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

Authors: Yijin Zhang; Zongbing Lin
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## 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 inXρ space. Finally, ergodicity of the systems is achieved.


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## Body

## 1. Introduction

We explore the asymptotic behavior of a class of stochastic lattice systems with time delay driven by additive white noise:(1)duidt=νui−1−2ui+ui+1−λui+Fiui+fiuit−ρ+gi+εaidwidt,t>τ,with initial data(2)uiτ+s=uτ,is,s∈−ρ,0,wherei∈Z, Z denotes the integer set, τ∈R+. u=uisi∈Z is a sequence in Xρ space (defined later), ν, λ, and ε are positive constants, ε is the intensity of noise, Fu=Fiuii∈Z is a superlinear source term, fut−ρ=fiuit−ρi∈Z is a nonlinear function satisfying certain structural conditions and capturing the time delay ρ≥0, g=gii∈Z∈X (defined later), a=aii∈Z∈X, w=wii∈Z 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 ofstochastic lattice equations containing nonlinearity with time delayexcept [30–33].In the present paper, we consider the stochastic delay lattice equations withsuperlinear 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(3)ℓp=u=uii∈Z:∑i∈Zuip<∞ui∈R,p≥  1.LetX:=ℓ2Zℓp:p=  2. It is known that X is a Hilbert space with the inner product u,v=∑i∈Zuivi, and the norm uX2=∑i∈Zui2, where u=uii∈Z,v=vii∈Z∈X. Then we define a Banach space by Xρ:=C−ρ,  0,X with the norm(4)uXρ=sups∈−ρ,0usX,∀u=uss∈−ρ,0∈Xρ.In the sequel, we use· and ·,· to denote the norm and inner product of X, respectively. The norm of ℓp is written as ·ℓpp≠2.The lettersC and Cii=1,2,  … 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 Section2, 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].Letχ be a Banach space and Ω,F,P,θtt∈R be a metric dynamical system, χ,d a complete separable metric space with Borel σ-algebra Bχ. Suppose D is a collection of some families of nonempty subsets of χ.Definition 1.
A mappingΦ:R+×R×Ω×χ→χ is called a continuous random dynamical system (RDS) on χ over Ω,F,P,θtt∈R if for all ω∈Ω and t,s∈R+,(i)
Φ·,·,·:R+×Ω×χ→χ is BR+×F×Bχ,Bχ-measurable;(ii)
Φ0,ω,· is the identity on χ;(iii)
Φt+s,ω,·=Φt,θsω,·∘Φs,ω,·;(iv)
Φt,ω,·:χ→χ is continuous.Definition 2.
A familyK=Kω:ω∈Ω∈D is called a D-pullback absorbing set for Φ if for all ω∈Ω and for every D∈D, there exists T=TD,  ω>0 such that(5)Φt,θ−tω,Dθ−tω⊆Kωforallt≥T.
If, in addition, for allω∈Ω, Kω is a closed nonempty subset of χ and K is measurable in ω with respect to F, then we say K is a closed measurable D-pullback absorbing set for Φ.Definition 3.
The random dynamical systemΦ is said to be D-pullback asymptotically compact in χ if for ω∈Ω, the sequence(6)Φtn,θ−tnω,xnn=1∞hasaconvergentsubsequenceinχ.
whenevertn→∞, and xn∈Bθ−tn  ω with Bω:ω∈Ω∈D.Definition 4.
A familyA=Aω:ω∈Ω∈D is called a D-pullback attractor for Φ if for every ω∈Ω,(i)
A is measurable in ω with respect to F and Aω is compact in χ;(ii)
A is invariant: Φt,ω,Aω=At,θtω,∀t≥0;(iii)
A attracts every member of D: for every D∈D,
(7)
lim
t
→
∞
d
χ
Φ
t
,
θ
−
t
ω
,
D
θ
−
t
ω
,
A
ω
=
0
,
wheredχ is the Hausdorff semi-distance in χ.
We borrow the following result for random dynamical systems from [28, 34] and omit its proof.Proposition 5.
LetD be an inclusion closed collection of some families of nonempty subsets of χ, and Φ be a continuous RDS on χ over R and Ω,F,P,θtt∈R. Then Φ has a D-pullback attractor A in D if Φ is D-pullback asymptotically compact in χ and Φ has a closed measurable D-pullback absorbing set K in D. The D-pullback attractor A is unique and is given by, for each ω∈Ω,(8)Aω=∩κ≥tKω∪t≥κΦt,θ−tω,Kθ−tω¯.


