Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

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.

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 lattice ℤ [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:𝑢_𝑖̇=−𝜆𝑢_𝑖+∑^𝑞_𝑗=−𝑞𝜂_𝑖,𝑗𝜃_𝑡𝜔𝑢_𝑖+𝑗−𝑓_𝑖𝑢_𝑖−𝛼𝑣_𝑖+ℎ_𝑖+𝑢_𝑖∘𝑑𝑤(𝑡)𝑑𝑡,𝑣_𝑖̇‌=−𝜎𝑣_𝑖+𝜇𝑢_𝑖+𝑔_𝑖+𝑢_𝑖∘𝑑𝑤(𝑡)𝑑𝑡,𝑖∈ℤ,𝑡>0,(1.1)𝑢_𝑖̇=−𝜆𝑢_𝑖+∑^𝑞_𝑗=−𝑞𝜂_𝑖,𝑗𝜃_𝑡𝜔𝑢_𝑖+𝑗−𝑓_𝑖𝑢_𝑖−𝛼𝑣_𝑖+ℎ_𝑖+𝑎_𝑖𝑑𝑤_𝑖(𝑡)𝑑𝑡,𝑣_𝑖̇‌=−𝜎𝑣_𝑖+𝜇𝑢_𝑖+𝑔_𝑖+𝑏_𝑖𝑑𝑤_𝑖(𝑡)𝑑𝑡,𝑖∈ℤ,𝑡>0,(1.2) where 𝑢_𝑖,ℎ_𝑖,𝑔_𝑖,𝑎_𝑖,𝑏_𝑖∈ℝ𝑓_𝑖∈𝐶¹(ℝ,ℝ); (𝑖∈ℤ), 𝜆,𝛼,𝜎,𝜇>0 are positive constants; 𝐴 is the coupling operator, 𝜂_𝑖,−𝑞(𝜔),…,𝜂_𝑖,0(𝜔),…,𝜂_𝑖,+𝑞(𝜔), 𝑖∈ℤ, 𝑞∈ℕ, are random variables, and 𝑤(𝑡), {𝑤_𝑖(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, 𝜂_𝑖,±1(𝜔)≡1, 𝜂_𝑖,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 𝑙²_𝜌×𝑙²_𝜌, 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⊂𝑙²_𝜌. If ∑_𝑖∈ℤ𝜌(𝑖)<∞, then 𝑙²_𝜌 contains any infinite sequences whose components are just bounded and 𝑙²⊂𝑙^∞⊂𝑙²_𝜌. Note that when 𝜌(𝑖)≡1, our results recover the results obtained in [4, 14] while 𝑙²_𝜌 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 𝑙²_𝜌×𝑙²_𝜌 are weaker than those in 𝑙²×𝑙².

The rest of this paper is organized as follows. In Section 2, we present some preliminary results for global random attractors of continuous random dynamical systems in weighted spaces of infinite sequences. We then discuss the existence of random attractors for stochastic lattice systems (1.1) and (1.2) in Sections 3 and 4, respectively.

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.

Let 𝜌 be a positive function from ℤ to (0,𝑀₀]⊂ℝ⁺, where 𝑀₀ is a finite positive constant. Define for any 𝑖∈ℤ, 𝜌_𝑖=𝜌(𝑖) and𝑙²_𝜌=⎧⎨⎩𝑢=𝑢_𝑖_𝑖∈ℤ∶_𝑖∈ℤ𝜌_𝑖||𝑢_𝑖||²<∞,𝑢_𝑖∈ℝ⎫⎬⎭,(2.1) then 𝑙²_𝜌 is a separable Hilbert space with the inner product ⟨𝑢,𝑣⟩_𝜌=∑_𝑖∈ℤ𝜌_𝑖𝑢_𝑖𝑣_𝑖 and norm ‖𝑢‖²_𝜌=⟨𝑢,𝑢⟩_𝜌=∑_𝑖∈ℤ𝜌_𝑖|𝑢_𝑖|² for 𝑢=(𝑢_𝑖)_𝑖∈ℤ, 𝑣=(𝑣_𝑖)_𝑖∈ℤ∈𝑙²_𝜌. Moreover, define𝐻=̇𝑙²_𝜌×𝑙²_𝜌(2.2) with inner product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,(2.3) and norm‖(𝑢,𝑣)‖_𝐻=‖𝑢‖²_𝜌+‖𝑣‖²_𝜌^1/2for(𝑢,𝑣)∈𝐻,(2.4) then 𝐻 is also a separable Hilbert space.

