# Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

Authors: Xiaoying Han
Journal: International Journal of Differential Equations
 (2011)

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## Abstract

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.


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## Body

## 1. Introduction

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 latticeℤ [1]. 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 [8].In this work, following the idea of [8], 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:(1.1)u̇i=-λui+∑j=-qqηi,j(θtω)ui+j-fi(ui)-αvi+hi+ui∘dw(t)dt,v̇i=-σvi+μui+gi+ui∘dw(t)dt,i∈Z,t>0,(1.2)u̇i=-λui+∑j=-qqηi,j(θtω)ui+j-fi(ui)-αvi+hi+aidwi(t)dt,v̇i=-σvi+μui+gi+bidwi(t)dt,i∈Z,t>0,
where ui,hi,gi,ai,bi∈ℝfi∈C1(ℝ,ℝ); (i∈ℤ), λ,α,σ,μ>0 are positive constants; A is the coupling operator, ηi,-q(ω),…,ηi,0(ω),…,ηi,+q(ω), i∈ℤ,  q∈ℕ, are random variables, and w(t),  {wi(t):i∈ℤ} 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 q=1,  ηi,±1(ω)≡1, ηi,0(ω)≡-2 for all i∈ℤ, Huang [4] and Wang et al. [14] proved the existence of a global random attractor for the associated RDS in the regular phase space l2×l2. In this work we will consider the existence of a compact global random attractor in the weighted space lρ2×lρ2, 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 ρ:ℤ→(0,M0] such that l2⊂lρ2. If ∑i∈ℤρ(i)<∞, then lρ2 contains any infinite sequences whose components are just bounded and l2⊂l∞⊂lρ2. Note that when ρ(i)≡1, our results recover the results obtained in [4, 14] while lρ2 is reduced to the standard l2. Moreover, the required conditions in this work for the existence of a random attractor for system (1.2)-(1.1) in weighted space lρ2×lρ2 are weaker than those in l2×l2.The rest of this paper is organized as follows. In Section2, 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.


## 2. Preliminaries

In this section, we present some concepts related to random dynamical systems (RDSs) and random attractors [1, 8, 15] on weighted space of infinite sequences.Letρ be a positive function from ℤ to (0,M0]⊂ℝ+, where M0 is a finite positive constant. Define for any i∈ℤ, ρi=ρ(i) and(2.1)lρ2={u=(ui)i∈Z:∑i∈Zρi|ui|2<∞,ui∈R},
then lρ2 is a separable Hilbert space with the inner product 〈u,v〉ρ=∑i∈ℤρiuivi and norm ∥u∥ρ2=〈u,u〉ρ=∑i∈ℤρi|ui|2 for u=(ui)i∈ℤ,  v=(vi)i∈ℤ∈lρ2. Moreover, define(2.2)H=̇lρ2×lρ2
with inner product(2.3)〈(u(1),v(1)),(u(2),v(2))〉H=〈u(1),u(2)〉ρ+〈v(1),v(2)〉ρ,for(u(1),v(1)),(u(2),v(2))∈H,
and norm(2.4)‖(u,v)‖H=(‖u‖ρ2+‖v‖ρ2)1/2for(u,v)∈H,
then H is also a separable Hilbert space.Let(Ω,ℱ,ℙ) be a probability space and {θt:Ω→Ω,t∈ℝ} be a family of measure-preserving transformations such that (t,Ω)↦θtΩ is (ℬ(ℝ)×ℱ,ℱ)-measurable, θ0=IdΩ and θt+s=θtθs for all s,t∈ℝ. The space (Ω,ℱ,ℙ,(θt)t∈ℝ) is called a metric dynamical system. In the following, “property (P) holds for a.e. ω∈Ω with respect to (θt)t∈ℝ" means that there is Ω̃⊂Ω with ℙ(Ω̃)=1 and θtΩ̃=Ω̃ such that (P) holds for all ω∈Ω̃.Recall the following definitions from existing literature.(i)
A stochastic process{S(t,ω)}t≥0,ω∈Ω is said to be a continuous RDS over (Ω,ℱ,ℙ,(θt)t∈ℝ) with state space H, if S:ℝ+×Ω×H→H is (ℬ(ℝ+)×ℱ×ℬ(H),ℬ(H))-measurable, and for each ω∈Ω, the mapping S(t,ω):H→H,u↦S(t,ω)u is continuous for t≥0, S(0,ω)u=u and S(t+s,ω)=S(t,θsω)S(s,ω) for all u∈H and s,t≥0.(ii)
A set-valued mappingω↦D(ω)⊂H (may be written as D(ω) for short) is said to be a random set if the mapping ω↦distH(u,D(ω)) is measurable for any u∈H.(iii)
A random setD(ω) is called a closed (compact) random set if D(ω) is closed (compact) for each ω∈Ω.(iv)
A random setD(ω) is said to be bounded if there exist u0∈H and a random variable r(ω)>0 such that D(ω)⊂{u∈H:∥u-u0∥H≤r(ω)} for all ω∈Ω.(v)
A random bounded setD(ω) is said to be tempered if for a.e. ω∈Ω,
(2.5)lim⁡t→∞e-βtsup⁡{‖u‖H:u∈D(θ-tω)}=0,∀β>0.
Denote by 𝒟(H) the set of all tempered random sets of H.(vi)
A random setB(ω) is said to be a random absorbing set in𝒟(H) if for any D(ω)∈𝒟(H) and a.e. ω∈Ω, there exists TD(ω) such that S(t,θ-tω)D(θ-tω)⊂B(ω)forallt≥TD(ω).(vii)
A random setA(ω) is said to be a random attracting set if for any D(ω)∈𝒟(H), we have
(2.6)lim⁡t→∞⁡distHS(t,θ-tω)D(θ-tω,A(ω))=0,a.e.ω∈Ω,
in which distH is the Hausdorff semidistance defined via distH(E,F)=supu∈Einfv∈F∥u-v∥ρ for any E,F⊂lρ2.(viii)
A random compact setA(ω) is said to be arandom global𝒟attractor if it is a compact random attracting set andS(t,ω)A(ω)=A(θtω) for a.e.ω∈Ω andt≥0.Definition 2.1 (see [8]).
{S(t,ω)}t≥0,ω∈Ω is said to be random asymptotically null in 𝒟(H), if for any D(ω)∈𝒟(H), a.e. ω∈Ω, and any ɛ>0, there exist T(ɛ,ω,D(ω))>0 and I(ɛ,ω,D(ω))∈ℕ such that
(2.7)(∑|i|>I(ɛ,ω,D(ω))ρi|S(t,θ-tω)u(θ-tω))i|2)1/2≤ɛ,∀t≥T(ɛ,ω,D(ω)),∀u(ω)∈D(ω).Theorem 2.2 (see [8]).
Let{S(t,ω)}t≥0,ω∈Ω be a continuous RDS over (Ω,ℱ,ℙ,(θt)t∈ℝ) with state space H and suppose that(a)
there exists a random bounded closed absorbing setB(ω)∈𝒟(H) such that for a.e. ω∈Ω and any D(ω)∈𝒟(H), there exists TD(ω)>0 yielding S(t,θ-tω)D(θ-tω)⊂B(ω) for all t≥TD(ω);(b)
{S(t,ω)}t≥0,ω∈Ω is random asymptotically null on B(ω); that is, for a.e. ω∈Ω and for any ɛ>0, there exist T(ɛ,ω,B(ω))>0 and I(ɛ,ω,B(ω))∈ℕ such that(2.8)sup⁡u∈B(ω)∑|i|>I(ɛ,ω,B(ω))ρi|(S(t,θ-tω)u(θ-tω))i|2≤ɛ2,∀t≥T(ɛ,ω,B(ω)).
Then the RDS{S(t,ω)}t≥0,ω∈Ω possesses a unique global random 𝒟 attractor A(ω) given by
(2.9)A(ω)=⋂τ≥TB(ω)⋃t≥τS(t,θ-tω)B(θ-tω)¯.


## 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 H=lρ2×lρ2. 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 H. 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 functionρ:ℤ→ℝ+ is chosen to satisfy(P0)
0<ρ(i)≤M0 and ρ(i)≤c·ρ(i±1),foralli∈ℤ for some positive constants M0 and c.(e.g.,ρ(x)=1/(1+ϵ2x2)q,q>1/2 [16, 17] and ρ(x)=e-ϵ|x|,x∈ℤ where ϵ>0).


### 3.1. Mathematical Setting

DefineΩ1={ω∈C(ℝ,ℝ):ω(0)=0}=C0(ℝ,ℝ), and denote by ℱ1 the Borel σ-algebra on Ω1 generated by the compact open topology (see [2, 15]) and ℙ1 the corresponding Wiener measure on ℱ1. Defining (θt)t∈ℝ on Ω1 via θtω(·)=ω(·+t)-ω(t) for t∈ℝ, then (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) is a metric dynamical system.Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise:(3.1)du=(-λu+A(θtω)u-f(u)-αv+h)dt+u∘dw(t),dv=(-σv+μu+g)+u∘dw(t),i∈Z,t>0,
where u=(ui)i∈ℤ, v=(vi)i∈ℤ; f(u)=(fi(ui))i∈ℤ with fi∈C1(ℝ,ℝ) (i∈ℤ), g=(gi)i∈ℤ, h=(hi)i∈ℤ; λ,α,σ,μ are positive constants; ηi,-q(ω),…,ηi,0(ω),…,ηi,+q(ω)(q∈ℕ) are random variables on the probability space (Ω1,ℱ1,ℙ1); A(·) is a linear operator on lρ2 defined by(3.2)(A(⋅)u)i=∑j=-qqηi,j(⋅)ui+j;w(t) is a Brownian motion (Wiener process) on the probability space (Ω1,ℱ1,ℙ1); ∘ denotes the Stratonovich sense of the stochastic term.For convenience, we first transform (3.1) into a random differential equation without white noise. Let(3.3)δ(θtω)=-∫-∞0esθtω(s)ds,t∈R,ω∈Ω1,
then δ(θtω) is an Ornstein-Uhlenbeck process on (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details)(3.4)dδ+δdt=dw(t),t≥0,
where w(t)(ω)=w(t,ω)=ω(t) for ω∈Ω1,  t∈ℝ, and possesses the following properties.Lemma 3.1 (see [2, 15]).
There exists aθt-invariant set Ω̃1∈ℱ1 of Ω1 offull ℙ1 measure such that for ω∈Ω̃1, one has(i)
the random variable|δ(ω)| is tempered;(ii)
the mappingδ(θtω)(3.5)(t,ω)⟼δ(θtω)=-∫-∞0esω(t+s)ds+w(t)
is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)
(3.6)lim⁡t→±∞|δ(θtω)|t=lim⁡t→±∞1t∫0tδ(θsω)ds=0.The mapping ofθ on Ω̃1 possesses same properties as the original one if we choose the trace σ-algebra with respect to Ω̃1 to be denoted also by ℱ1. Therefore we can change our metric dynamical system with respect to Ω̃1, still denoted by the symbols (Ω1,ℱ1,ℙ1,(θt)t∈ℝ).Let(3.7)x(t,ω)=e-δ(θtω)u(t,ω),y(t,ω)=e-δ(θtω)v(t,ω),ω∈Ω1,
where (u(t,ω),v(t,ω)) is a solution of (3.1), then (u(t,ω),v(t,ω))↦(x(t,ω),y(t,ω)) is a homomorphism in H. System (3.1) can then be transformed to the following random system with random coefficients but without white noise:(3.8)dxdt=-λx+A(θtω)x-e-δ(θtω)f(eδ(θtω)x)+δ(θtω)x-αy+e-δ(θtω)h,dydt=-σy+δ(θtω)y+μx+e-δ(θtω)gt>0,
Letting z=(x,y), (3.8) are equivalent to(3.9)dzdt=F(z,θtω),t>0,
where
(3.10)F(z,θtω)=(-λx+A(θtω)x-e-δ(θtω)f(eδ(θtω)x)+δ(θtω)x-αy+e-δ(θtω)h-σy+δ(θtω)y+μx+e-δ(θtω)g).We now make the following standing assumptions onfi, gi, hi, and ηi,j,(j=-q,…,q)i∈ℤ and study in the following subsections asymptotic behavior of system (3.9).(H1)
g=(gi)i∈ℤ,  h=(hi)i∈ℤ∈lρ2.(H2)
Let(3.11)η(ω)=sup⁡{|ηi,-q(ω)|,…,|ηi,0(ω)|,…,|ηi,+q(ω)|:i∈Z}≥0,q∈N.η(θtω)(<∞)  belongs toLloc1(ℝ)  with respect tot∈ℝ  for eachω∈Ω1. 										(3.12)E(η)=lim⁡t→±∞1t∫0tη(θtω)ds<∞,andη(ω)  is tempered, that is, there exists aθt-invariant setΩ10∈ℱ1of  fullℙ1  measure such that forω∈Ω10,
(3.13)lim⁡t→+∞⁡e-βtsup⁡t∈R⁡|η(θ-tω)|=0,∀β>0.In the following, we will consider ω∈Ω10∩Ω̃1 and still write Ω10∩Ω̃1 as Ω1.(H3)
min{λ,σ}>q̃E|η(ω)|=limt→±∞(1/t)∫0t(q+∑k=0qck)η(θtω)ds, whereq̃=q+∑k=0qck.(H4)
There exists a functionR∈C(ℝ+,ℝ+)such that(3.14)sup⁡i∈Zmax⁡s∈[-r,r]|fi′(s)|≤R(r),∀r∈R+.(H5)
fi∈C1(ℝ,ℝ),fi(0)=0,sfi(s)≥-bi2,b=(bi)i∈ℤ∈lρ2, and there exists a constanta≥0such thatfi′(s)≥-a,foralls∈ℝ,i∈ℤ.