## 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 functionsF,f are assumed to satisfy the following conditions with positive constants p,α1,α2,Cf,Lf,LF, and p≥2.(A1)
For alls,t,β1,i,β2,i∈R,(9)
F
i
s
s
≤
−
α
1
s
p
+
β
1
,
i
,
(10)
F
i
s
≤
α
2
s
p
−
1
+
β
2
,
i
,and∑i∈Zβ1,i<∞,∑i∈Zβ2,i<∞. F also possesses the local Lipschitz condition, i.e., for any bounded interval I⊂R, there exists a positive constant LF, such that for every s,t∈I,(11)Fit−Fis≤LFt−s,(A2)
f:R→R is continuous and for all uii∈Z,vii∈Z∈ℓ2,(12)
f
i
u
i
−
f
i
v
i
≤
L
f
u
i
−
v
i
,
(13)
f
i
u
i
2
≤
C
f
2
u
i
2
+
η
i
2
,whereηii∈Z∈X, and Cη is a constant satisfying ∑i∈Zηi2=Cη.(A3)
We also need this assumption:(14)
λ
≥
2
2
C
f
.(A4)
For sufficiently largeλ>0, there is a positive constant ϵ small enough, such that(15)
λ
−
ϵ
−
4
λ
C
f
2
e
ϵ
ρ
≥
0
.(A5)
We can choose a positive constantϑ such that(16)
3
λ
2
−
ϑ
−
8
λ
C
f
2
e
ϑ
ρ
≥
0
.For convenience, we now formulate system (1) as stochastic differential equations in Xρ. Denote by B,B∗, and A the linear operators from Xρ into Xρ in the following way: for any u=uii∈Z∈Xρ,(17)Bui=ui+1−ui,B∗ui=ui−1−ui,and(18)Aui=−ui−1+2ui−ui+1.Then we haveA=BB∗=B∗B, B∗u,v=u,Bv, and Au,u≥0 for all u,v∈Xρ.In the sequel, we consider the probability spaceΩ,F,  P where(19)Ω=ω∈CR,X:ω0=0,F is the Borel σ-algebra induced by the compact-open topology of Ω, and F0 is the Borel σ-algebra on Ω. P is the corresponding Wiener measure on Ω,F.Let us recall a filtration over the parametric spaceΩ,F,P,θtt∈R(20)Ft0t=σWt2−Wt1:t0≤t1≤t2≤t,which is the smallestσ-algebra generated by random variable Wt2−Wt1 for every t0≤t1≤t2≤t. This σ-algebra has the property: θh−1Ft0t=Ft0+ht+h. So Wt is adapted to Ft0t.There is a classical groupθtt∈R acting on Ω,F,P, which is defined by(21)θtω·=ω·+t−ωt,ω∈Ω,t∈R.ThenΩ,F,P,θtt∈R is a parametric dynamical system (see [9] for more details).Letei∈X,i∈Z denote the element having 1 at position i and all the other components 0. We have(22)∑i∈Zaieiwit=Wtwithaii∈Z∈X,is the white noise taking valus inX defined on the probability space Ω,F,P.Then problem (1) and (2) can be written as the following abstract form:(23)dudt+νAu+λu=Fu+fut−ρ+g+εdWdt,t>τ,with the initial conditions(24)uτ+s=uτs,s∈−ρ,0.Next, we define a continuous RDS for lattice system (1) and (2) in Xρ. This can be achieved by transferring the stochastic lattice system into a deterministic one with random parameters in a standard manner. Let zθt  ω satisfies the one-dimensional stochastic differential equation(25)dzθtω+αzθtωdt=dwt.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)(26)zω=−α∫−∞0eαsωsds.In fact, we have that there exists aθtt∈R-invariant subset Ω~⊆Ω of full measure such that zθtω is continuous in t for every ω∈Ω~, and the random variable |zω| is tempered. Let F~ and P~ be the restrictions of F and P, respectively. We will define a continuous RDS for lattice system (1) and (2) in Xρ over R and Ω˜,F˜,P˜,θtt∈R. For convenience, from now on, we will abuse the notation slightly and write the space Ω˜,F˜,P˜ as Ω,F,P.Given a bounded nonempty subsetD of Xρ, the Hausdorff semidistance between D and the origin in Xρ is denoted by D=supψ∈Dψ. Let D=Dω:ω∈Ω be a family of nonempty subsets of Xρ. Such a D is said to be tempered in Xρ if for every γ>0,(27)limt→−∞eγtDθtω=0.Throughout the rest of this paper, we always useD to denote the collection of all families of tempered nonempty subsets of Xρ.The system (23) may be rewritten as an integral equation in Xρ,(28)ut=uτ0+∫τt−νAur−λur+Fur+fur−ρ+gdr+εWt−εWτ,t≥τ,w∈Ω.Theorem 6.