Let (Ω,ℱ,ℙ) be a probability space and {𝜃_𝑡∶Ω→Ω,𝑡∈ℝ} be a family of measure-preserving transformations such that (𝑡,Ω)↦𝜃_𝑡Ω is (ℬ(ℝ)×ℱ,ℱ)-measurable, 𝜃₀=Id_Ω and 𝜃_𝑡+𝑠=𝜃_𝑡𝜃_𝑠 for all 𝑠,𝑡∈ℝ. The space (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) is called a metric dynamical system. In the following, “property (P) holds for a.e. 𝜔∈Ω with respect to (𝜃_𝑡)_𝑡∈ℝ" means that there is Ω⊂Ω with ℙ(Ω)=1 and 𝜃_𝑡Ω=Ω such that (P) holds for all 𝜔∈Ω.

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,(𝜃_𝑡)_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,𝜃_𝑠𝜔)𝑆(𝑠,𝜔) for all 𝑢∈𝐻 and 𝑠,𝑡≥0.(ii)A set-valued mapping 𝜔↦𝐷(𝜔)⊂𝐻 (may be written as 𝐷(𝜔) for short) is said to be a random set if the mapping 𝜔↦dist_𝐻(𝑢,𝐷(𝜔)) is measurable for any 𝑢∈𝐻.(iii)A random set 𝐷(𝜔) is called a closed (compact) random set if 𝐷(𝜔) is closed (compact) for each 𝜔∈Ω.(iv)A random set 𝐷(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 set 𝐷(𝜔) is said to be tempered if for a.e. 𝜔∈Ω, lim_𝑡→∞𝑒^−𝛽𝑡sup‖𝑢‖_𝐻∶𝑢∈𝐷𝜃_−𝑡𝜔=0,∀𝛽>0.(2.5) Denote by 𝒟(𝐻) the set of all tempered random sets of 𝐻.(vi)A random set 𝐵(𝜔) is said to be a random absorbing set in 𝒟(𝐻) if for any 𝐷(𝜔)∈𝒟(𝐻) and a.e. 𝜔∈Ω, there exists 𝑇_𝐷(𝜔) such that 𝑆(𝑡,𝜃_−𝑡𝜔)𝐷(𝜃_−𝑡𝜔)⊂𝐵(𝜔)forall𝑡≥𝑇_𝐷(𝜔).(vii)A random set 𝐴(𝜔) is said to be a random attracting set if for any 𝐷(𝜔)∈𝒟(𝐻), we have lim_𝑡→∞dist_𝐻𝑆𝑡,𝜃_−𝑡𝜔𝐷𝜃_−𝑡𝜔,𝐴(𝜔)=0,a.e.𝜔∈Ω,(2.6) in which dist_𝐻 is the Hausdorff semidistance defined via dist_𝐻(𝐸,𝐹)=sup_𝑢∈𝐸inf_𝑣∈𝐹‖𝑢−𝑣‖_𝜌 for any 𝐸,𝐹⊂𝑙²_𝜌.(viii)A random compact set𝐴(𝜔) is said to be a random global 𝒟 attractor if it is a compact random attracting set and𝑆(𝑡,𝜔)𝐴(𝜔)=𝐴(𝜃_𝑡𝜔) 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 𝜀>0, there exist 𝑇(𝜀,𝜔,𝐷(𝜔))>0 and 𝐼(𝜀,𝜔,𝐷(𝜔))∈ℕ such that ⎛⎜⎝_{|𝑖|>𝐼(𝜀}_{,𝜔,𝐷(𝜔))}𝜌_𝑖||𝑆𝑡,𝜃_−𝑡𝜔𝑢𝜃_−𝑡𝜔)_𝑖||²⎞⎟⎠^1/2≤𝜀,∀𝑡≥𝑇(𝜀,𝜔,𝐷(𝜔)),∀𝑢(𝜔)∈𝐷(𝜔).(2.7)

