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

In this paper, we focus on the asymptotic behavior of solutions to stochastic delay lattice equations with additive noise and deterministic forcing. We rst show the existence of a continuous random dynamical system for the equations. en 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.


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 ∈ Z, Z denotes the integer set, ∈ R + . = ( ) ∈Z is a sequence in space (de ned later), , , and are positive constants, is the intensity of noise, ( ) = ∈Z is a superlinear source term, − = − ∈Z is a nonlinear function satisfying certain structural conditions and capturing the time delay ≥ 0, = ∈Z ∈ (de ned later), = ∈Z ∈ , = ∈Z is a two-side real valued Wiener process on a probability space.
e theory of attractors is a powerful tool to depict the asymptotic dynamics of an in nite-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 uctuations. Until now, random attractors have been investigated by many researchers, e.g., in [9][10][11][12][13] for autonomous stochastic equations, and in [14][15][16][17][18][19][20][21] for nonautonomous stochastic ones.
Lots of work have been done regarding the existence of global random attractors for SLDS with white noises of in nite sequences, see e.g., [22][23][24][25][26][27][28][29] and the references therein. Note that the stochastic equations considered in these papers do not contain nonlinearity with time delay. e di erential equations with delays arise, for instance, from population dynamics where a time lag or a er-e ect 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][31][32][33]. (1) In the present paper, we consider the stochastic delay lattice equations with superlinear nonlinearity (delay terms). More precisely, we rst prove the existence and uniqueness of tempered random attractors of equation (1). en we show the ergodicity of the systems. Denote Let := ℓ 2 (Z) ℓ : = 2 . It is known that is a Hilbert space with the inner product ( , v) = ∑ ∈Z v , and the norm In the sequel, we use ‖⋅‖ and (⋅, ⋅) to denote the norm and inner product of , respectively. e norm of ℓ is written as ‖⋅‖ ℓ ̸ = 2 . e letters and ( = 1, 2, ⋅ ⋅ ⋅) are general positive constants, taking di erent values from line to line. Since their values are not signi cant, we do not care about their values and relationship between one and another.
is 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 di erential equation (1) generates an in nite dimensional random dynamical system. e existence of the global random attractor is given in Section 4. Finally, the proof of ergodicity of the systems is nished in Section 5.

Preliminaries
In the following, we recall some basic concepts on random dynamical systems and pullback attractors which are mentioned in [8,22]. Let be a Banach space and Ω, F , , ∈R be a metric dynamical system, , a complete separable metric space with Borel -algebra B . Suppose D is a collection of some families of nonempty subsets of .

De nition 2.
A family = { ( ) : ∈ Ω} ∈ D is called a D -pullback absorbing set for Φ if for all ∈ Ω and for every ∈ D, there exists = ( , ) > 0 such that If, in addition, for all ∈ Ω, ( ) is a closed nonempty subset of and is measurable in with respect to F, then we say is a closed measurable D-pullback absorbing set for Φ.

De nition 4. A family
(iii) A attracts every member of D: for every ∈ D, where is the Hausdor semi-distance in . We borrow the following result for random dynamical systems from [28,34] and omit its proof.

Proposition 5.
Let D be an inclusion closed collection of some families of nonempty subsets of , and Φ be a continuous RDS on over R and Ω, F , , ∈R . en Φ has a D-pullback attractor A in D if Φ is D-pullback asymptotically compact in and Φ has a closed measurable D-pullback absorbing set in D. e D-pullback attractor A is unique and is given by, for each ∈ Ω,