LetT>0, then the following three properties hold:(1)
Equation (28) possesses a unique solution ut,uτ∈L2Ω,C0,T,X;(2)
We have the following estimate, for everyω∈Ω:
(29)
sup
t
∈
τ
,
T
u
t
2
≤
C
u
τ
0
−
ε
W
τ
2
+
ε
2
sup
t
∈
τ
,
T
W
t
2
+
∫
τ
T
W
r
2
+
W
r
ℓ
p
p
+
W
r
ℓ
1
+
g
2
d
r
;
(3)
The solution of (28) depends continuously on the initial data uτ, that is to say, for all ω∈Ω, the mapping uτ∈X↦u·,ω,uτ∈C0,T,X is continuous.Proof.
(1)
Setzt=ut−εWt. For every ω∈Ω, equation (28) has a solution u∈L2Ω,Cτ,T,X if and only if
(30)
z
t
=
u
τ
0
+
∫
τ
t
−
ν
A
z
r
−
λ
z
r
+
F
z
r
+
ε
W
r
+
f
z
r
−
ρ
+
ε
W
r
−
ρ
+
g
−
ν
ε
A
W
r
−
λ
ε
W
r
d
r
−
ε
W
τ
,
has a solutionz∈L2Ω,Cτ,T,X for every t∈0,  T and every ω∈Ω. For each fixed ω∈Ω, (30) becomes a deterministic equation. As we know, it has a local solution z∈Cτ,Tmax,X, where τ,Tmax is the maximal interval of the existence of the solution. Next we prove this local solution is a global one.
Supposeω∈Ω, from (30) we have(31)zt2=uτ0−εWτ2+2∫τt−νAzr,zr−λzr2+Fzr+εWr,zr+fzr−ρ+εWr−ρ,zr+g,zr−νεAWr,zr−λεWr,zrdr.
By the assumption (A1) and Young’s inequality, we obtain (32)Fzr+εWr,zr=Fzr+εWr,zr+εWr−Fzr+εWr,εWr≤∑i∈Z−α1zi+εaieiwirp+β1,i+ε∑i∈Zα2zi+εaieiwirp−1+β2,iaieiwir≤∑i∈Zβ1,i+C1∑i∈Zaieiwirp+ε∑i∈Zβ2,iaieiwir≤C21+Wrℓpp+Wrℓ1,
It follows from assumption (A2) and Young’s inequality that(33)fzr−ρ+εWr−ρ,zr≤λ8zr2+4λCf2∑i∈Zzir−ρ2+4λCfε2∑i∈Zaieiwir−ρ2+2λCη.
Utilizing Young’s inequality, we gain the following three inequalities:(34)g,zr≤λ8zr2+2λg2,
(35)
−
ν
ε
A
W
r
,
z
r
≤
λ
8
z
r
2
+
2
λ
ν
ε
2
A
2
W
r
2
,
(36)
−
λ
ε
W
r
,
z
r
≤
λ
8
z
r
2
+
2
λ
λ
ε
2
W
r
2
.
By (31)–(36), we get(37)zt2≤uτ0−εWτ2−λ∫τtzr2dr+8λCf2∫τ−ρt−ρzr2dr+8λCfε2∫τ−ρt−ρWr2dr+4λCηt−τ+4λ∫τtg2dr+4λνε2∫τtA2Wr2dr+4λλε2∫τtWr2dr+2C2∫τt1+Wrℓpp+Wrℓ1dr,
where(38)8λCf2∫τ−ρt−ρzr2dr≤8λCf2∫τ−ρtzr2dr=8λCf2∫τ−ρtzr2dr+∫τtzr2dr≤8λCf2ρsupr∈τ−ρ,τzr+∫τtzr2dr.