Theorem 2.2 (see [8]). Let {𝑆(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω} be a continuous RDS over (Ω,ℱ,ℙ,(𝜃_𝑡)_𝑡∈ℝ) with state space 𝐻 and suppose that(a)there exists a random bounded closed absorbing set 𝐵(𝜔)∈𝒟(𝐻) such that for a.e. 𝜔∈Ω and any 𝐷(𝜔)∈𝒟(𝐻), there exists 𝑇_𝐷(𝜔)>0 yielding 𝑆(𝑡,𝜃_−𝑡𝜔)𝐷(𝜃_−𝑡𝜔)⊂𝐵(𝜔) for all 𝑡≥𝑇_𝐷(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 𝜀>0, there exist 𝑇(𝜀,𝜔,𝐵(𝜔))>0 and 𝐼(𝜀,𝜔,𝐵(𝜔))∈ℕ such thatsup_{𝑢∈𝐵(𝜔)}_{|𝑖|>𝐼(𝜀,𝜔,𝐵(𝜔))}𝜌_𝑖||𝑆𝑡,𝜃_−𝑡𝜔𝑢𝜃_−𝑡𝜔_𝑖||²≤𝜀²,∀𝑡≥𝑇(𝜀,𝜔,𝐵(𝜔)).(2.8)
Then the RDS {𝑆(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω} possesses a unique global random 𝒟 attractor 𝐴(𝜔) given by 𝐴(𝜔)=_𝜏≥_{𝑇𝐵(𝜔)}_𝑡≥𝜏𝑆𝑡,𝜃_−𝑡𝜔𝐵𝜃_−𝑡𝜔.(2.9)

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=𝑙²_𝜌×𝑙²_𝜌. 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 function 𝜌∶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

Define Ω₁={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 (𝜃_𝑡)_𝑡∈ℝ on Ω₁ via 𝜃_𝑡𝜔(⋅)=𝜔(⋅+𝑡)−𝜔(𝑡) for 𝑡∈ℝ, then (Ω₁,ℱ₁,ℙ₁,(𝜃_𝑡)_𝑡∈ℝ) is a metric dynamical system.

Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise:𝑑𝑢=−𝜆𝑢+𝐴𝜃_𝑡𝜔𝑢−𝑓(𝑢)−𝛼𝑣+ℎ𝑑𝑡+𝑢∘𝑑𝑤(𝑡),𝑑𝑣=(−𝜎𝑣+𝜇𝑢+𝑔)+𝑢∘𝑑𝑤(𝑡),𝑖∈ℤ,𝑡>0,(3.1) where 𝑢=(𝑢_𝑖)_𝑖∈ℤ, 𝑣=(𝑣_𝑖)_𝑖∈ℤ; 𝑓(𝑢)=(𝑓_𝑖(𝑢_𝑖))_𝑖∈ℤ with 𝑓_𝑖∈𝐶¹(ℝ,ℝ) (𝑖∈ℤ), 𝑔=(𝑔_𝑖)_𝑖∈ℤ, ℎ=(ℎ_𝑖)_𝑖∈ℤ; 𝜆,𝛼,𝜎,𝜇 are positive constants; 𝜂_𝑖,−𝑞(𝜔),…,𝜂_𝑖,0(𝜔),…,𝜂_𝑖,+𝑞(𝜔)(𝑞∈ℕ) are random variables on the probability space (Ω₁,ℱ₁,ℙ₁); 𝐴(⋅) is a linear operator on 𝑙²_𝜌 defined by(𝐴(⋅)𝑢)_𝑖=^𝑞_𝑗_=−𝑞𝜂_𝑖,𝑗(⋅)𝑢_𝑖+𝑗;(3.2)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. Let𝛿𝜃_𝑡𝜔=−⁰_−∞𝑒^𝑠𝜃_𝑡𝜔(𝑠)𝑑𝑠,𝑡∈ℝ,𝜔∈Ω₁,(3.3) then 𝛿(𝜃_𝑡𝜔) is an Ornstein-Uhlenbeck process on (Ω₁,ℱ₁,ℙ₁,(𝜃_𝑡)_𝑡∈ℝ) that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details)𝑑𝛿+𝛿𝑑𝑡=𝑑𝑤(𝑡),𝑡≥0,(3.4) 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 a 𝜃_𝑡-invariant set Ω₁∈ℱ₁ of Ω₁ offull ℙ₁ measure such that for 𝜔∈Ω₁, one has(i)the random variable |𝛿(𝜔)| is tempered;(ii)the mapping𝛿(𝜃_𝑡𝜔)(𝑡,𝜔)⟼𝛿𝜃_𝑡𝜔=−⁰_−∞𝑒^𝑠𝜔(𝑡+𝑠)d𝑠+𝑤(𝑡)(3.5) is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)lim_𝑡→±∞||𝛿𝜃_𝑡𝜔||𝑡=lim_𝑡→±∞1𝑡^𝑡₀𝛿𝜃_𝑠𝜔𝑑𝑠=0.(3.6)