### 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 H.Definition 3.2.
We callz:[0,T)→H a solution of the following random differential equation
(3.15)dzdt=F(z,θtω),z=(zi)i∈Z,F=(Fi)i∈Z,
where ω∈Ω0, if z∈C([0,T),H) satisfies
(3.16)zi(t)=zi(0)+∫0tFi(z(s),θsω)ds,fori∈Z,t∈[0,T).Theorem 3.3.
LetT>0 and (P0), (H1), (H2), (H4), and (H5) hold. Then for any ω∈Ω1 and any initial data z0=(x(0),y(0))∈H, (3.9) admits a unique solution z(·;ω,z0)=(x(·;ω,z0),y(·;ω,z0))∈C([0,T),H) with z(0;ω,z0)=z0.Proof.
(1) DenoteE=l2×l2, we first show that if z0∈E and (h,g)∈E, then (3.9) admits a unique solution z(t;ω,z0,h,g)∈E on [0,T) with z(0;ω,z0,g,h)=z0. Given z∈E, ω∈Ω1, and (h,g)∈E, note that F(z,ω) is continuous in z and measurable in ω from E×Ω1 to E.
By (3.2) and (H2),
(3.17)‖A(ω)x‖=[∑i∈Z(∑j=-qqηi,j(ω)xi+j)2]1/2≤(2q+1)η(ω)⋅‖x‖.
By (H4),
(3.18)‖f(eδ(ω)x)‖≤max⁡{R(eδ(ω)‖x‖),a}⋅eδ(ω)‖x‖,
and therefore
(3.19)‖F(z,ω)‖E≤(λ+(2q+1)η(ω)+max⁡{R(eδ(ω)‖x‖),a}+|δ(ω)|+μ)⋅‖x‖+(α+σ+|δ(ω)|)⋅‖y‖+|e-δ(ω)|⋅(‖h‖+‖g‖).
For any z(1)=(x(1),y(1)),  z(2)=(x(2),y(2))∈E, and for some ϑ∈(0,1)(3.20)‖f(eδ(ω)x(1))-f(eδ(ω)x(2))‖≤max⁡{R(eδ(ω)((1-ϑ)‖x(1)‖+ϑ‖x(2)‖)),a}⋅eδ(ω)‖x(1)-x(2)‖.
Also
(3.21)‖A(ω)x(1)-A(ω)x(2)‖=∑i∈Z{∑j=-qqηi,j(ω)(xi+j(1)-xi+j(2))}1/2≤(2q+1)η(ω)⋅‖x(1)-x(2)‖.
It then follows that
(3.22)‖F(z(1),ω)-F(z(2),ω)‖E≤(α+σ+|δ(ω)|)⋅‖y(1)-y(2)‖+[(eδ(ω)((1-ϑ)‖v(1)‖+ϑ‖v(2)‖))λ+(2q+1)η(ω)+μ+|δ(ω)|+eδ(ω)max⁡{R(eδ(ω)((1-ϑ)‖v(1)‖+ϑ‖v(2)‖)),a}]⋅‖x(1)-x(2)‖.
For any compact set D⊂E with supz∈D∥z∥≤r, defining random variable ζD(ω)≥0 via
(3.23)ζD(ω)=(λ+(2q+1)η(ω)+μ+|δ(ω)|+max⁡{R(eδ(ω)r),a}⋅eδ(ω))r+(α+σ+|δ(ω)|)r+|e-δ(ω)|⋅(‖h‖+‖g‖),
we have
(3.24)∫ττ+1ζD(θtω)dt<∞,∀τ∈R,
and for any z,z(1),z(2)∈D,
(3.25)‖F(z,ω)‖E≤ζD(ω),‖F(z(1),ω)-F(z(2),ω)‖E≤ζD(ω)‖z(1)-z(2)‖E.
According to [15, 19, 20], problem (3.9) possesses a unique local solution z(·;ω,z0,g,h)∈C([0,Tmax),E) (0<Tmax≤T) satisfying the integral equation
(3.26)z(t)=z0+∫0tF(z(s),ω)ds,fort∈[0,Tmax⁡).
We will next show thatTmax=T. Since the set C0(ℝ) of continuous random process in t is dense in the set L1(ℝ) (see [18, 21]), for each ω∈Ω1, there exists a sequence {ηi,j(m)(t,ω)}m=1∞ of continuous random process in t∈ℝ such that
(3.27)lim⁡m→∞∫0T(ηi,j(m)(s,ω)-ηi,j(s,ω))ds=0,∀T>0,|ηi,j(m)(t,ω)|≤|ηi,j(θtω)|≤|η(θtω)|,∀t∈R.
Consider the random differential equation with initial dataz0∈E:(3.28)dz(m)dt=(-λx(m)+Am(t,ω)x(m)-e-δ(θtω)f(eδ(θtω)x(m))+δ(θtω)x(m)-αy(m)+e-δ(θtω)h-σy(m)+δ(θtω)y(m)+μx(m)+e-δ(θtω)g),
where
(3.29)(Am(t,ω)x(m))i=∑j=-qqηi,j(m)(t,ω)xi+j.
Follow the same procedure as above, (3.28) has a unique solution z(m)(·;ω,z0,g,h)∈C([0,Tmax(m)),E), that is,
(3.30)zi(m)(t)=z0+∫0tFi(m)(z(m)(s),ω)ds,fort∈[0,Tmax⁡(m)),
and by the continuity of Am(s,ω) in s, there holds
(3.31)dzi(m)dt=(-λxi(m)+(Am(θtω)x(m))i-e-δ(θtω)fi(eδ(θtω)xi(m))+δ(θtω)xi(m)-αyi(m)+e-δ(θtω)hi-σyi(m)+δ(θtω)yi(m)+μxi(m)+e-δ(θtω)g).
Note that
(3.32)(Am(θtω)x(m))i⋅xi(m)≤η(θtω)⋅|xi(m)xi-q(m)+⋯+xi(m)xi(m)+⋯+xi(m)xi+q(m)|,-ae2δ(θtω)⋅(xi(m))2≤(eδ(θtω)xi(m))⋅fi(eδ(θtω)xi(m))≤R(eδ(θtω)‖x(m)‖)⋅e2δ(θtω)(xi(m))2,t∈[0,T],
multiplying (3.31) by (μxi(m)00αyi(m)) and sum over i∈ℤ results in
(3.33)ddt(μ‖x(m)‖2+α‖y(m)‖2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+2(2q+1)η(θtω)]⋅(μ‖x(m)‖2+α‖y(m)‖2)+(2μλ∥h∥2+2ασ∥g∥2)e-2δ(θtω).
Applying Gronwall’s inequality to (3.33) we obtain that
(3.34)μ‖x(m)‖2+α‖y(m)‖2≤e2at+∫0t[2δ(θsω)+2(2q+1)η(θsω)]ds(μ‖x(0)‖2+α‖y(0)‖2)+(2μλ‖h‖2+2ασ‖g‖2)(e2at+∫0t[2δ(θsω)+2(2q+1)η(θsω)]ds)⋅(∫0te(min⁡{λ,σ}-2a)s-2δ(θsω)+∫0s[2δ(θrω)+2(2q+1)η(θrω)]drds)≐κ(t),t∈[0,Tmax⁡(m)),
where κ(t)∈C([0,T),ℝ) is independent of m, which implies that the interval of existence of z(m)(t) is [0,T), and z(m)(·;ω,z0,g,h)∈C1([0,T),E).
By (3.34),
(3.35)|xi(m)|2+|yi(m)|2≤κ(t)min⁡{μ,α},∀m∈N,t∈[0,T).
Since |Fi(m)(z(m)(t),θtω)|2≤K2(T,ω) for some K(T,ω)>0 and t∈[0,T), then for any t,τ∈[0,T),m∈ℕ,
(3.36)|zi(m)(t)-zi(m)(τ)|=∫τt|Fi(m)(z(m)(s),θsω)|ds≤K(T,ω)⋅|t-τ|.
By the Arzela-Acoli Theorem, there exists a convergent subsequence {zi(mk)(t),t∈[0,T)} of {zi(m)(t),t∈[0,T)} such that
(3.37)zi(mk)(t)⟶z¯i(t)ask⟶∞,t∈[0,T),i∈Z,
and z¯i(t) is continuous on t∈[0,T). Moreover, |z¯i|2≤κ(t)/min{μ,α} for t∈[0,T). By (3.27), (3.35), assumption (H2), and the Lebesgue Dominated Convergence Theorem we have
(3.38)lim⁡k→∞∫0t(ηi,j(mk)(s,ω)-ηi,j(θsω))ds=∫0tlim⁡k→∞(ηi,j(mk)(s,ω)-ηi,j(θsω))ds=0,lim⁡k→∞(ηi,j(mk)(s,ω)-ηi,j(θsω))=0,fora.e.s∈[0,T],lim⁡k→∞∫0t(ηi,j(mk)(s,ω)xi(mk)(s)-ηi,j(θsω)x¯i(s))ds=0.
Thus replacing m by mk in (3.31) and letting k→∞ give
(3.39)z¯i(t)=(z0)i+∫0tFi(z¯i,θsω)dsfort∈[0,T).
By the uniquness of the solutions of (3.9), we have z¯i(t)=zi(t) for t∈[0,Tmax). By (3.34), ∥z(t)∥E2≤κ(t)/min{μ,α} for t∈[0,Tmax), which implies that the solution z(t) of (3.9) exists globally on t∈[0,T).
(2) Next we prove that for anyz0∈H and (h,g)∈H, (3.9) has a solution z(t;ω,z0,h,g) on [0,T) with z(0;ω,z0,h,g)=z0. Let z1,0,z2,0∈E and h1=(h1,i)i∈ℤ,  h2=(h2,i)i∈ℤ,g1=(g1,i)i∈ℤ,g2=(g2,i)i∈ℤ∈l2. Let z1(m)(t,ω),z2(m)(t,ω) be two solutions of (3.28) with initial data z1,0,z2,0 and h,g replaced by h1, h2, g1, g2, respectively. Set d(m)(t)=z1(m)(t)-z2(m)(t)=(d1(m)(t),d2(m)(t))∈E⊂H. Take inner product 〈·,·〉H of (d/dt)d(m) with d(m) and evaluate each term as follows. By (P0), (H1), (H2), and (H4),
(3.40)|〈Am(θtω)d1(m),d1(m)〉ρ|≤q̃η(θtω)‖d1(m)‖ρ2,〈f(eδ(θtω)x1(m))-f(eδ(θtω)x2(m)),d1(m)〉ρ≥-aeδ(θtω)‖d1(m)‖ρ2,〈f(eδ(θtω)x1(m))-f(eδ(θtω)x2(m)),d1(m)〉ρ≤R(eδ(θtω)(‖x1(m)‖+‖x2(m)‖))eδ(θtω)‖d1(m)‖ρ2,〈h1-h2,d1(m)〉ρ≤‖h1-h2‖ρ2⋅‖d1(m)‖ρ2;〈g1-g2,d2(m)〉ρ≤‖g1-g2‖ρ2⋅‖d2(m)‖ρ2.
It then follows that
(3.41)ddt(μ‖d1(m)‖ρ2+α‖d2(m)‖ρ2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+q̃η(θtω)]⋅(μ‖d1(m)‖ρ2+α‖d2(m)‖ρ2)+(2μλ‖h1-h2‖ρ2+2ασ‖g1-g2‖ρ2)e-2δ(θtω).
For T>0, applying Gronwall’s inequality to (3.41) on [0,T] implies that
(3.42)μ‖d1(m)(t)‖ρ2+α‖d2(m)(t)‖ρ2≤CT(μ‖d1(m)(0)‖ρ2+α‖d2(m)(0)‖ρ2+‖h1-h2‖tρ2+‖g1-g2‖ρ2)
for some constant CT depending on T, and thus
(3.43)‖z1(m)(t)-z2(m)(t)‖H2≤C̃T(‖z1(m)(0)-z2(m)(0)‖H2+‖h1-h2‖ρ2+‖g1-g2‖ρ2),
where C̃T is a constant depending on T. Denote by Ẽ=l̃2×l̃2, where l̃2=l2 with the norm ∥·∥ρ. By (3.43), there exists a mapping Φ(m)∈C(Ẽ×Ẽ,C([0,T],H)) such that Φ(m)(z0,g,h)=z(m)(t;ω,z0,g,h), where z(m)(t;ω,z0,g,h) is the solution of (3.28) on [0,T) with z(m)(0;ω,z0,g,h)=z0. Since l̃2 is dense in lρ2, the mapping Φ(m) can be extended uniquely to a continuous mapping Φ̃(m):H×H→C([0,T],H).
For givenz0∈H and (g,h)∈H, Φ̃(m)(z0,g,h)=z(m)(·;ω,z0,g,h)∈C([0,T],H) for T>0. There exist sequences {z0n}⊂Ẽ, {(hn,gn)}⊂Ẽ such that
(3.44)‖z0n-z0‖E⟶0,‖hn-h‖ρ⟶0,‖gn-g‖ρ⟶0asn⟶∞.
Let zn(m)(t)=Φ̃(m)(z0n,hn,gn)=Φ(m)(z0n,hn,gn)=z(m)(t;ω,z0n,hn,gn)∈Ẽ be the solution of (3.28), then it satisfies the integral equation
(3.45)(zn(m))i(t)=(z0n)i+∫0tFi(zn(m),θsω)ds.
By the continuity of Φ̃(m), we have for t∈[0,T),
(3.46)z(m)(t;ω,z0n,hn,gn)=Φ̃(m)(z0n,hn,gn)⟶n→∞Φ̃(m)(z0,h,g)=z(m)(t;ω,z0,h,g)∈H.
Thus for each i∈ℤ,
(3.47)(zn(m))i(t)⟶zi(m)(t):=zi(m)(t;ω,z0,h,g)asn⟶∞uniformlyon⁡t∈[0,T).
Moreover, {(zn(m))i(t)} is bounded in n. Let n→∞, then we have
(3.48)zi(m)(t)=(z0)i+∫0tFi(z(m),θsω)ds,
and zi(m)(t) satisfies the differential equation (3.31).
Multiply equation (3.31) by (μρixi(m)00αρiyi(m)) and sum over i∈ℤ, we obtain
(3.49)ddt(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+q̃η(θtω)]⋅(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)+(2μλ‖h‖ρ2+2ασ‖g‖ρ2)e-2δ(θtω),μ‖x(m)‖ρ2+α‖y(m)‖ρ2≤e2at+∫0t[2δ(θsω)+q̃η(θsω)]ds(μ‖x(0)‖ρ2+α‖y(0)‖ρ2)+(2μλ‖h‖ρ2+2ασ‖g‖ρ2)(e2at+∫0t[2δ(θsω)+q̃η(θsω)]ds)⋅(∫0te(λ+σ-2a)s-2δ(θsω)+∫0s[2δ(θrω)+q̃η(θrω)]drds)≐κρ(t),t∈[0,Tmax⁡)
Similar to the process (3.35)–(3.39) in part (1), we obtain the existence of a unique solution z(t;ω,z0,g,h)∈H of (3.9) with initial data z0∈H, which is the limit function of a subsequence of {z(m)(t;ω,z0,g,h)} in H for t∈[0,T). In the latter part of this paper, we may write z(t;ω,z0,h,g) as z(t;ω,z0) for simplicity.Theorem 3.4.
Assume that (P0), (H1), (H2), (H4), and (H5) hold. Then (3.9) generates a continuous RDS {ψ(t,ω)}t≥0,ω∈Ω1 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) with state space H:
(3.50)ψ(t,ω)z0:=z(t;ω,z0)forz0∈H,t≥0,ω∈Ω1.
Moreover,
(3.51)φ(t,ω)(u0,v0):=eδ(θtω)ψ(t,ω)e-δ(ω)(u0,v0)for(u0,v0)∈H,t≥0,ω∈Ω1,
defines a continuous RDS {φ(t,ω)}t≥0,ω∈Ω1 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) associated with (3.1).Proof.
By Theorem3.3, the solution z(t;ω,z0) of (3.9) with z(0;ω,z0)=z0 exists globally on [0,∞). It is then left to show that z(t;ω,z0)=z(t;ω,z0,h,g) is measurable in (t,ω,z0).
In fact, forz0∈E and (h,g)∈E, the solution of (3.9) z(t;ω,z0,h,g)∈E for t∈[0,∞). In this case, function F(z,t,ω,h,g)=F(z,t,ω) is continuous in z,h, g and measurable in t,ω, which implies that z:[0,∞)×Ω1×E×E→E,(t;ω,z0,h,g)↦z(t;ω,z0,h,g) is (ℬ([0,∞)×ℱ1×ℬ(E)×ℬ(E),ℬ(E))-measurable.
Forz0∈H and (h,g)∈H, the solution z(t;ω,z0,h,g)∈H for t∈[0,∞). For any given N>0, define TN:H→E, (u,v)=((ui),(vi))i∈ℤ→TN(u,v)=((TN(u,v))i)i∈ℤ by
(3.52)(TN(u,v))j={(uj,vj)if|j|≤N,0if|j|>N,
and write
(3.53)zN(t;ω,z0,h,g)=z(t;ω,TNz0,TN(h,g)).
Then TN is continuous and for any z0∈H,(h,g)∈H, and
(3.54)z(t;ω,z0,h,g)=lim⁡N→∞z(t;ω,TNz0,TN(h,g)).
Thus z:[0,∞)×Ω1×E×E→H is (ℬ([0,∞))×ℱ0×ℬ(E)×ℬ(E),ℬ(H))-measurable. Observe also that (Id,Id,TN,TN):[0,∞)×Ω1×H×H→[0,∞)×Ω0×E×E is (ℬ([0,∞))×ℱ0×ℬ(H)×ℬ(H),ℬ([0,∞))×ℱ0×ℬ(E)×ℬ(E))-measurable. Hence zN=z∘(Id,Id,TN,TN):[0,∞)×Ω1×H×H→H is (ℬ([0,∞))×ℱ0×ℬ(H)×ℬ(H),ℬ(H))-measurable. It then follows from (3.54) that z:[0,∞)×Ω1×H×H→H is (ℬ([0,∞))×ℱ1×ℬ(H)×ℬ(H),ℬ(H))-measurable. Therefore, fix (h,g)∈H we have that z(t;ω,z0)=z(t;ω,z0,h,g) is measurable in (t,ω,z0). The other statements then follow directly.Remark 3.5.
If(h,g)∈E, system (3.1) defines a continuous RDS {φ(t)}t≥0 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) in both state spaces E and H.