By the similar way, we receive(39)8λCfε2∫τ−ρt−ρWr2dr≤8λCfε2ρsupr∈τ−ρ,τWr+∫τtWr2dr.
It follows from (37)–(39) and assumption (A3) that(40)zt2≤C2uτ0−εWτ2+C1∫τtWr2+Wrℓpp+Wrℓ1+g2+1dr,
whereC1,C2 are positive constants depending on ν,λ,ε,ρ,Cf,Cη,supr∈τ−ρ,τzr,supr∈τ−ρ,τ‖Wr‖. This tells us that zt is bounded by a continuous function, hence there exists a global solution on any τ,  T. For every ω∈Ω,(41)supt∈τ,Tzt2≤Cuτ0−εWτ2+∫τTWr2+Wrℓpp+Wrℓ1+g2dr.
Taking expectation on both sides of the above inequality, we known thatzt∈L2Ω,Cτ,T,X. It implies that (28) has a global solution ut∈L2Ω,Cτ,T,X.(2)
By (41) and zt=ut−εWt, for every ω∈Ω, we can conclude (29).(3)
Letuτ,Y1s,uτ,Ys∈Xρ,uτ,Y1s,uτ,Ys<r0 for some r0>0, and Y1t=:ut,uτ,Y1s,Yt=:ut,uτ,Ys be the corresponding solutions of (28). Then it follows from (28) that
(42)
Y
1
t
−
Y
t
2
=
u
τ
,
Y
1
s
−
u
τ
,
Y
s
2
+
2
∫
τ
t
−
ν
A
Y
1
r
−
Y
r
,
Y
1
r
−
Y
r
−
λ
Y
1
r
−
Y
r
2
+
F
Y
1
r
−
F
Y
r
,
Y
1
r
−
Y
r
+
f
Y
1
r
−
ρ
−
f
Y
r
−
ρ
,
Y
1
r
−
Y
r
d
r
.
Set(43)R=Cr02+ε2supt∈τ,TWt2+∫τTWr2+Wrℓpp+Wrℓ1+g2dr.
By the assumption of local Lipschitz condition ofF, we know there exists a constant LR such that on the ball B0,  R,(44)FY1r−FYr≤LRY1r−Yr.
So(45)FY1r−FYr,Y1r−Yr≤LRY1r−Yr2.
By Schwarz inequality, Young’s inequality and the assumption (A2), we get(46)∫τtfY1r−ρ−fYr−ρ,Y1r−Yrdr≤12∫τtfY1r−ρ−fYr−ρ2+Y1r−Yr2dr≤12∫τtLf2Y1r−ρ−Yr−ρ2+Y1r−Yr2dr≤12∫τtLf2+1Y1r−Yr2dr+C1,
whereC1=1/2∫τ−ρτLf2Y1r−Yr2dr. Therefore, we infer(47)Y1t−Yt2≤ρLf2+1uτ,Y1s−uτ,Ys2+2LR+Lf2−2λ+1∫τtY1r−Yr2dr.By Gronwall’s inequality, we find(48)Y1t−Yt2≤uτ,Y1s−uτ,Ys2e2LR+Lf2−2λ+1t−τ,t∈τ,T.
Hence(49)supt∈τ,TY1t−Yt2≤uτ,Y1s−uτ,Ys2e2LR+Lf2−2λ+1T−τ.
This 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 Theorem7 in [8] with minor modifications, we can prove.Theorem 7.
System (27) generates a continuous RDS Φt,st≥τ over Ω,F,P,θtt∈R and R, where(50)Φt,s,ω,uτs=ut,s,ω,uτss∈−ρ,0,
for eacht≥τ,ω∈Ω.


## 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 Xρ. 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 aθt-invariant set Ω′⊂Ω with full P measure and an absorbing set Kω,ω∈Ω′, for Φt,s,ω,uτs, that is to say, there is an absorbing time T=TD,ω>0, for each D∈D,ω∈Ω′ and t≥T such that(51)Φt,θ−tω,Dθ−tω⊂Kω,
what is more,K∈D.Proof.
We apply an Ornstein–Uhlenbeck process on metric dynamical systemΩ,F,P,θt in X. Suppose(52)zθtω≕−λ∫−∞0eλsθtωsdst>τ,λ>0,
(53)
d
z
+
λ
z
d
t
=
d
w
t
t
>
τ
.