The mapping of 𝜃 on Ω₁ possesses same properties as the original one if we choose the trace 𝜎-algebra with respect to Ω₁ to be denoted also by ℱ₁. Therefore we can change our metric dynamical system with respect to Ω₁, still denoted by the symbols (Ω₁,ℱ₁,ℙ₁,(𝜃_𝑡)_𝑡∈ℝ).

Let𝑥(𝑡,𝜔)=𝑒^{−𝛿(𝜃𝑡𝜔)}𝑢(𝑡,𝜔),𝑦(𝑡,𝜔)=𝑒^{−𝛿(𝜃𝑡𝜔)}𝑣(𝑡,𝜔),𝜔∈Ω₁,(3.7) 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:𝑑𝑥𝑑𝑡=−𝜆𝑥+𝐴𝜃_𝑡𝜔𝑥−𝑒^{−𝛿(𝜃𝑡𝜔)}𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥+𝛿𝜃_𝑡𝜔𝑥−𝛼𝑦+𝑒^{−𝛿(𝜃𝑡𝜔)}ℎ,𝑑𝑦𝑑𝑡=−𝜎𝑦+𝛿𝜃_𝑡𝜔𝑦+𝜇𝑥+𝑒^{−𝛿(𝜃𝑡𝜔)}𝑔𝑡>0,(3.8) Letting 𝐳=(𝑥,𝑦), (3.8) are equivalent to𝑑𝐳𝑑𝑡=𝐅𝐳,𝜃_𝑡𝜔,𝑡>0,(3.9) where 𝐅𝐳,𝜃_𝑡𝜔=⎛⎜⎜⎝−𝜆𝑥+𝐴𝜃_𝑡𝜔𝑥−𝑒^{−𝛿(𝜃𝑡𝜔)}𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥+𝛿𝜃_𝑡𝜔𝑥−𝛼𝑦+𝑒^{−𝛿(𝜃𝑡𝜔)}ℎ−𝜎𝑦+𝛿𝜃_𝑡𝜔𝑦+𝜇𝑥+𝑒^{−𝛿(𝜃𝑡𝜔)}𝑔⎞⎟⎟⎠.(3.10)