### 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{φ(t,ω)}t≥0,ω∈Ω1 generated by (3.1) in weighted space H.Theorem 3.6.
Assume that (P0), (H1)–(H5) hold, then there exists a closed tempered random bounded absorbing setB1ρ(ω)∈𝒟(H) of {φ(t,ω)}t≥0,ω∈Ω1 such that for any D(ω)∈𝒟(H) and each ω∈Ω1, there exists TD(ω)>0 yielding φ(t,θ-tω)D(θ-tω)⊂B1ρ(ω) for all t≥TD(ω). In particular, there exists T1ρ(ω)>0 such that φ(t,θ-tω)B1ρ(θ-tω)⊂B1ρ(ω) for allt≥T1ρ(ω).Proof.
(1) For initial conditionz0∈E and (h,g)∈E, let z(m)(t,ω)=z(m)(t;ω,z0(ω),h,g) be a solution of (3.28) with z0(ω)=e-δ(ω)z0∈E, where ω∈Ω1, then z(m)(t,ω)∈E for all t≥0. Let ϵ1>0 be such that
(3.55)λ1=2min⁡{λ,σ}-ϵ1>2q̃E|η(ω)|.
By (H4) and (H5), we have
(3.56)-‖b‖ρ2≤∑i∈Zρi(eδ(θtω)xi(m))⋅fi(eδ(θtω)xi(m))<∞,forfixedt≥0,(3.57)ddt(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)≤[-λ1+2δ(θtω)+2q̃η(θtω)]⋅(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)+2e-2δ(θtω)(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2).
Applying Gronwall’s inequality to (3.57), we obtain that for t>0,
(3.58)μ‖x(m)(t,ω)‖ρ2+α‖y(m)(t,ω)‖ρ2≤e-λ1t+∫0t[2δ(θsω)+2q̃η(θsω)]ds(μ‖x(0)‖ρ2+α‖y(0)‖ρ2)+2(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)e-λ1t+∫0t[2δ(θsω)+2q̃η(θsω)]ds⋅∫0teλ1s-2δ(θsω)-∫0s[2δ(θrω)+2q̃η(θrω)]drds.
(2) For anyz0∈H and (h,g)∈H, let {z0n}⊂E and {(hn,gn)}⊂E be sequences such that
(3.59)‖z0n-z0‖E⟶0,‖hn-h‖ρ⟶0,‖gn-g‖ρ⟶0asn⟶∞.
Then z(m)(t;ω,z0n,hn,gn)→z(m)(t;ω,z0,h,g) as n→∞ in H, and (3.58) holds for z0∈H. Therefore,
(3.60)μ‖x(m)(t,θ-tω,z0(θ-tω))‖ρ2+α‖y(m)(t,θ-tω,z0(θ-tω))‖ρ2≤e-λ1t+∫0t[2δ(θs-tω)+2q̃η(θs-tω)]ds(μ‖x0(θ-tω)‖ρ2+α‖y0(θ-tω)‖ρ2)+12r1ρ2(ω)e-2δ(ω),
where
(3.61)r1ρ2(ω)=4e2δ(ω)(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)∫-∞0eλ1s-2δ(θsω)+∫s0[2δ(θrω)+2q̃η(θrω)]drds<∞.
For any β>0, since
(3.62)(e-βtr0ρ(θ-tω))2(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)-1=4e-2βt+δ(θ-tω)∫-∞-teλ1(s+t)-2δ(θsω)+∫s-t[2δ(θrω)+2q̃|η(θrω)|]drds⟶t→+∞0,
then r1ρ(ω) is tempered.
Letz(t,ω)=ψ(t,ω)z0(ω)=z(t;ω,z0(ω),h,g) be a solution of equation (3.9) with z0(ω)=e-δ(ω)(u0,v0)∈H, where ω∈Ω1 and (h,g)∈H, then z(t,ω)∈H, and there exists a subsequence z(mk)(t,ω) converging to z(t,ω) as mk→∞ for all t≥0. Inequality (3.60) still holds after replacing z(m)(t,ω) by z(t,ω) since the right hand of (3.60) is independent of m. Thus for (u0,v0)∈D(θ-tω),
(3.63)z0(θ-tω)=(x0(θ-tω),y0(θ-tω))=e-δ(θ-tω)(u0,v0),μ‖u(t,θ-tω,(u0,v0))‖ρ2+α‖v(t,θ-tω,(u0,v0))‖ρ2≤e2δ(ω)e-λ1t-2δ(θ-tω)+∫0t[2δ(θs-tω)+2q̃η(θs-tω)]dssup⁡(u,v)∈D(θ-tω)(μ‖u‖ρ2+α‖v‖ρ2)+12r1ρ2(ω).
Let ϵ2=λ1-2q̃E|η(ω)|>0. By properties of η(θ±tω) and D∈𝒟(H), we have
(3.64)e-[ϵ2/2+2q̃E|η(ω)|]t-δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]ds⟶0ast⟶+∞,
and hence
(3.65)lim⁡t→+∞e2δ(ω)e-λ1t-δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]dssup⁡(u,v)∈D(θ-tω)(μ‖u‖ρ2+α‖v‖ρ2)=0.
Denote by r̃1ρ=r1ρ/min{μ,α}, it follows that
(3.66)B1ρ(ω)={(u,v)∈H:‖(u,v)‖H≤r̃1ρ(ω)}¯=BH(0,r̃1ρ(ω))¯⊂H
is a tempered closed random absorbing set for {φ(t,ω)}t≥0,ω∈Ω1.Theorem 3.7.
Assume that (P0), (H1)–(H5) hold, then the RDS{φ(t,ω)}t≥0,ω∈Ω1 generated by (3.1) possesses a unique global random 𝒟 attractor given by
(3.67)A1(ω)=⋂τ≥TB1ρ(ω)⋃t≥τφ(t,θ-tω)B1ρ(θ-tω)¯⊂H.Proof.
According to Theorem2.2, it remains to prove the asymptotically nullness of {φ(t,ω)}t≥0,ω∈Ω1; that is, for any ɛ>0, there exists T(ɛ,ω,B1ρ)>T1ρ(ω) and I(ɛ,ω)∈ℕ such that when t≥T(ɛ,ω,B1ρ), the solution φ(t,ω)(u0,v0)=((ui,vi)(t;ω,u0,v0))i∈ℤ∈H of (3.1) with (u0,v0)∈B1ρ(θ-tω) satisfies
(3.68)∑|i|≥I(ɛ,ω)ρi|(φ(t,ω)(u0,v0))i|2=∑|i|≥I(ɛ,ω)ρi|(ui,vi)(t;ω,u0,v0)|2≤ɛ.
Choose a smooth increasing function ξ∈C1(ℝ+,[0,1]) such that
(3.69)ξ(s)=0,0≤s≤1,0≤ξ(s)≤1,1≤s≤2,|ξ′(s)|≤C0(constant)fors∈R+,ξ(s)=1,s≥2.
Let (u,v)(t;ω,u0,v0,h,g)=((ui,vi)(t;ω,u0,v0,h,g))i∈ℤ be a solution of (3.1), then
(3.70)z(t;ω,z0(ω),h,g)=(zi(t;ω,z0(ω),h,g))i∈Z=e-δ(θtω)φ(t,ω)(u0,v0)
is a solution of (3.9) with z0(ω)=e-δ(ω)(u0,v0)∈H.
Letz0n=Tnz0,(hn,gn)=Tn(h,g), where Tn is as in (3.52). Then z0n∈E,(hn,gn)∈E and z(t;ω,z0n,hn,gn)→z(t;ω,z0,h,g) in H. For any n≥1, let z(m)(t)=z(m)(t;ω,z0n(ω),hn,gn) be the solution of (3.28), where z(m)(0)=z0n(ω). By Theorem 3.4, z(m)(·)∈C([0,∞),E)∩C1((0,∞),E). Let M be a suitable large integer (will be specified later); multiply (3.31) by (μρiξ(|i|/M)xi(m)00αρiξ(|i|/M)yi(m)) and sum over i∈ℤ, we obtain
(3.71)ddt∑i∈Zξ(|i|M)ρi(μ(xi(m))2+α(yi(m))2)≤[-λ1+2δ(θtω)+2q̃η(θtω)]⋅∑i∈Zξ(|i|M)ρi(μ(xi(m))2+α(yi(m))2)+η(θtω)⋅cC0M‖x(m)‖ρ2+2e-2δ(θtω)∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2).
Applying Gronwall’s inequality to (3.71) from T1ρ=TB1ρ(ω) to t gives
(3.72)∑i∈Zξ(|i|M)ρi(μ(xi(m)(t;ω,z0n,hn,gn))2+α(yi(m)(t;ω,z0n,hn,gn))2)≤e-λ1(t-T1ρ)+∫T1ρt[2δ(θsω)+2q̃η(θsω)]ds⋅(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)+∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2)∫T1ρte-λ1(t-τ)+∫τt[2δ(θsω)+2q̃η(θsω)]ds-2δ(θτω)dτ+cC0M∫T1ρte-λ1(t-τ)+∫τt[2δ(θsω)+2q̃|η(θsω)|]ds-2δ(θτω)η(θτω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ.
Therefore for (u0,v0)∈B1ρ(θ-tω)∩H,
(3.73)∑i∈Zξ(|i|M)ρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤e-λ1(t-T1ρ)+∫T1ρt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)︸(i)+∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds-2δ(θτ-tω)dτ︸(ii)+cC0M∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds-2δ(θτ-tω)η(θτ-tω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ︸(iii).
We next estimate terms (i),(ii),(iii) on the right-hand side of (3.73). By (3.61),
(3.74)e-λ1(t-T1ρ)+∫T1ρt[2δ(θs-tω)+2q̃η(θs-tω)]ds(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)⟶0ast⟶+∞,
which implies that for all ɛ>0, there exists T1(ɛ,ω,B1ρ)≥T1ρ such that for t≥T1(ɛ,ω,B1ρ),
(3.75)(i)≤ɛ3min⁡{μ,α}e-2δ(ω).
By h,g,b∈lρ2 and
(3.76)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃η(θs-tω)]ds-2δ(θτ-tω)dτ≤∫-∞0eλ1τ-∫τ0[δ(θsω)+2q̃η(θsω)]ds-2δ(θτω)dτ<∞,
there exists I1(ɛ,ω)∈ℕ such that for M>I1(ɛ,ω),
(3.77)(ii)≤ɛ3min⁡{μ,α}e-2δ(ω).
Note thatϵ2=λ1-2q̃E|η(ω)|>0, then by (H2), η(ω) is tempered and it follows that there exists a T1′>0 such that
(3.78)η(θτ-tω)≤e(ϵ2/3)(t-τ),∀t-τ≥T1′.
Therefore
(3.79)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃η(θs-tω)]ds-2δ(θτ-tω)η(θτ-tω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ≤(μ‖x0‖ρ2+α‖y0‖ρ2)e-(λ1-(ϵ2/3))t-4δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]ds∫T1ρte(-ϵ2/3)τdτ+12r1ρ2(ω)e-2δ(ω)∫0t-T1ρe-(λ1-(ɛ2/3))τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ.
For t≫T1ρ, by (H2) and (3.6), there exists 0<T2′=T2′(ω)<t-T1ρ such that
(3.80)1τ∫0τ[2δ(θsω)+2q̃η(θsω)]ds<λ1-2ϵ23,forτ≥T2′.
Let T2=max{T1′,T2′}<∞, and t>T2+T1ρ, write
(3.81)∫0t-T1ρe-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ=(∫0T2+∫T2t-T1ρ)e-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ,
of which
(3.82)∫0T2e-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ<∞,∫T2t-T1ρe-(λ-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ≤3ϵ2e(-ϵ2/3)T2.
Equation (3.79) together with (3.82) implies that there exist T3(ɛ,ω,B1ρ)≥T2 and I2(ɛ,ω)∈ℕ such that for M>I2(ɛ,ω),t≥T3(ɛ,ω,B1ρ),
(3.83)(iii)≤ɛ3min⁡{μ,α}e-2δ(ω).
In summary, let
(3.84)T(ɛ,ω,B1ρ)=max⁡{T1(ɛ,ω,B1ρ),T2(ɛ,ω,B1ρ),T3(ɛ,ω,B1ρ)},I(ɛ,ω)=max⁡{I1(ɛ,ω),I2(ɛ,ω)}
Then for t>T(ɛ,ω,B1ρ) and M≥I(ɛ,ω), we have
(3.85)∑|i|≥2Mρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤ɛmin⁡{μ,α}e-2δ(ω).
Since zi(mk)(t;θ-tω,z0n(θ-tω),hn,gn)→zi(t;θ-tω,z0n(θ-tω),hn,gn) as mk→∞, by (3.85),
(3.86)∑|i|≥2Mρi|(φ(t,ω)(u0,v0))i|2=∑|i|≥2Mρie2δ(ω)|zi(t;θ-tω,z0n,gn,hn)|2≤e2δ(ω)min⁡{μ,α}∑|i|≥2Mρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤ɛ.
Letting n→∞ in (3.86), we obtain
(3.87)∑|i|≥2Mρi|(φ(t,ω)(u0,v0))i|2≤ɛ
That is, {φ(t,ω)}t≥0,ω∈Ω1 is asymptotically null on B1ρ(ω), which completes the proof.