Moreover, there is aθt-invariant set Ω′⊂Ω with full P measure such that, for all ω∈Ω′, (1) the mapping t→yθt  ω is continuous; (2) the random variable yω is tempered.
Setvt=ut−εyθt  ω, where ut is a solution of (23). We have(54)dvdt=−νAv−λv+Fvt+εyθtω+ft,vt−ρ+εyθt−ρω+g−νεAyθtω,t>τ,
with the initial conditions(55)vτ+s=vτs,s∈−ρ,0.
Taking the inner product of (54) with vt, we can obtain(56)12ddtv2+νAv,v+λv2=Fv+εy,v+ft,vt−ρ+εyθt−ρω,v+g,v−νεAy,v.
By using the assumption (A1) and Young’s inequality, we arrive at(57)Fv+εy,v=Fv+εy,v+εy−Fv+εy,εy≤∑i∈Z−α1vi+εyip+β1,i+∑i∈Zα2vi+εyip−1+β2,iεyi≤C1+yℓpp+yℓ1,
whereC is a positive constant depending only on ε,p,α1,α2,∑i∈Zβ1,i,∑i∈Zβ2,i.
Young’s inequality and the assumption (A2) also yield(58)fvt−ρ+εyθt−ρω,v≤1λfvt−ρ+εyθt−ρω2+λ4v2≤λ4v2+1λ∑i∈ZCf2vit−ρ+εyiθt−ρωi2+ηi2≤λ4v2+2λCf2vt−ρ2+2λCfε2∑i∈Zyiθt−ρωi2+1λCη,
(59)
g
,
v
≤
λ
8
v
2
+
2
λ
g
2
,
(60)
−
ν
ε
A
y
,
v
≤
λ
8
v
2
+
2
λ
ν
ε
2
A
2
y
2
.
Using (56)–(60), we have(61)ddteϵtv2+λ−ϵeϵtv2≤eϵt2C1+yℓpp+yℓ1+4λCf2vt−ρ2+4λCfε2yθt−ρω2+2λCη+4λg2+4λνε2A2y2.
Integrating (61) from τ to t and estimating the following terms, we find that(62)4λCf2∫τteϵrvr−ρ2dr≤4λCf2∫τ−ρτeϵr+ρvr2dr+∫τteϵr+ρvr2dr≤4λCf2ρsupr∈τ−ρ,τeϵr+ρvr2+4λCf2∫τteϵr+ρvr2dr,
and similarly,(63)4λCfε2∫τteϵryθr−ρω2dr≤C1+4λCfε2∫τteϵr+ρyθrω2dr.
It follows from (61)–(63) and the assumption (A4) that(64)eϵtvt2≤eϵτvτ2+C∫τteϵr1+yℓpp+yℓ1+y2dr+2λϵCη+2g2eϵt.
Therefore,(65)vt,ω,vτs,ω2≤vτs,ω2eϵτ−t+C∫τteϵr−t1+yθrωℓpp+yθrωℓ1+yθrω2dr+2λϵCη+2g2.
Sinceyθtω is continuous and yω is tempered, 1+yℓpp+yℓ1+y2 is tempered. From proposition 4.3.3 [13] P187, there is a tempered function rω>0 such that(66)1+yθtωℓpp+yθtωℓ1+yθtω2≤rθtω≤rωeϵ/2t.
In (65), with ω replaced by θ−tω, from (66), we have(67)vt,θ−tω,vτs,θ−tω2≤vτs,θ−tω2eϵτ−t+C∫τteϵr−t1+yθr−tωℓpp+yθr−tωℓ1+yθr−tω2d+2λϵCη+2g2≤vτs,θ−tω2e−ϵt−τ+C∫τ−t0eϵς1+yθςωℓpp+yθςωℓ1+yθςω2dς+2λϵCη+2g2≤vτs,θ−tω2e−ϵt−τ+2Crωϵ+2λϵCη+2g2.
Choose(68)Rω=4Crωϵ+4λϵCη+2g2.
Then,Rω is tempered and K~ω=v∈Xρ:v2≤Rω is an absorbing set for vt,ω,vτs,  ω. i.e., there is an absorbing time T=TD,  ω>0, for each D∈D,ω∈Ω′, and t≥T such that vt,θ−tω,Dθ−t  ω⊂K~ω. Furthermore, K~∈D.
Let(69)Kω=u∈Xρ:u2≤2Rω+2εyω2.