We now make the following standing assumptions on 𝑓_𝑖, 𝑔_𝑖, ℎ_𝑖, and 𝜂_𝑖,𝑗,(𝑗=−𝑞,…,𝑞)𝑖∈ℤ and study in the following subsections asymptotic behavior of system (3.9).(H1) 𝑔=(𝑔_𝑖)_𝑖∈ℤ, ℎ=(ℎ_𝑖)_𝑖∈ℤ∈𝑙²_𝜌. (H2)Let𝜂(𝜔)=sup||𝜂_𝑖,−𝑞(𝜔)||,…,||𝜂_𝑖,0(𝜔)||,…,||𝜂_𝑖,+𝑞(𝜔)||∶𝑖∈ℤ≥0,𝑞∈ℕ.(3.11)𝜂(𝜃_𝑡𝜔)(<∞)  belongs to 𝐿¹_loc(ℝ)  with respect to 𝑡∈ℝ  for each 𝜔∈Ω₁. 𝐄(𝜂)=lim_𝑡→±∞1𝑡^𝑡₀𝜂𝜃_𝑡𝜔𝑑𝑠<∞,(3.12)and 𝜂(𝜔)  is tempered, that is, there exists a 𝜃_𝑡-invariant set Ω₁₀∈ℱ₁ of full ℙ₁  measure such that for 𝜔∈Ω₁₀, lim_𝑡→+∞𝑒^−𝛽𝑡sup_𝑡∈ℝ||𝜂𝜃_−𝑡𝜔||=0,∀𝛽>0.(3.13)In the following, we will consider 𝜔∈Ω₁₀∩Ω₁ and still write Ω₁₀∩Ω₁ as Ω₁.(H3)min{𝜆,𝜎}>𝑞𝐄̃|𝜂(𝜔)|=lim_𝑡→±∞(1/𝑡)∫^𝑡₀(𝑞+∑^𝑞_𝑘=0𝑐^𝑘)𝜂(𝜃_𝑡𝜔)𝑑𝑠, where 𝑞̃=𝑞+∑^𝑞_𝑘=0𝑐^𝑘.(H4)There exists a function 𝑅∈𝐶(ℝ⁺,ℝ⁺) such thatsup_𝑖∈ℤmax_{𝑠∈[−𝑟,𝑟]}||𝑓^_𝑖(𝑠)||≤𝑅(𝑟),∀𝑟∈ℝ⁺.(3.14)(H5)𝑓_𝑖∈𝐶¹(ℝ,ℝ), 𝑓_𝑖(0)=0, 𝑠𝑓_𝑖(𝑠)≥−𝑏²_𝑖, 𝑏=(𝑏_𝑖)_𝑖∈ℤ∈𝑙²_𝜌, and there exists a constant 𝑎≥0 such that 𝑓^_𝑖(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 call 𝐳∶[0,𝑇)→𝐻 a solution of the following random differential equation 𝑑𝐳𝑑𝑡=𝐅𝐳,𝜃_𝑡𝜔,𝐳=𝐳_𝑖_𝑖∈ℤ,𝐅=𝐅_𝑖_𝑖∈ℤ,(3.15) where 𝜔∈Ω₀, if 𝐳∈𝐶([0,𝑇),𝐻) satisfies 𝐳_𝑖(𝑡)=𝐳_𝑖(0)+^𝑡₀𝐅_𝑖𝐳(𝑠),𝜃_𝑠𝜔𝑑𝑠,for𝑖∈ℤ,𝑡∈[0,𝑇).(3.16)