## 3.1. Mathematical Setting

DefineΩ1={ω∈C(ℝ,ℝ):ω(0)=0}=C0(ℝ,ℝ), and denote by ℱ1 the Borel σ-algebra on Ω1 generated by the compact open topology (see [2, 15]) and ℙ1 the corresponding Wiener measure on ℱ1. Defining (θt)t∈ℝ on Ω1 via θtω(·)=ω(·+t)-ω(t) for t∈ℝ, then (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) is a metric dynamical system.Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise:(3.1)du=(-λu+A(θtω)u-f(u)-αv+h)dt+u∘dw(t),dv=(-σv+μu+g)+u∘dw(t),i∈Z,t>0,
where u=(ui)i∈ℤ, v=(vi)i∈ℤ; f(u)=(fi(ui))i∈ℤ with fi∈C1(ℝ,ℝ) (i∈ℤ), g=(gi)i∈ℤ, h=(hi)i∈ℤ; λ,α,σ,μ are positive constants; ηi,-q(ω),…,ηi,0(ω),…,ηi,+q(ω)(q∈ℕ) are random variables on the probability space (Ω1,ℱ1,ℙ1); A(·) is a linear operator on lρ2 defined by(3.2)(A(⋅)u)i=∑j=-qqηi,j(⋅)ui+j;w(t) is a Brownian motion (Wiener process) on the probability space (Ω1,ℱ1,ℙ1); ∘ denotes the Stratonovich sense of the stochastic term.For convenience, we first transform (3.1) into a random differential equation without white noise. Let(3.3)δ(θtω)=-∫-∞0esθtω(s)ds,t∈R,ω∈Ω1,
then δ(θtω) is an Ornstein-Uhlenbeck process on (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details)(3.4)dδ+δdt=dw(t),t≥0,
where w(t)(ω)=w(t,ω)=ω(t) for ω∈Ω1,  t∈ℝ, and possesses the following properties.Lemma 3.1 (see [2, 15]).
There exists aθt-invariant set Ω̃1∈ℱ1 of Ω1 offull ℙ1 measure such that for ω∈Ω̃1, one has(i)
the random variable|δ(ω)| is tempered;(ii)
the mappingδ(θtω)(3.5)(t,ω)⟼δ(θtω)=-∫-∞0esω(t+s)ds+w(t)
is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)
(3.6)lim⁡t→±∞|δ(θtω)|t=lim⁡t→±∞1t∫0tδ(θsω)ds=0.The mapping ofθ on Ω̃1 possesses same properties as the original one if we choose the trace σ-algebra with respect to Ω̃1 to be denoted also by ℱ1. Therefore we can change our metric dynamical system with respect to Ω̃1, still denoted by the symbols (Ω1,ℱ1,ℙ1,(θt)t∈ℝ).Let(3.7)x(t,ω)=e-δ(θtω)u(t,ω),y(t,ω)=e-δ(θtω)v(t,ω),ω∈Ω1,
where (u(t,ω),v(t,ω)) is a solution of (3.1), then (u(t,ω),v(t,ω))↦(x(t,ω),y(t,ω)) is a homomorphism in H. System (3.1) can then be transformed to the following random system with random coefficients but without white noise:(3.8)dxdt=-λx+A(θtω)x-e-δ(θtω)f(eδ(θtω)x)+δ(θtω)x-αy+e-δ(θtω)h,dydt=-σy+δ(θtω)y+μx+e-δ(θtω)gt>0,
Letting z=(x,y), (3.8) are equivalent to(3.9)dzdt=F(z,θtω),t>0,
where
(3.10)F(z,θtω)=(-λx+A(θtω)x-e-δ(θtω)f(eδ(θtω)x)+δ(θtω)x-αy+e-δ(θtω)h-σy+δ(θtω)y+μx+e-δ(θtω)g).We now make the following standing assumptions onfi, gi, hi, and ηi,j,(j=-q,…,q)i∈ℤ and study in the following subsections asymptotic behavior of system (3.9).(H1)
g=(gi)i∈ℤ,  h=(hi)i∈ℤ∈lρ2.(H2)
Let(3.11)η(ω)=sup⁡{|ηi,-q(ω)|,…,|ηi,0(ω)|,…,|ηi,+q(ω)|:i∈Z}≥0,q∈N.η(θtω)(<∞)  belongs toLloc1(ℝ)  with respect tot∈ℝ  for eachω∈Ω1. 										(3.12)E(η)=lim⁡t→±∞1t∫0tη(θtω)ds<∞,andη(ω)  is tempered, that is, there exists aθt-invariant setΩ10∈ℱ1of  fullℙ1  measure such that forω∈Ω10,
(3.13)lim⁡t→+∞⁡e-βtsup⁡t∈R⁡|η(θ-tω)|=0,∀β>0.In the following, we will consider ω∈Ω10∩Ω̃1 and still write Ω10∩Ω̃1 as Ω1.(H3)
min{λ,σ}>q̃E|η(ω)|=limt→±∞(1/t)∫0t(q+∑k=0qck)η(θtω)ds, whereq̃=q+∑k=0qck.(H4)
There exists a functionR∈C(ℝ+,ℝ+)such that(3.14)sup⁡i∈Zmax⁡s∈[-r,r]|fi′(s)|≤R(r),∀r∈R+.(H5)
fi∈C1(ℝ,ℝ),fi(0)=0,sfi(s)≥-bi2,b=(bi)i∈ℤ∈lρ2, and there exists a constanta≥0such thatfi′(s)≥-a,foralls∈ℝ,i∈ℤ.