ThenKω is an absorbing set for Φt,ω,uτs because Φt,ω,uτs=vt,ω,uτs−εyθτω+εyθtω and K∈D. This completes the proof of the theorem.In order to prove that the random dynamical systemΦt,ω,uτs,ω is asymptotic compact, we require the following lemma.Lemma 9.
LetKω be the absorbing set and uτs,  ω∈Kω the initial data. Then for each η>0, there exist an absorbing time Tη,  ω>0 and Nη,  ω>0 such that for every t≥Tη,  ω, the solution of (1) meets(70)sup−ρ≤s≤0∑i≥Nuis,θ−tω,uτs,θ−tω2≤η.Proof.
Letξs be a smooth cut-off function with(71)ξs=0,0≤s≤1,ξs,1≤s≤2,1,s≥2,
where0≤ξs≤1, s∈R+, and with a constant C such that ξ′s≤C.
Withk∈Z+, taking the inner product of (54) with the sequence ξi/kvii∈Z , we obtain(72)12ddt∑i∈Zξikvi2+ν∑i∈ZξikAvivi+λ∑i∈Zξikvi2=∑i∈ZξikFivi+εyivi+∑i∈Zξikfivit−ρ+εyiθt−ρωivi+∑i∈Zθikgivi−νε∑i∈ZθikAyivi
We now make the following estimate.(73)ν∑i∈ZξikAvivi=ν∑i∈Zvi+1−viξi+1k−ξikvi+1+ξikvi+1−vi=ν∑i∈Zξi+1k−ξikvi+1−vivi+1+ν∑i∈Zξikvi+1−vi2≥ν∑i∈Zξi+1k−ξikvi+1−vivi+1.
Using the property of the smooth cut-off functionξs, we have(74)∑i∈Zξi+1k−ξikvi+1−vivi+1≤∑i∈Zξ′ςikvi+1−vivi+1≤Ck∑i∈Zvi+12+vivi+1≤2Ckv2,
which tells us that(75)ν∑i∈ZξikAvivi≥−2Cνkv2.
It follows from the assumption (A1) and Young’s inequality that(76)∑i∈ZξikFivi+εyivi=∑i∈ZξikFivi+εyivi+εyi−∑i∈ZξikFivi+εyiεyi≤∑i∈Zξik−α1vi+εyip+β1,i+ε∑i∈Zξikα2vi+εyip−1+β2,iyi≤C∑i∈Zξik1+yip+yi,
whereC is a positive constant depending on ε,p,α1,α2,∑i∈Zβ1,i,∑i∈Zβ2,i,ξi/k only.
By the assumption (A2) and Young’s inequality, we get(77)∑i∈Zξikfivit−ρ+εyiθt−ρωivi≤λ8∑i∈Zξikvi2+4λ∑i∈ZξikCf2vit−ρ2+εyiθt−ρωi2+2λ∑i∈Zξikηi2,
(78)
∑
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Z
ξ
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k
g
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∑
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.
By (72), (73), (75)–(78), we arrive at(79)12ddteϑt∑i∈Zξikvi2+3λ4−ϑ2eϑt∑i∈Zξikvi2≤eϑt2Cνkv2+C∑i∈Zξik1+yip+yi+4λCf2∑i∈Zξikvit−ρ2+4λCfε2∑i∈Zξikyiθt−ρωi2+2λCη+2λ∑i≥kgi2,
where∑i∈Zξi/kηi2≤∑i∈Zηi2=Cη.
Integrating the above inequality fromTk to t, we have(80)eϑt∑i∈Zξikvi2+3λ2−ϑ∫Tkteϑr∑i∈Zξikvi2dr≤eϑTk∑i∈ZξikviTk,ω,vτs,ω2+4Cνk∫Tkteϑrvr,ω,vτs,ω2dr+2C∫Tkteϑr∑i∈Zξik1+yip+yidr+8λCf2∫Tkteϑr∑i∈Zξikvir−ρ2dr+8λCfε2∫Tkteϑr∑i∈Zξikyiθr−ρωi2dr+4λ∫TkteϑrCη+∑i≥kgi2dr.