RDSs for Stochastic Delay Lattice Systems
In this section, we rst state some assumptions that will be used throughout this paper. en 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 , 1 , 2 , , , , and ≥ 2.
3 Discrete Dynamics in Nature and Society where ∈Z ∈ , and is a constant satisfying ∑ ∈Z (A 3 ) We also need this assumption: (A 4 ) For su ciently large > 0, there is a positive constant small enough, such that (A 5 ) We can choose a positive constant such that For convenience, we now formulate system (1) as stochastic di erential equations in . Denote by , * , and the linear operators from into in the following way: for any = ∈Z ∈ , and en we have In the sequel, we consider the probability space (Ω, F , ) where F is the Borel -algebra induced by the compact-open topology of Ω, and F 0 is the Borel -algebra on Ω. is the corresponding Wiener measure on (Ω, F ).
Let us recall a ltration over the parametric space Ω, F , , ∈R which is the smallest -algebra generated by random variable ∈R is a parametric dynamical system (see [9] for more details).
Let ∈ , ∈ Z denote the element having 1 at position and all the other components 0. We have is the white noise taking valus in de ned on the probability space (Ω, F , ). en problem (1) and (2) can be written as the following abstract form: with the initial conditions Next, we de ne a continuous RDS for lattice system (1) and (2) in . is can be achieved by transferring the stochastic lattice system into a deterministic one with random parameters in a standard manner. Let satis es the one-dimensional stochastic di erential equation is equation has a random xed 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 ∈R -invariant subset Ω ⊆ Ω of full measure such that is continuous in for every ∈Ω, and the random variable | ( )| is tempered. Let F and ∼ be the restrictions of F and , respectively. We will de ne a continuous RDS for lattice system (1) and (2) in over R and Ω ,F ,̃ , ∈R . For convenience, from now on, we will abuse the notation slightly and write the space Ω ,F ,̃ as (Ω, F , ).
Given a bounded nonempty subset of , the Hausdor semidistance between and the origin in is denoted by ∈ Ω} be a family of nonempty subsets of . Such a is said to be tempered in if for every > 0, roughout the rest of this paper, we always use D to denote the collection of all families of tempered nonempty subsets of . e system (23) may be rewritten as an integral equation in , Theorem 6. Let > 0, then the following three properties hold: (1) Equation (28)  (2) We have the following estimate, for every ∈ Ω: Taking expectation on both sides of the above inequality, we known that ( ) ∈ L 2 (Ω, C ([ , ], )). It implies that (28) has a global solution ( ) ∈ L 2 (Ω, C ([ , ], )).  (28). en it follows from (28) that (37) (3) e solution of (28) depends continuously on the initial data , that is to say, for all ∈ Ω, the mapping ∈ → ⋅, , ∈ C ([0, ], ) is continuous. Proof.
Suppose ∈ Ω, from (30) we have By the assumption (A 1 ) and Young's inequality, we obtain It follows from assumption (A 2 ) and Young's inequality that Utilizing Young's inequality, we gain the following three inequalities:

Existence of Pullback Attractors
is section is devoted to the proof of existence of tempered pullback attractors for the systems (1) and (2) in . We rst show the existence of the absorbing set for the system (23).
en 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. ere exists a -invariant set Ω ὔ ⊂ Ω with full P measure and an absorbing set ( ), ∈ Ω ὔ , for Φ , , , ( ) , that is to say, there is an absorbing time = ( , ) > 0, for each ∈ D, ∈ Ω ὔ and ≥ such that what is more, ∈ D.
Proof. We apply an Ornstein-Uhlenbeck process on metric dynamical system Ω, F , , in . Suppose Moreover, there is a -invariant set Ω ὔ ⊂ Ω with full P measure such that, for all ∈ Ω ὔ , (1) the mapping → is continuous; (2) the random variable where ( ) is a solution of (23). We have with the initial conditions Taking the inner product of (54) with v( ), we can obtain By using the assumption (A 1 ) and Young's inequality, we arrive at where is a positive constant depending only on , , 1 , 2 , ∑ ∈Z 1, , ∑ ∈Z 2, .

Set
By the assumption of local Lipschitz condition of , we know there exists a constant such that on the ball (0, ),

So
By Schwarz inequality, Young's inequality and the assumption (A 2 ), we get By Gronwall's inequality, we nd Hence is inequality implies the uniqueness and continuous dependence on the initial data of the solution of (30). is proof is completed.
Similar to the proof of the eorem 7 in [8] with minor modi cations, we can prove. ☐ (42 (48) In order to prove that the random dynamical system Φ , , ( , ) is asymptotic compact, we require the following lemma. where 0 ≤ ( ) ≤ 1, ∈ R + , and with a constant such that With ∈ Z + , taking the inner product of (54) with the sequence (| |/ )v ∈Z , we obtain (68) ( ) = 4 ( ) Young's inequality and the assumption (A 2 ) also yield Using (56)-(60), we have Integrating (61) from to and estimating the following terms, we nd that and similarly, It follows from (61)-(63) and the assumption (A 4 ) that erefore, Since is continuous and ᐉ ᐉ ᐉ ᐉ ( ) ᐉ ᐉ ᐉ ᐉ is tempered, [13] P187, there is a tempered function ( ) > 0 such that (61) (62) (65) 7 Discrete Dynamics in Nature and Society By (72), (73), (75)-(78), we arrive at Integrating the above inequality from to , we have It's easy to know that And Hence, from (80)-(82) and the assumption (A 5 ), then multiplying both sides of it by − , we get, for ≥ = ( ) > , (108) lim where C is a space of bounded and uniformly continuous function on . e related transition probability ( , ⋅) is where L (⋅) is a Banach algebra of all linear bounded operators from to , and B is a -eld of all Borel subsets of . Now we prove equation (23)  Theorem 12. e system (23) and (24) is ergodic.
Proof. We need to prove that there is a unique invariant measure for Markov semigroup { } ≥0 . By the same discussion with a little change like (67), we get.