## 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 H.Definition 3.2.
We callz:[0,T)→H a solution of the following random differential equation
(3.15)dzdt=F(z,θtω),z=(zi)i∈Z,F=(Fi)i∈Z,
where ω∈Ω0, if z∈C([0,T),H) satisfies
(3.16)zi(t)=zi(0)+∫0tFi(z(s),θsω)ds,fori∈Z,t∈[0,T).Theorem 3.3.
LetT>0 and (P0), (H1), (H2), (H4), and (H5) hold. Then for any ω∈Ω1 and any initial data z0=(x(0),y(0))∈H, (3.9) admits a unique solution z(·;ω,z0)=(x(·;ω,z0),y(·;ω,z0))∈C([0,T),H) with z(0;ω,z0)=z0.Proof.
(1) DenoteE=l2×l2, we first show that if z0∈E and (h,g)∈E, then (3.9) admits a unique solution z(t;ω,z0,h,g)∈E on [0,T) with z(0;ω,z0,g,h)=z0. Given z∈E, ω∈Ω1, and (h,g)∈E, note that F(z,ω) is continuous in z and measurable in ω from E×Ω1 to E.
By (3.2) and (H2),
(3.17)‖A(ω)x‖=[∑i∈Z(∑j=-qqηi,j(ω)xi+j)2]1/2≤(2q+1)η(ω)⋅‖x‖.
By (H4),
(3.18)‖f(eδ(ω)x)‖≤max⁡{R(eδ(ω)‖x‖),a}⋅eδ(ω)‖x‖,
and therefore
(3.19)‖F(z,ω)‖E≤(λ+(2q+1)η(ω)+max⁡{R(eδ(ω)‖x‖),a}+|δ(ω)|+μ)⋅‖x‖+(α+σ+|δ(ω)|)⋅‖y‖+|e-δ(ω)|⋅(‖h‖+‖g‖).
For any z(1)=(x(1),y(1)),  z(2)=(x(2),y(2))∈E, and for some ϑ∈(0,1)(3.20)‖f(eδ(ω)x(1))-f(eδ(ω)x(2))‖≤max⁡{R(eδ(ω)((1-ϑ)‖x(1)‖+ϑ‖x(2)‖)),a}⋅eδ(ω)‖x(1)-x(2)‖.
Also
(3.21)‖A(ω)x(1)-A(ω)x(2)‖=∑i∈Z{∑j=-qqηi,j(ω)(xi+j(1)-xi+j(2))}1/2≤(2q+1)η(ω)⋅‖x(1)-x(2)‖.
It then follows that
(3.22)‖F(z(1),ω)-F(z(2),ω)‖E≤(α+σ+|δ(ω)|)⋅‖y(1)-y(2)‖+[(eδ(ω)((1-ϑ)‖v(1)‖+ϑ‖v(2)‖))λ+(2q+1)η(ω)+μ+|δ(ω)|+eδ(ω)max⁡{R(eδ(ω)((1-ϑ)‖v(1)‖+ϑ‖v(2)‖)),a}]⋅‖x(1)-x(2)‖.
For any compact set D⊂E with supz∈D∥z∥≤r, defining random variable ζD(ω)≥0 via
(3.23)ζD(ω)=(λ+(2q+1)η(ω)+μ+|δ(ω)|+max⁡{R(eδ(ω)r),a}⋅eδ(ω))r+(α+σ+|δ(ω)|)r+|e-δ(ω)|⋅(‖h‖+‖g‖),
we have
(3.24)∫ττ+1ζD(θtω)dt<∞,∀τ∈R,
and for any z,z(1),z(2)∈D,
(3.25)‖F(z,ω)‖E≤ζD(ω),‖F(z(1),ω)-F(z(2),ω)‖E≤ζD(ω)‖z(1)-z(2)‖E.
According to [15, 19, 20], problem (3.9) possesses a unique local solution z(·;ω,z0,g,h)∈C([0,Tmax),E) (0<Tmax≤T) satisfying the integral equation
(3.26)z(t)=z0+∫0tF(z(s),ω)ds,fort∈[0,Tmax⁡).
We will next show thatTmax=T. Since the set C0(ℝ) of continuous random process in t is dense in the set L1(ℝ) (see [18, 21]), for each ω∈Ω1, there exists a sequence {ηi,j(m)(t,ω)}m=1∞ of continuous random process in t∈ℝ such that
(3.27)lim⁡m→∞∫0T(ηi,j(m)(s,ω)-ηi,j(s,ω))ds=0,∀T>0,|ηi,j(m)(t,ω)|≤|ηi,j(θtω)|≤|η(θtω)|,∀t∈R.
Consider the random differential equation with initial dataz0∈E:(3.28)dz(m)dt=(-λx(m)+Am(t,ω)x(m)-e-δ(θtω)f(eδ(θtω)x(m))+δ(θtω)x(m)-αy(m)+e-δ(θtω)h-σy(m)+δ(θtω)y(m)+μx(m)+e-δ(θtω)g),
where
(3.29)(Am(t,ω)x(m))i=∑j=-qqηi,j(m)(t,ω)xi+j.
Follow the same procedure as above, (3.28) has a unique solution z(m)(·;ω,z0,g,h)∈C([0,Tmax(m)),E), that is,
(3.30)zi(m)(t)=z0+∫0tFi(m)(z(m)(s),ω)ds,fort∈[0,Tmax⁡(m)),
and by the continuity of Am(s,ω) in s, there holds
(3.31)dzi(m)dt=(-λxi(m)+(Am(θtω)x(m))i-e-δ(θtω)fi(eδ(θtω)xi(m))+δ(θtω)xi(m)-αyi(m)+e-δ(θtω)hi-σyi(m)+δ(θtω)yi(m)+μxi(m)+e-δ(θtω)g).
Note that
(3.32)(Am(θtω)x(m))i⋅xi(m)≤η(θtω)⋅|xi(m)xi-q(m)+⋯+xi(m)xi(m)+⋯+xi(m)xi+q(m)|,-ae2δ(θtω)⋅(xi(m))2≤(eδ(θtω)xi(m))⋅fi(eδ(θtω)xi(m))≤R(eδ(θtω)‖x(m)‖)⋅e2δ(θtω)(xi(m))2,t∈[0,T],
multiplying (3.31) by (μxi(m)00αyi(m)) and sum over i∈ℤ results in
(3.33)ddt(μ‖x(m)‖2+α‖y(m)‖2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+2(2q+1)η(θtω)]⋅(μ‖x(m)‖2+α‖y(m)‖2)+(2μλ∥h∥2+2ασ∥g∥2)e-2δ(θtω).
Applying Gronwall’s inequality to (3.33) we obtain that
(3.34)μ‖x(m)‖2+α‖y(m)‖2≤e2at+∫0t[2δ(θsω)+2(2q+1)η(θsω)]ds(μ‖x(0)‖2+α‖y(0)‖2)+(2μλ‖h‖2+2ασ‖g‖2)(e2at+∫0t[2δ(θsω)+2(2q+1)η(θsω)]ds)⋅(∫0te(min⁡{λ,σ}-2a)s-2δ(θsω)+∫0s[2δ(θrω)+2(2q+1)η(θrω)]drds)≐κ(t),t∈[0,Tmax⁡(m)),
where κ(t)∈C([0,T),ℝ) is independent of m, which implies that the interval of existence of z(m)(t) is [0,T), and z(m)(·;ω,z0,g,h)∈C1([0,T),E).
By (3.34),
(3.35)|xi(m)|2+|yi(m)|2≤κ(t)min⁡{μ,α},∀m∈N,t∈[0,T).
Since |Fi(m)(z(m)(t),θtω)|2≤K2(T,ω) for some K(T,ω)>0 and t∈[0,T), then for any t,τ∈[0,T),m∈ℕ,
(3.36)|zi(m)(t)-zi(m)(τ)|=∫τt|Fi(m)(z(m)(s),θsω)|ds≤K(T,ω)⋅|t-τ|.
By the Arzela-Acoli Theorem, there exists a convergent subsequence {zi(mk)(t),t∈[0,T)} of {zi(m)(t),t∈[0,T)} such that
(3.37)zi(mk)(t)⟶z¯i(t)ask⟶∞,t∈[0,T),i∈Z,
and z¯i(t) is continuous on t∈[0,T). Moreover, |z¯i|2≤κ(t)/min{μ,α} for t∈[0,T). By (3.27), (3.35), assumption (H2), and the Lebesgue Dominated Convergence Theorem we have
(3.38)lim⁡k→∞∫0t(ηi,j(mk)(s,ω)-ηi,j(θsω))ds=∫0tlim⁡k→∞(ηi,j(mk)(s,ω)-ηi,j(θsω))ds=0,lim⁡k→∞(ηi,j(mk)(s,ω)-ηi,j(θsω))=0,fora.e.s∈[0,T],lim⁡k→∞∫0t(ηi,j(mk)(s,ω)xi(mk)(s)-ηi,j(θsω)x¯i(s))ds=0.
Thus replacing m by mk in (3.31) and letting k→∞ give
(3.39)z¯i(t)=(z0)i+∫0tFi(z¯i,θsω)dsfort∈[0,T).
By the uniquness of the solutions of (3.9), we have z¯i(t)=zi(t) for t∈[0,Tmax). By (3.34), ∥z(t)∥E2≤κ(t)/min{μ,α} for t∈[0,Tmax), which implies that the solution z(t) of (3.9) exists globally on t∈[0,T).