It’s easy to know that(81)∫Tkteϑr∑i∈Zξikvir−ρ2dr≤∫Tk−ρteϑr+ρ∑i∈Zξikvir2dr=∫Tk−ρTkeϑr+ρ∑i∈Zξikvir2dr+∫Tkteϑr+ρ∑i∈Zξikvir2dr≤C1+∫Tkteϑr+ρ∑i∈Zξikvir2dr,
whereC1=ρ∑i∈Zξi/ksupr∈Tk−ρ,Tkeϑr+ρvir2. And(82)∫Tkteϑr∑i∈Zξikyiθr−ρωi2dr≤C2+∫Tkteϑr+ρ∑i∈Zξikyiθrωi2dr,
whereC2=ρ∑i∈Zξi/ksupr∈Tk−ρ,Tkeϑr+ρyiθrωi2.
Hence, from (80)–(82) and the assumption (A5), then multiplying both sides of it by e−ϑt, we get, for t≥Tk=Tkω>τ,(83)∑i∈Zξikvit,ω,vτs,ω2≤eϑTk−t∑i∈ZξikviTk,ω,vτs,ω2+4Cνke−ϑt∫Tkteϑrvr,ω,vτs,ω2dr+2Ce−ϑt∫Tkteϑr∑i∈Zξik1+yip+yidr+8λCfε2e−ϑt∫Tkteϑr+ρ∑i∈Zξikyi2dr+4ϑλCη+∑i≥kgi21−eϑTk−t+C3+C4e−ϑt,
whereC3=8/λCf2C1,C4=8/λCfε2C2.
We replaceω by θ−tω, and estimate each term on the right-hand side of the above inequality. With t instead of Tk in (65), we have(84)eϑTk−t∑i∈ZξikviTk,θ−tω,vτs,θ−tω2≤eϑTk−tvTk,τ−Tk,θ−tω,vτs,θ−tω2≤eϑTk−tvτs,θ−tω2e−ϵTk−τ+2Crωϵ+2λϵCη+2g2.
As we know,vτs,θ−t  ω∈Kθ−t  ω, namely, vτs,θ−tω≤Rθ−tω (tempered), so there exists a T1η,  ω>Tkω, such that when t>T1,(85)eϑTk−t∑i∈ZξikviTk,θ−tω,vτs,θ−tω2≤128η.
Also by (65), we estimate(86)4Cνke−ϑt∫Tkteϑrvr,θ−tω,vτs,θ−tω2dr≤4Cνke−ϑteτϵϑ−ϵvτs,θ−tω2eϑ−ϵt−eϑ−ϵTk+2Crωϑϵ+2λϑϵCη+2g2eϑt−eϑTk,
whereϑ−ϵ≠0. Therefore there exist T2η,  ω>Tkω and N1η,  ω>0, such that as t>T2, k>N1,(87)4Cνke−ϑt∫Tkteϑrvr,θ−tω,vτs,θ−tω2dr≤128η.
Using (66), for k>N2η,  ω>0 we deduce(88)2Ce−ϑt∫Tkteϑr∑i∈Zξik1+yiθ−tωp+yiθ−tωdr≤2Crωϑe−ϵ/2t1−eϑTk−t,
so there is aT3η,  ω>Tkω, such that as t>T3,(89)2Ce−ϑt∫Tkteϑr∑i∈Zξik1+yiθ−tωp+yiθ−tωdr≤128η.
Applying (66), we also get(90)8λCfε2e−ϑt∫Tkteϑr+ρ∑i∈Zξikyiθ−tω2dr≤8λCfε2e−ϑt∫Tkteϑr+ρrωe−ϵ/2tdr=8ϑλCfε2rωe−ϑt−ϵ/2teϑt+ρ−eϑTk+ρ.
Thus there existsT4η,  ω>Tkω, such that as t>T4,(91)8λCfε2e−ϑt∫Tkteϑr+ρ∑i∈Zξikyiθ−tω2dr≤128η.
Sinceg∈X, there is N3η,  ω>0, we obtain, for k>N3η,  ω>0,(92)4ϑλCη+∑i≥kgi2≤128η.
By the same argument, we haveT5η,  ω>Tkω, when t>T5, satisfying(93)−4ϑλCη+∑i≥kgi2eϑTk−t≤128η.
Obviously, there is aT6η,  ω>Tkω, when t>T6, fulfilling(94)C3+C4e−ϑt≤128η.
TakeTη,  ω=max{T1,T2,T3,T4,T5,T6}>Tkω,Nη,  ω=max{N1,N2,N3}>0. We collect (83)–(94) to yield, for every t>T,k>N,(95)sup−ρ≤s≤0∑i≥2kuis,θ−tω,uτs,θ−tω2≤sup−ρ≤s≤0∑i∈Zξikuis,θ−tω,uτs,θ−tω2≤2sup−ρ≤s≤0∑i≥Nvis,θ−tω,vτs,θ−tω2+ε2yiθ−tω2≤η,
which concludes the proof.
