7 in [8] with minor modifications, we can prove.

Theorem 7. System (27) generates a continuous RDS over and , wherefor each .

4. Existence of Pullback Attractors

This section is devoted to the proof of existence of tempered pullback attractors for the systems (1) and (2) in . We first show the existence of the absorbing set for the system (23). Then we make uniform estimate on the tails of solutions of systems (1) and (2). Finally we derive the theorem for the existence of the tempered pullback attractors.

Theorem 8. There exists a -invariant set with full measure and an absorbing set , for , that is to say, there is an absorbing time , for each and such thatwhat is more, .

Proof. We apply an Ornstein–Uhlenbeck process on metric dynamical system in . SupposeMoreover, there is a -invariant set with full measure such that, for all , (1) the mapping is continuous; (2) the random variable is tempered.
Set , where is a solution of (23). We havewith the initial conditionsTaking the inner product of (54) with , we can obtainBy using the assumption (A₁) and Young’s inequality, we arrive atwhere is a positive constant depending only on .
Young’s inequality and the assumption (A₂) also yieldUsing (56)–(60), we haveIntegrating (61) from to and estimating the following terms, we find thatand similarly,It follows from (61)–(63) and the assumption (A₄) thatTherefore,Since is continuous and is tempered, is tempered. From proposition 4.3.3 [13] P187, there is a tempered function such thatIn (65), with replaced by , from (66), we haveChooseThen, is tempered and is an absorbing set for . i.e., there is an absorbing time , for each , and such that Furthermore,
LetThen is an absorbing set for because and This completes the proof of the theorem.

In order to prove that the random dynamical system is asymptotic compact, we require the following lemma.

Lemma 9. Let be the absorbing set and the initial data. Then for each , there exist an absorbing time and such that for every , the solution of (1) meets

Proof. Let be a smooth cut-off function withwhere , , and with a constant such that .
With , taking the inner product of (54) with the sequence , we obtainWe now make the following estimate.Using the property of the smooth cut-off function , we havewhich tells us thatIt follows from the assumption (A₁) and Young’s inequality thatwhere is a positive constant depending on only.
By the assumption (A₂) and Young’s inequality, we getBy (72), (73), (75)–(78), we arrive atwhere .
Integrating the above inequality from to , we haveIt’s easy to know thatwhere . Andwhere .
Hence, from (80)–(82) and the assumption (A₅), then multiplying both sides of it by , we get, for ,where .
We replace by , and estimate each term on the right-hand side of the above inequality. With instead of in (65), we haveAs we know, namely, (tempered), so there exists a , such that when ,Also by (65), we estimatewhere . Therefore there exist and , such that as , Using (66), for we deduceso there is a , such that as ,Applying (66), we also getThus there exists , such that as ,Since , there is , we obtain, for By the same argument, we have , when , satisfyingObviously, there is a , when , fulfillingTake . We collect (83)–(94) to yield, for every ,which concludes the proof.
With the same method like the proof of the Theorem 11 in [8], we can obtain the asymptotic compactness of the continuous RDS .

Theorem 10. is asymptotically compact for all : each sequence has a convergent subsequence in as .
We are now in a position to present our main result about the existence of -pullback attractor.

Theorem 11. Suppose that assumptions (A₁)–(A₆) hold. Then the continuous RDS associated with problems (1) and (2) has a unique -pullback attractor, which is characterized by, for each ,

Proof. Note that the existence, uniqueness and characterization of the -pullback attractor of follows from Proposition 5 immediately, based on Theorem 8. Lemma 9. and Theorem 10.

5. Ergodicity of the Systems

We discuss the ergodicity of the systems in this section.

There exists at least an invariant measure for the random dynamical system induced by (23) and (24) (see [10]). The Markov semigroup associated with them is denoted by

where is a space of bounded and uniformly continuous function on .

The related transition probability is

where is a Banach algebra of all linear bounded operators from to , and is a -field of all Borel subsets of .

Now we prove equation (23) is ergodic, i.e., the invariant measure is unique.

Assume to be a mild solution of equation

with the initial conditions

Suppose is a semigroup formed by , from Theorem 6, we have

Put and . Then is a mild solution of

Theorem 12. The system (23) and (24) is ergodic.

Proof. We need to prove that there is a unique invariant measure for Markov semigroup . By the same discussion with a little change like (67), we get.We know thenLet then andSo, for arbitrary , we induceFor and , we obtainHence, we know there is a random variable such thatFor arbitrary we are led toThis tells us that, the law is the unique invariant measure for Markov semigroup . This accomplishes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

We express our sincere thanks to the editors and the anonymous reviewers for their constructive suggestions and helpful comments. Zhang was supported partially by Natural Science Foundation Project of Chongqing (Grant no. cstc2019jcyj-msxmX0716). Lin was supported partially by Science and Technology Project of Sichuan Province (2019YJ0683), Science and Technology Project of Panzhihua City (2019ZD-R-1) and Doctoral Research Initiation Fund Project of Panzhihua University.

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Copyright © 2020 Yijin Zhang and Zongbing Lin. 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.

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