(2) Next we prove that for anyz0∈H and (h,g)∈H, (3.9) has a solution z(t;ω,z0,h,g) on [0,T) with z(0;ω,z0,h,g)=z0. Let z1,0,z2,0∈E and h1=(h1,i)i∈ℤ,  h2=(h2,i)i∈ℤ,g1=(g1,i)i∈ℤ,g2=(g2,i)i∈ℤ∈l2. Let z1(m)(t,ω),z2(m)(t,ω) be two solutions of (3.28) with initial data z1,0,z2,0 and h,g replaced by h1, h2, g1, g2, respectively. Set d(m)(t)=z1(m)(t)-z2(m)(t)=(d1(m)(t),d2(m)(t))∈E⊂H. Take inner product 〈·,·〉H of (d/dt)d(m) with d(m) and evaluate each term as follows. By (P0), (H1), (H2), and (H4),
(3.40)|〈Am(θtω)d1(m),d1(m)〉ρ|≤q̃η(θtω)‖d1(m)‖ρ2,〈f(eδ(θtω)x1(m))-f(eδ(θtω)x2(m)),d1(m)〉ρ≥-aeδ(θtω)‖d1(m)‖ρ2,〈f(eδ(θtω)x1(m))-f(eδ(θtω)x2(m)),d1(m)〉ρ≤R(eδ(θtω)(‖x1(m)‖+‖x2(m)‖))eδ(θtω)‖d1(m)‖ρ2,〈h1-h2,d1(m)〉ρ≤‖h1-h2‖ρ2⋅‖d1(m)‖ρ2;〈g1-g2,d2(m)〉ρ≤‖g1-g2‖ρ2⋅‖d2(m)‖ρ2.
It then follows that
(3.41)ddt(μ‖d1(m)‖ρ2+α‖d2(m)‖ρ2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+q̃η(θtω)]⋅(μ‖d1(m)‖ρ2+α‖d2(m)‖ρ2)+(2μλ‖h1-h2‖ρ2+2ασ‖g1-g2‖ρ2)e-2δ(θtω).
For T>0, applying Gronwall’s inequality to (3.41) on [0,T] implies that
(3.42)μ‖d1(m)(t)‖ρ2+α‖d2(m)(t)‖ρ2≤CT(μ‖d1(m)(0)‖ρ2+α‖d2(m)(0)‖ρ2+‖h1-h2‖tρ2+‖g1-g2‖ρ2)
for some constant CT depending on T, and thus
(3.43)‖z1(m)(t)-z2(m)(t)‖H2≤C̃T(‖z1(m)(0)-z2(m)(0)‖H2+‖h1-h2‖ρ2+‖g1-g2‖ρ2),
where C̃T is a constant depending on T. Denote by Ẽ=l̃2×l̃2, where l̃2=l2 with the norm ∥·∥ρ. By (3.43), there exists a mapping Φ(m)∈C(Ẽ×Ẽ,C([0,T],H)) such that Φ(m)(z0,g,h)=z(m)(t;ω,z0,g,h), where z(m)(t;ω,z0,g,h) is the solution of (3.28) on [0,T) with z(m)(0;ω,z0,g,h)=z0. Since l̃2 is dense in lρ2, the mapping Φ(m) can be extended uniquely to a continuous mapping Φ̃(m):H×H→C([0,T],H).
For givenz0∈H and (g,h)∈H, Φ̃(m)(z0,g,h)=z(m)(·;ω,z0,g,h)∈C([0,T],H) for T>0. There exist sequences {z0n}⊂Ẽ, {(hn,gn)}⊂Ẽ such that
(3.44)‖z0n-z0‖E⟶0,‖hn-h‖ρ⟶0,‖gn-g‖ρ⟶0asn⟶∞.
Let zn(m)(t)=Φ̃(m)(z0n,hn,gn)=Φ(m)(z0n,hn,gn)=z(m)(t;ω,z0n,hn,gn)∈Ẽ be the solution of (3.28), then it satisfies the integral equation
(3.45)(zn(m))i(t)=(z0n)i+∫0tFi(zn(m),θsω)ds.
By the continuity of Φ̃(m), we have for t∈[0,T),
(3.46)z(m)(t;ω,z0n,hn,gn)=Φ̃(m)(z0n,hn,gn)⟶n→∞Φ̃(m)(z0,h,g)=z(m)(t;ω,z0,h,g)∈H.
Thus for each i∈ℤ,
(3.47)(zn(m))i(t)⟶zi(m)(t):=zi(m)(t;ω,z0,h,g)asn⟶∞uniformlyon⁡t∈[0,T).
Moreover, {(zn(m))i(t)} is bounded in n. Let n→∞, then we have
(3.48)zi(m)(t)=(z0)i+∫0tFi(z(m),θsω)ds,
and zi(m)(t) satisfies the differential equation (3.31).
Multiply equation (3.31) by (μρixi(m)00αρiyi(m)) and sum over i∈ℤ, we obtain
(3.49)ddt(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)≤[-min⁡{λ,σ}+2a+2δ(θtω)+q̃η(θtω)]⋅(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)+(2μλ‖h‖ρ2+2ασ‖g‖ρ2)e-2δ(θtω),μ‖x(m)‖ρ2+α‖y(m)‖ρ2≤e2at+∫0t[2δ(θsω)+q̃η(θsω)]ds(μ‖x(0)‖ρ2+α‖y(0)‖ρ2)+(2μλ‖h‖ρ2+2ασ‖g‖ρ2)(e2at+∫0t[2δ(θsω)+q̃η(θsω)]ds)⋅(∫0te(λ+σ-2a)s-2δ(θsω)+∫0s[2δ(θrω)+q̃η(θrω)]drds)≐κρ(t),t∈[0,Tmax⁡)
Similar to the process (3.35)–(3.39) in part (1), we obtain the existence of a unique solution z(t;ω,z0,g,h)∈H of (3.9) with initial data z0∈H, which is the limit function of a subsequence of {z(m)(t;ω,z0,g,h)} in H for t∈[0,T). In the latter part of this paper, we may write z(t;ω,z0,h,g) as z(t;ω,z0) for simplicity.Theorem 3.4.
Assume that (P0), (H1), (H2), (H4), and (H5) hold. Then (3.9) generates a continuous RDS {ψ(t,ω)}t≥0,ω∈Ω1 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) with state space H:
(3.50)ψ(t,ω)z0:=z(t;ω,z0)forz0∈H,t≥0,ω∈Ω1.
Moreover,
(3.51)φ(t,ω)(u0,v0):=eδ(θtω)ψ(t,ω)e-δ(ω)(u0,v0)for(u0,v0)∈H,t≥0,ω∈Ω1,
defines a continuous RDS {φ(t,ω)}t≥0,ω∈Ω1 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) associated with (3.1).Proof.
By Theorem3.3, the solution z(t;ω,z0) of (3.9) with z(0;ω,z0)=z0 exists globally on [0,∞). It is then left to show that z(t;ω,z0)=z(t;ω,z0,h,g) is measurable in (t,ω,z0).
In fact, forz0∈E and (h,g)∈E, the solution of (3.9) z(t;ω,z0,h,g)∈E for t∈[0,∞). In this case, function F(z,t,ω,h,g)=F(z,t,ω) is continuous in z,h, g and measurable in t,ω, which implies that z:[0,∞)×Ω1×E×E→E,(t;ω,z0,h,g)↦z(t;ω,z0,h,g) is (ℬ([0,∞)×ℱ1×ℬ(E)×ℬ(E),ℬ(E))-measurable.
Forz0∈H and (h,g)∈H, the solution z(t;ω,z0,h,g)∈H for t∈[0,∞). For any given N>0, define TN:H→E, (u,v)=((ui),(vi))i∈ℤ→TN(u,v)=((TN(u,v))i)i∈ℤ by
(3.52)(TN(u,v))j={(uj,vj)if|j|≤N,0if|j|>N,
and write
(3.53)zN(t;ω,z0,h,g)=z(t;ω,TNz0,TN(h,g)).
Then TN is continuous and for any z0∈H,(h,g)∈H, and
(3.54)z(t;ω,z0,h,g)=lim⁡N→∞z(t;ω,TNz0,TN(h,g)).
Thus z:[0,∞)×Ω1×E×E→H is (ℬ([0,∞))×ℱ0×ℬ(E)×ℬ(E),ℬ(H))-measurable. Observe also that (Id,Id,TN,TN):[0,∞)×Ω1×H×H→[0,∞)×Ω0×E×E is (ℬ([0,∞))×ℱ0×ℬ(H)×ℬ(H),ℬ([0,∞))×ℱ0×ℬ(E)×ℬ(E))-measurable. Hence zN=z∘(Id,Id,TN,TN):[0,∞)×Ω1×H×H→H is (ℬ([0,∞))×ℱ0×ℬ(H)×ℬ(H),ℬ(H))-measurable. It then follows from (3.54) that z:[0,∞)×Ω1×H×H→H is (ℬ([0,∞))×ℱ1×ℬ(H)×ℬ(H),ℬ(H))-measurable. Therefore, fix (h,g)∈H we have that z(t;ω,z0)=z(t;ω,z0,h,g) is measurable in (t,ω,z0). The other statements then follow directly.Remark 3.5.
If(h,g)∈E, system (3.1) defines a continuous RDS {φ(t)}t≥0 over (Ω1,ℱ1,ℙ1,(θt)t∈ℝ) in both state spaces E and H.