With the same method like the proof of the Theorem11 in [8], we can obtain the asymptotic compactness of the continuous RDS Φ.Theorem 10.
Φ is asymptotically compact for all ω∈Ω′: each sequence qn∈Φtn,θ−tnω,Kτ,θ−tn  ω has a convergent subsequence in Xρ as tn→∞.
We are now in a position to present our main result about the existence ofD-pullback attractor.Theorem 11.
Suppose that assumptions (A1)–(A6) hold. Then the continuous RDS Φ associated with problems (1) and (2) has a unique D-pullback attractor, which is characterized by, for each ω∈Ω, (96)Aω=∩κ≥tKω∪t≥κΦt,θ−tω,Kθ−tω¯.Proof.
Note that the existence, uniqueness and characterization of theD-pullback attractor of Φ 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 systemΦt,ω,uτs,  ω induced by (23) and (24) (see [10]). The Markov semigroup {Pt}t≥0 associated with them is denoted by(97)Ptφx=Eφut,0,x,t≥0x∈Xρφ∈CXρ,whereCXρ is a space of bounded and uniformly continuous function on Xρ.The related transition probabilityPtx,  · is(98)Ptx,Γ=Lut,0,xΓ,t≥0x∈XρΓ∈BXρ,whereL· is a Banach algebra of all linear bounded operators from Xρ to Xρ, and BXρ is a σ-field of all Borel subsets of Xρ.Now we prove equation (23) is ergodic, i.e., the invariant measure is unique.Assumeut,ϱ,uϱ,ϱ∈R,t>ϱ to be a mild solution of equation(99)dudt+νAu+λu=Fu+fut+g+εdWdt,with the initial conditions(100)uϱ=uϱ+s,s∈−ϱ,0.Suppose{St}t≥ϱ is a semigroup formed by −νA−λ, from Theorem 6, we have(101)ut,ϱ,uϱ=St−ϱuϱ−∫ϱtSt−hFu+futdh+ε∫ϱtSt−hdWh.Putζt=ε∫ϱtSt−  hdWh and zt=ut,ϱ,uϱ−ε∫ϱtSt−  hdWh. Then zt is a mild solution of(102)dzdt=−νA+λzt+Fzt+fzt+g.Theorem 12.
The system (23) and (24) is ergodic.Proof.
We need to prove that there is a unique invariant measureμ for Markov semigroup {Pt}t≥0. By the same discussion with a little change like (67), we get.(103)zt≤zτeϵ/2ϱ−t+2Crωϵ+2λϵCη+2g2.
We knowsupt≥ϱEζt<+∞, then(104)Eut≤Ezt+supt≥ϱEζt≤zτeϵ/2ϱ−t+supt≥ϱEζt+2Crωϵ+2λϵCη+2g2.
Lett≥2ϱ,s∈−ϱ,  0, then t+s≥ϱ and(105)Eut≤zτeϵ/2ϱ−t+supt≥ϱEζt+2Crωϵ+2λϵCη+2g2.
So, for arbitraryuϱ,1,uϱ,2∈Xρ,t+s≥ϱ, we induce(106)ut+s,ϱ,uϱ,1−ut+s,ϱ,uϱ,2≤uϱ,1−uϱ,2eϵ/2ϱ−t.
Forϱ0∈R and ϱ0≤ϱ<2ϱ≤t, we obtain(107)Eut+s,ϱ,uϱ−ut+s,ϱ0,uϱ≤Euϱ−uϱ,ϱ0,uϱeϵ/2ϱ−t≤2zτ+supt≥ϱEζt+2Crωϵ+2λϵCη+2g2eϵ/2ϱ−t.
Hence, we know there is a random variableϖ such that(108)limϱ→−∞Eu0,ϱ,uϱ−ϖ=0.
For arbitraryuϱ∈Xρ, we are led to(109)Ptuϱ,·=Lut,uϱ=Lu0,−t,uϱ→μ.
This tells us that, the lawμ=Lϖ is the unique invariant measure for Markov semigroup Ptt≥0. This accomplishes the proof.



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*Source: 5405947-2020-01-05.xml*