## 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{φ(t,ω)}t≥0,ω∈Ω1 generated by (3.1) in weighted space H.Theorem 3.6.
Assume that (P0), (H1)–(H5) hold, then there exists a closed tempered random bounded absorbing setB1ρ(ω)∈𝒟(H) of {φ(t,ω)}t≥0,ω∈Ω1 such that for any D(ω)∈𝒟(H) and each ω∈Ω1, there exists TD(ω)>0 yielding φ(t,θ-tω)D(θ-tω)⊂B1ρ(ω) for all t≥TD(ω). In particular, there exists T1ρ(ω)>0 such that φ(t,θ-tω)B1ρ(θ-tω)⊂B1ρ(ω) for allt≥T1ρ(ω).Proof.
(1) For initial conditionz0∈E and (h,g)∈E, let z(m)(t,ω)=z(m)(t;ω,z0(ω),h,g) be a solution of (3.28) with z0(ω)=e-δ(ω)z0∈E, where ω∈Ω1, then z(m)(t,ω)∈E for all t≥0. Let ϵ1>0 be such that
(3.55)λ1=2min⁡{λ,σ}-ϵ1>2q̃E|η(ω)|.
By (H4) and (H5), we have
(3.56)-‖b‖ρ2≤∑i∈Zρi(eδ(θtω)xi(m))⋅fi(eδ(θtω)xi(m))<∞,forfixedt≥0,(3.57)ddt(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)≤[-λ1+2δ(θtω)+2q̃η(θtω)]⋅(μ‖x(m)‖ρ2+α‖y(m)‖ρ2)+2e-2δ(θtω)(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2).
Applying Gronwall’s inequality to (3.57), we obtain that for t>0,
(3.58)μ‖x(m)(t,ω)‖ρ2+α‖y(m)(t,ω)‖ρ2≤e-λ1t+∫0t[2δ(θsω)+2q̃η(θsω)]ds(μ‖x(0)‖ρ2+α‖y(0)‖ρ2)+2(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)e-λ1t+∫0t[2δ(θsω)+2q̃η(θsω)]ds⋅∫0teλ1s-2δ(θsω)-∫0s[2δ(θrω)+2q̃η(θrω)]drds.
(2) For anyz0∈H and (h,g)∈H, let {z0n}⊂E and {(hn,gn)}⊂E be sequences such that
(3.59)‖z0n-z0‖E⟶0,‖hn-h‖ρ⟶0,‖gn-g‖ρ⟶0asn⟶∞.
Then z(m)(t;ω,z0n,hn,gn)→z(m)(t;ω,z0,h,g) as n→∞ in H, and (3.58) holds for z0∈H. Therefore,
(3.60)μ‖x(m)(t,θ-tω,z0(θ-tω))‖ρ2+α‖y(m)(t,θ-tω,z0(θ-tω))‖ρ2≤e-λ1t+∫0t[2δ(θs-tω)+2q̃η(θs-tω)]ds(μ‖x0(θ-tω)‖ρ2+α‖y0(θ-tω)‖ρ2)+12r1ρ2(ω)e-2δ(ω),
where
(3.61)r1ρ2(ω)=4e2δ(ω)(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)∫-∞0eλ1s-2δ(θsω)+∫s0[2δ(θrω)+2q̃η(θrω)]drds<∞.
For any β>0, since
(3.62)(e-βtr0ρ(θ-tω))2(μ‖b‖ρ2+μϵ1‖h‖ρ2+ασ‖g‖ρ2)-1=4e-2βt+δ(θ-tω)∫-∞-teλ1(s+t)-2δ(θsω)+∫s-t[2δ(θrω)+2q̃|η(θrω)|]drds⟶t→+∞0,
then r1ρ(ω) is tempered.
Letz(t,ω)=ψ(t,ω)z0(ω)=z(t;ω,z0(ω),h,g) be a solution of equation (3.9) with z0(ω)=e-δ(ω)(u0,v0)∈H, where ω∈Ω1 and (h,g)∈H, then z(t,ω)∈H, and there exists a subsequence z(mk)(t,ω) converging to z(t,ω) as mk→∞ for all t≥0. Inequality (3.60) still holds after replacing z(m)(t,ω) by z(t,ω) since the right hand of (3.60) is independent of m. Thus for (u0,v0)∈D(θ-tω),
(3.63)z0(θ-tω)=(x0(θ-tω),y0(θ-tω))=e-δ(θ-tω)(u0,v0),μ‖u(t,θ-tω,(u0,v0))‖ρ2+α‖v(t,θ-tω,(u0,v0))‖ρ2≤e2δ(ω)e-λ1t-2δ(θ-tω)+∫0t[2δ(θs-tω)+2q̃η(θs-tω)]dssup⁡(u,v)∈D(θ-tω)(μ‖u‖ρ2+α‖v‖ρ2)+12r1ρ2(ω).
Let ϵ2=λ1-2q̃E|η(ω)|>0. By properties of η(θ±tω) and D∈𝒟(H), we have
(3.64)e-[ϵ2/2+2q̃E|η(ω)|]t-δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]ds⟶0ast⟶+∞,
and hence
(3.65)lim⁡t→+∞e2δ(ω)e-λ1t-δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]dssup⁡(u,v)∈D(θ-tω)(μ‖u‖ρ2+α‖v‖ρ2)=0.
Denote by r̃1ρ=r1ρ/min{μ,α}, it follows that
(3.66)B1ρ(ω)={(u,v)∈H:‖(u,v)‖H≤r̃1ρ(ω)}¯=BH(0,r̃1ρ(ω))¯⊂H
is a tempered closed random absorbing set for {φ(t,ω)}t≥0,ω∈Ω1.Theorem 3.7.
Assume that (P0), (H1)–(H5) hold, then the RDS{φ(t,ω)}t≥0,ω∈Ω1 generated by (3.1) possesses a unique global random 𝒟 attractor given by
(3.67)A1(ω)=⋂τ≥TB1ρ(ω)⋃t≥τφ(t,θ-tω)B1ρ(θ-tω)¯⊂H.Proof.
According to Theorem2.2, it remains to prove the asymptotically nullness of {φ(t,ω)}t≥0,ω∈Ω1; that is, for any ɛ>0, there exists T(ɛ,ω,B1ρ)>T1ρ(ω) and I(ɛ,ω)∈ℕ such that when t≥T(ɛ,ω,B1ρ), the solution φ(t,ω)(u0,v0)=((ui,vi)(t;ω,u0,v0))i∈ℤ∈H of (3.1) with (u0,v0)∈B1ρ(θ-tω) satisfies
(3.68)∑|i|≥I(ɛ,ω)ρi|(φ(t,ω)(u0,v0))i|2=∑|i|≥I(ɛ,ω)ρi|(ui,vi)(t;ω,u0,v0)|2≤ɛ.
Choose a smooth increasing function ξ∈C1(ℝ+,[0,1]) such that
(3.69)ξ(s)=0,0≤s≤1,0≤ξ(s)≤1,1≤s≤2,|ξ′(s)|≤C0(constant)fors∈R+,ξ(s)=1,s≥2.
Let (u,v)(t;ω,u0,v0,h,g)=((ui,vi)(t;ω,u0,v0,h,g))i∈ℤ be a solution of (3.1), then
(3.70)z(t;ω,z0(ω),h,g)=(zi(t;ω,z0(ω),h,g))i∈Z=e-δ(θtω)φ(t,ω)(u0,v0)
is a solution of (3.9) with z0(ω)=e-δ(ω)(u0,v0)∈H.
Letz0n=Tnz0,(hn,gn)=Tn(h,g), where Tn is as in (3.52). Then z0n∈E,(hn,gn)∈E and z(t;ω,z0n,hn,gn)→z(t;ω,z0,h,g) in H. For any n≥1, let z(m)(t)=z(m)(t;ω,z0n(ω),hn,gn) be the solution of (3.28), where z(m)(0)=z0n(ω). By Theorem 3.4, z(m)(·)∈C([0,∞),E)∩C1((0,∞),E). Let M be a suitable large integer (will be specified later); multiply (3.31) by (μρiξ(|i|/M)xi(m)00αρiξ(|i|/M)yi(m)) and sum over i∈ℤ, we obtain
(3.71)ddt∑i∈Zξ(|i|M)ρi(μ(xi(m))2+α(yi(m))2)≤[-λ1+2δ(θtω)+2q̃η(θtω)]⋅∑i∈Zξ(|i|M)ρi(μ(xi(m))2+α(yi(m))2)+η(θtω)⋅cC0M‖x(m)‖ρ2+2e-2δ(θtω)∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2).
Applying Gronwall’s inequality to (3.71) from T1ρ=TB1ρ(ω) to t gives
(3.72)∑i∈Zξ(|i|M)ρi(μ(xi(m)(t;ω,z0n,hn,gn))2+α(yi(m)(t;ω,z0n,hn,gn))2)≤e-λ1(t-T1ρ)+∫T1ρt[2δ(θsω)+2q̃η(θsω)]ds⋅(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)+∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2)∫T1ρte-λ1(t-τ)+∫τt[2δ(θsω)+2q̃η(θsω)]ds-2δ(θτω)dτ+cC0M∫T1ρte-λ1(t-τ)+∫τt[2δ(θsω)+2q̃|η(θsω)|]ds-2δ(θτω)η(θτω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ.
Therefore for (u0,v0)∈B1ρ(θ-tω)∩H,
(3.73)∑i∈Zξ(|i|M)ρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤e-λ1(t-T1ρ)+∫T1ρt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)︸(i)+∑|i|≥Mρi(μbi2+μϵ1(hn)i2+ασ(gn)i2)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds-2δ(θτ-tω)dτ︸(ii)+cC0M∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃|η(θs-tω)|]ds-2δ(θτ-tω)η(θτ-tω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ︸(iii).
We next estimate terms (i),(ii),(iii) on the right-hand side of (3.73). By (3.61),
(3.74)e-λ1(t-T1ρ)+∫T1ρt[2δ(θs-tω)+2q̃η(θs-tω)]ds(μ‖x(m)(T1ρ)‖ρ2+α‖y(m)(T1ρ)‖ρ2)⟶0ast⟶+∞,
which implies that for all ɛ>0, there exists T1(ɛ,ω,B1ρ)≥T1ρ such that for t≥T1(ɛ,ω,B1ρ),
(3.75)(i)≤ɛ3min⁡{μ,α}e-2δ(ω).
By h,g,b∈lρ2 and
(3.76)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃η(θs-tω)]ds-2δ(θτ-tω)dτ≤∫-∞0eλ1τ-∫τ0[δ(θsω)+2q̃η(θsω)]ds-2δ(θτω)dτ<∞,
there exists I1(ɛ,ω)∈ℕ such that for M>I1(ɛ,ω),
(3.77)(ii)≤ɛ3min⁡{μ,α}e-2δ(ω).
Note thatϵ2=λ1-2q̃E|η(ω)|>0, then by (H2), η(ω) is tempered and it follows that there exists a T1′>0 such that
(3.78)η(θτ-tω)≤e(ϵ2/3)(t-τ),∀t-τ≥T1′.
Therefore
(3.79)∫T1ρte-λ1(t-τ)+∫τt[2δ(θs-tω)+2q̃η(θs-tω)]ds-2δ(θτ-tω)η(θτ-tω)(μ‖x(m)(τ)‖ρ2+α‖y(m)(τ)‖ρ2)dτ≤(μ‖x0‖ρ2+α‖y0‖ρ2)e-(λ1-(ϵ2/3))t-4δ(θ-tω)+∫-t0[2δ(θsω)+2q̃η(θsω)]ds∫T1ρte(-ϵ2/3)τdτ+12r1ρ2(ω)e-2δ(ω)∫0t-T1ρe-(λ1-(ɛ2/3))τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ.
For t≫T1ρ, by (H2) and (3.6), there exists 0<T2′=T2′(ω)<t-T1ρ such that
(3.80)1τ∫0τ[2δ(θsω)+2q̃η(θsω)]ds<λ1-2ϵ23,forτ≥T2′.
Let T2=max{T1′,T2′}<∞, and t>T2+T1ρ, write
(3.81)∫0t-T1ρe-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ=(∫0T2+∫T2t-T1ρ)e-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ,
of which
(3.82)∫0T2e-(λ1-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ<∞,∫T2t-T1ρe-(λ-ϵ2/3)τ+∫0τ[2δ(θsω)+2q̃η(θsω)]dsdτ≤3ϵ2e(-ϵ2/3)T2.
Equation (3.79) together with (3.82) implies that there exist T3(ɛ,ω,B1ρ)≥T2 and I2(ɛ,ω)∈ℕ such that for M>I2(ɛ,ω),t≥T3(ɛ,ω,B1ρ),
(3.83)(iii)≤ɛ3min⁡{μ,α}e-2δ(ω).
In summary, let
(3.84)T(ɛ,ω,B1ρ)=max⁡{T1(ɛ,ω,B1ρ),T2(ɛ,ω,B1ρ),T3(ɛ,ω,B1ρ)},I(ɛ,ω)=max⁡{I1(ɛ,ω),I2(ɛ,ω)}
Then for t>T(ɛ,ω,B1ρ) and M≥I(ɛ,ω), we have
(3.85)∑|i|≥2Mρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤ɛmin⁡{μ,α}e-2δ(ω).
Since zi(mk)(t;θ-tω,z0n(θ-tω),hn,gn)→zi(t;θ-tω,z0n(θ-tω),hn,gn) as mk→∞, by (3.85),
(3.86)∑|i|≥2Mρi|(φ(t,ω)(u0,v0))i|2=∑|i|≥2Mρie2δ(ω)|zi(t;θ-tω,z0n,gn,hn)|2≤e2δ(ω)min⁡{μ,α}∑|i|≥2Mρi(μ(xi(m)(t;θ-tω,z0n,hn,gn))2+α(yi(m)(t;θ-tω,z0n,hn,gn))2)≤ɛ.
Letting n→∞ in (3.86), we obtain
(3.87)∑|i|≥2Mρi|(φ(t,ω)(u0,v0))i|2≤ɛ
That is, {φ(t,ω)}t≥0,ω∈Ω1 is asymptotically null on B1ρ(ω), which completes the proof.


## 4. Stochastic Partly Dissipative Lattice Systems with Additive White Noise in Weighted Spaces

This section is devoted to the study of asymptotic behavior for system (1.2) in weighted space H=lρ2×lρ2. The structure and the idea of proofs are similar to that of Section 3, and we will present our major results without elaborting the details of proofs in this section.


### 4.1. Mathematical Setting

DefineΩ2={ω∈C(ℝ,l2):ω(0)=0}, and denote by ℱ2 the Borel σ-algebra on Ω2 generated by the compact open topology [1] and ℙ2 is the corresponding Wiener measure on ℱ2, then (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) is a metric dynamical system. Let the infinite sequence ei (i∈ℤ) denote the element having 1 at position i and 0 for all other components. Write(4.1)W1(t,ω)=∑i∈Zaiwi(t)ei,W2(t,ω)=∑i∈Zbiwi(t)ei,
where {wi(t):i∈ℤ} are independent two-sided Brownian motions on probability space (Ω2,ℱ2,ℙ2); then W1(t,ω) and W2(t,ω) are Wiener processes with values in l2 defined on the probability space (Ω2,ℱ2,ℙ2).Consider stochastic lattice system (1.2) with random coupled coefficients and additive independent white noises:(4.2)u̇=-λu+A(θtω)u-f(u)+αv+h+dW1(t)dt,v̇=-σv+μu+g+dW2(t)dt,i∈Z,t>0,
where ui,vi,hi,gi,ai,bi∈ℝ; u=(ui)i∈ℤ, v=(vi)i∈ℤ, f(u)=(fi(ui))i∈ℤ, g=(gi)i∈ℤ,h=(hi)i∈ℤ, a=(ai)i∈ℤ∈l2,  b=(bi)i∈ℤ∈l2, fi∈C1(ℝ,ℝ) (i∈ℤ); λ,α,σ,μ are positive constants; ηi,-q(ω),…,  ηi,0(ω),…,ηi,+q(ω),i∈ℤ, q∈ℕ are random variables; A is defined as in (3.2).To transform (4.2) into a random equation without white noise, let(4.3)δ1(θtω)=-λ∫-∞0eλsθtω(s)ds,δ2(θtω)=-σ∫-∞0eσsθtω(s)ds,t∈R,ω∈Ω2.
Then δ1(θtω),  δ2(θtω) are both Ornstein-Uhlenbeck processes on (Ω2,ℱ2,ℙ2) and solve the following Ornstein-Uhlenbeck equations (see [1]), respectively,(4.4)dδ1+λδ1dt=dW1(t),dδ2+σδ2dt=dSW2(t),t≥0.Lemma 4.1 (see [1]).
There exists aθt-invariant set  Ω̃2∈ℱ2 of Ω2 of full ℙ measure such that for ω∈Ω̃2,(i)
limt→±∞∥ω(t)∥/t=0;(ii)
the random variables∥δj(ω)∥ are tempered and the mappings
(4.5)(t,ω)⟶δj(θtω)∈l2,j=1,2,
are stationary solutions of Ornstein-Uhlenbeck equations (4.4) in l2 with continuous trajectories;(iii)
(4.6)lim⁡t→±∞‖δj(θtω)‖t=lim⁡t→±∞1t∫0tδj(θsω)ds=0,j=1,2.In the following, we consider the completion of the probability spaceω∈Ω̃2, still denoted by (Ω2,ℱ2,ℙ2),Let(4.7)x̃(t,ω)=u(t,ω)-δ1(θtω),ỹ(t,ω)=v(t,ω)-δ2(θtω),ω∈Ω2,t∈R.
then system (4.2) becomes the following random system with random coefficients but without white noise:(4.8)dx̃dt=-λx̃+A(θtω)x̃-f(x̃+δ1(θtω))-αx̃+A(θtω)δ1(θtω)-αδ2(θtω)+h,dỹdt=-σỹ+μx̃+μδ1(θtω)+g,i∈Z,t>0.
In addition, we make the following assumptions on functions fi,i∈ℤ:(H6)
fi∈C1(ℝ,ℝ)satisfy(4.9)sfi(s)≥μs2(p+1)-di2,|fi(s)|≤df|s|(|s|2p+1),∀s∈R,i∈Z,whereμ,
di,
dfare positive constants,p∈ℕ, andd=(di)i∈ℤ∈l2.


### 4.2. Random Dynamical System Generated by Random Lattice System

Denote byz̃=(x̃,ỹ), we have the following.Theorem 4.2.
LetT>0 and assume that (P0), (H1), (H2), (H4), and (H6) hold. Then for every ω∈Ω2 and any initial data z̃0=(x̃0,ỹ0)∈H, problem (4.8) admits a unique solution z̃(·;ω,z̃0)∈C([0,T),H) with z̃(0;ω,z̃0)=z̃0.Proof.
Similar to the proof of Theorem3.3.Theorem 4.3.
Assume that (P0), (H1), (H2), (H4), and (H6) hold. Then system (4.8) generates a continuous RDS {ψ̃(t,ω)}t≥0,ω∈Ω2 over (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) with state space H:
(4.10)ψ̃(t,ω)z0̃∶=z̃(t;ω,z0̃),forz0̃∈H,t≥0,ω∈Ω2.
Moreover,
(4.11)φ̃(t,ω)(u0,v0):=ψ̃(t,ω)(u0-δ1(ω)v0-δ2(ω))+(δ1(θtω)δ2(θtω)),
where (u0,v0)∈H,t≥0,ω∈Ω2, defines a continuous RDS {φ̃(t,ω)}t≥0,ω∈Ω2 over (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) associated with (4.2).Proof.
It follows immediately from similar arguments to the proof of Theorem3.4.


### 4.3. Existence of Tempered Bounded Random Absorbing Set and Random Attractor 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{φ̃(t,ω)}t≥0,ω∈Ω2 generated by (4.2) in weighted space H.Theorem 4.4.
Assume that (P0), (H1)–(H4), and (H6) hold. Then(a)
there exists a closed tempered bounded random absorbing setB2ρ(ω)∈𝒟(H) of RDS {φ̃(t,ω)}t≥0,ω∈Ω2 such that for any D∈𝒟(H) and each ω∈Ω2, there exists T̃D(ω)>0 yielding φ̃(t,θ-tω)D(θ-tω)⊂B2ρ(ω),  forallt≥T̃D(ω). In particular, there exists T2ρ(ω)>0 such that φ̃(t,θ-tω)B2ρ(θ-tω)⊂B2ρ(ω),  forallt≥T2ρ(ω);(b)
the RDS{φ̃(t,ω2)}t≥0,ω∈Ω2 generated by equations (4.2) possesses a unique global random 𝒟 attractor given by(4.12)A2ρ(ω)=⋂τ≥T2ρ(ω)⋃t≥τφ̃(t,θ-tω)B2ρ(θ-tω)¯∈H.Proof.
Similar to the proofs of Theorems3.6 and 3.7.


## 4.1. Mathematical Setting

DefineΩ2={ω∈C(ℝ,l2):ω(0)=0}, and denote by ℱ2 the Borel σ-algebra on Ω2 generated by the compact open topology [1] and ℙ2 is the corresponding Wiener measure on ℱ2, then (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) is a metric dynamical system. Let the infinite sequence ei (i∈ℤ) denote the element having 1 at position i and 0 for all other components. Write(4.1)W1(t,ω)=∑i∈Zaiwi(t)ei,W2(t,ω)=∑i∈Zbiwi(t)ei,
where {wi(t):i∈ℤ} are independent two-sided Brownian motions on probability space (Ω2,ℱ2,ℙ2); then W1(t,ω) and W2(t,ω) are Wiener processes with values in l2 defined on the probability space (Ω2,ℱ2,ℙ2).Consider stochastic lattice system (1.2) with random coupled coefficients and additive independent white noises:(4.2)u̇=-λu+A(θtω)u-f(u)+αv+h+dW1(t)dt,v̇=-σv+μu+g+dW2(t)dt,i∈Z,t>0,
where ui,vi,hi,gi,ai,bi∈ℝ; u=(ui)i∈ℤ, v=(vi)i∈ℤ, f(u)=(fi(ui))i∈ℤ, g=(gi)i∈ℤ,h=(hi)i∈ℤ, a=(ai)i∈ℤ∈l2,  b=(bi)i∈ℤ∈l2, fi∈C1(ℝ,ℝ) (i∈ℤ); λ,α,σ,μ are positive constants; ηi,-q(ω),…,  ηi,0(ω),…,ηi,+q(ω),i∈ℤ, q∈ℕ are random variables; A is defined as in (3.2).To transform (4.2) into a random equation without white noise, let(4.3)δ1(θtω)=-λ∫-∞0eλsθtω(s)ds,δ2(θtω)=-σ∫-∞0eσsθtω(s)ds,t∈R,ω∈Ω2.
Then δ1(θtω),  δ2(θtω) are both Ornstein-Uhlenbeck processes on (Ω2,ℱ2,ℙ2) and solve the following Ornstein-Uhlenbeck equations (see [1]), respectively,(4.4)dδ1+λδ1dt=dW1(t),dδ2+σδ2dt=dSW2(t),t≥0.Lemma 4.1 (see [1]).
There exists aθt-invariant set  Ω̃2∈ℱ2 of Ω2 of full ℙ measure such that for ω∈Ω̃2,(i)
limt→±∞∥ω(t)∥/t=0;(ii)
the random variables∥δj(ω)∥ are tempered and the mappings
(4.5)(t,ω)⟶δj(θtω)∈l2,j=1,2,
are stationary solutions of Ornstein-Uhlenbeck equations (4.4) in l2 with continuous trajectories;(iii)
(4.6)lim⁡t→±∞‖δj(θtω)‖t=lim⁡t→±∞1t∫0tδj(θsω)ds=0,j=1,2.In the following, we consider the completion of the probability spaceω∈Ω̃2, still denoted by (Ω2,ℱ2,ℙ2),Let(4.7)x̃(t,ω)=u(t,ω)-δ1(θtω),ỹ(t,ω)=v(t,ω)-δ2(θtω),ω∈Ω2,t∈R.
then system (4.2) becomes the following random system with random coefficients but without white noise:(4.8)dx̃dt=-λx̃+A(θtω)x̃-f(x̃+δ1(θtω))-αx̃+A(θtω)δ1(θtω)-αδ2(θtω)+h,dỹdt=-σỹ+μx̃+μδ1(θtω)+g,i∈Z,t>0.
In addition, we make the following assumptions on functions fi,i∈ℤ:(H6)
fi∈C1(ℝ,ℝ)satisfy(4.9)sfi(s)≥μs2(p+1)-di2,|fi(s)|≤df|s|(|s|2p+1),∀s∈R,i∈Z,whereμ,
di,
dfare positive constants,p∈ℕ, andd=(di)i∈ℤ∈l2.


## 4.2. Random Dynamical System Generated by Random Lattice System

Denote byz̃=(x̃,ỹ), we have the following.Theorem 4.2.
LetT>0 and assume that (P0), (H1), (H2), (H4), and (H6) hold. Then for every ω∈Ω2 and any initial data z̃0=(x̃0,ỹ0)∈H, problem (4.8) admits a unique solution z̃(·;ω,z̃0)∈C([0,T),H) with z̃(0;ω,z̃0)=z̃0.Proof.
Similar to the proof of Theorem3.3.Theorem 4.3.
Assume that (P0), (H1), (H2), (H4), and (H6) hold. Then system (4.8) generates a continuous RDS {ψ̃(t,ω)}t≥0,ω∈Ω2 over (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) with state space H:
(4.10)ψ̃(t,ω)z0̃∶=z̃(t;ω,z0̃),forz0̃∈H,t≥0,ω∈Ω2.
Moreover,
(4.11)φ̃(t,ω)(u0,v0):=ψ̃(t,ω)(u0-δ1(ω)v0-δ2(ω))+(δ1(θtω)δ2(θtω)),
where (u0,v0)∈H,t≥0,ω∈Ω2, defines a continuous RDS {φ̃(t,ω)}t≥0,ω∈Ω2 over (Ω2,ℱ2,ℙ2,(θt)t∈ℝ) associated with (4.2).Proof.
It follows immediately from similar arguments to the proof of Theorem3.4.


## 4.3. Existence of Tempered Bounded Random Absorbing Set and Random Attractor 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{φ̃(t,ω)}t≥0,ω∈Ω2 generated by (4.2) in weighted space H.Theorem 4.4.
Assume that (P0), (H1)–(H4), and (H6) hold. Then(a)
there exists a closed tempered bounded random absorbing setB2ρ(ω)∈𝒟(H) of RDS {φ̃(t,ω)}t≥0,ω∈Ω2 such that for any D∈𝒟(H) and each ω∈Ω2, there exists T̃D(ω)>0 yielding φ̃(t,θ-tω)D(θ-tω)⊂B2ρ(ω),  forallt≥T̃D(ω). In particular, there exists T2ρ(ω)>0 such that φ̃(t,θ-tω)B2ρ(θ-tω)⊂B2ρ(ω),  forallt≥T2ρ(ω);(b)
the RDS{φ̃(t,ω2)}t≥0,ω∈Ω2 generated by equations (4.2) possesses a unique global random 𝒟 attractor given by(4.12)A2ρ(ω)=⋂τ≥T2ρ(ω)⋃t≥τφ̃(t,θ-tω)B2ρ(θ-tω)¯∈H.Proof.
Similar to the proofs of Theorems3.6 and 3.7.



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*Source: 628459-2011-12-20.xml*