Theorem 3.3. Let 𝑇>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) Denote 𝐸=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), ‖𝐴(𝜔)𝑥‖=⎡⎢⎣_𝑖∈ℤ⎛⎜⎝^𝑞_𝑗_=−𝑞𝜂_𝑖,𝑗(𝜔)𝑥_𝑖+𝑗⎞⎟⎠²⎤⎥⎦^1/2≤(2𝑞+1)𝜂(𝜔)⋅‖𝑥‖.(3.17) By (H4), ‖‖𝑓𝑒^𝛿(𝜔)𝑥‖‖≤max𝑅𝑒^𝛿(𝜔)‖𝑥‖,𝑎⋅𝑒^𝛿(𝜔)‖𝑥‖,(3.18) and therefore ‖𝐅(𝐳,𝜔)‖_𝐸≤𝜆+(2𝑞+1)𝜂(𝜔)+max𝑅𝑒^𝛿(𝜔)‖𝑥‖,𝑎+||𝛿(𝜔)||+𝜇⋅‖𝑥‖+𝛼+𝜎+||𝛿(𝜔)||⋅‖𝑦‖+||𝑒^{−𝛿(𝜔)}||⋅(‖ℎ‖+‖𝑔‖).(3.19) For any 𝐳⁽¹⁾=(𝑥⁽¹⁾,𝑦⁽¹⁾), 𝐳⁽²⁾=(𝑥⁽²⁾,𝑦⁽²⁾)∈𝐸, and for some 𝜗∈(0,1)‖‖𝑓𝑒^𝛿(𝜔)𝑥⁽¹⁾−𝑓𝑒^𝛿(𝜔)𝑥⁽²⁾‖‖≤max𝑅𝑒^𝛿(𝜔)(1−𝜗)‖‖𝑥⁽¹⁾‖‖+𝜗‖‖𝑥⁽²⁾‖‖,𝑎⋅𝑒^𝛿(𝜔)‖‖𝑥⁽¹⁾−𝑥⁽²⁾‖‖.(3.20) Also ‖‖𝐴(𝜔)𝑥⁽¹⁾−𝐴(𝜔)𝑥⁽²⁾‖‖=_𝑖∈ℤ⎧⎨⎩^𝑞_𝑗_=−𝑞𝜂_𝑖,𝑗(𝜔)𝑥⁽¹⁾_𝑖+𝑗−𝑥⁽²⁾_𝑖+𝑗⎫⎬⎭^1/2≤(2𝑞+1)𝜂(𝜔)⋅‖‖𝑥⁽¹⁾−𝑥⁽²⁾‖‖.(3.21) It then follows that ‖‖𝐅𝐳⁽¹⁾,𝜔−𝐅𝐳⁽²⁾,𝜔‖‖_𝐸≤𝛼+𝜎+||𝛿(𝜔)||⋅‖‖𝑦⁽¹⁾−𝑦⁽²⁾‖‖+𝜆+(2𝑞+1)𝜂(𝜔)+𝜇+||𝛿(𝜔)||+𝑒^𝛿(𝜔)max𝑅𝑒^𝛿(𝜔)(1−𝜗)‖‖𝑣⁽¹⁾‖‖+𝜗‖‖𝑣⁽²⁾‖‖,𝑎⋅‖‖𝑥⁽¹⁾−𝑥⁽²⁾‖‖.(3.22) For any compact set 𝐷⊂𝐸 with sup_𝐳∈𝐷‖𝐳‖≤𝑟, defining random variable 𝜁_𝐷(𝜔)≥0 via 𝜁_𝐷(𝜔)=𝜆+(2𝑞+1)𝜂(𝜔)+𝜇+||𝛿(𝜔)||+max𝑅𝑒^𝛿(𝜔)𝑟,𝑎⋅𝑒^𝛿(𝜔)𝑟+𝛼+𝜎+||𝛿(𝜔)||𝑟+||𝑒^{−𝛿(𝜔)}||⋅(‖ℎ‖+‖𝑔‖),(3.23) we have ^𝜏+1_𝜏𝜁_𝐷𝜃_𝑡𝜔𝑑𝑡<∞,∀𝜏∈ℝ,(3.24) and for any 𝐳,𝐳⁽¹⁾,𝐳⁽²⁾∈𝐷, ‖𝐅(𝐳,𝜔)‖_𝐸≤𝜁_𝐷(𝜔),‖‖𝐅𝐳⁽¹⁾,𝜔−𝐅𝐳⁽²⁾,𝜔‖‖_𝐸≤𝜁_𝐷(𝜔)‖‖𝐳⁽¹⁾−𝐳⁽²⁾‖‖_𝐸.(3.25) According to [15, 19, 20], problem (3.9) possesses a unique local solution 𝐳(⋅;𝜔,𝐳₀,𝑔,ℎ)∈𝐶([0,𝑇_max),𝐸) (0<𝑇_max≤𝑇) satisfying the integral equation 𝐳(𝑡)=𝐳₀+^𝑡₀𝐅(𝐳(𝑠),𝜔)𝑑𝑠,for𝑡∈0,𝑇_max.(3.26)
We will next show that 𝑇_max=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 {𝜂^(𝑚)_𝑖,𝑗(𝑡,𝜔)}^∞_𝑚=1 of continuous random process in 𝑡∈ℝ such that lim_𝑚→∞^𝑇₀𝜂^(𝑚)_𝑖,𝑗(𝑠,𝜔)−𝜂_𝑖,𝑗(𝑠,𝜔)𝑑𝑠=0,∀𝑇>0,|||𝜂^(𝑚)_𝑖,𝑗(𝑡,𝜔)|||≤||𝜂_𝑖,𝑗𝜃_𝑡𝜔||≤||𝜂𝜃_𝑡𝜔||,∀𝑡∈ℝ.(3.27)
Consider the random differential equation with initial data 𝐳₀∈𝐸∶𝑑𝐳^(𝑚)𝑑𝑡=⎛⎜⎜⎝−𝜆𝑥^(𝑚)+𝐴_𝑚(𝑡,𝜔)𝑥^(𝑚)−𝑒^{−𝛿(𝜃t𝜔)}𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)+𝛿𝜃_𝑡𝜔𝑥^(𝑚)−𝛼𝑦^(𝑚)+𝑒^{−𝛿(𝜃𝑡𝜔)}ℎ−𝜎𝑦^(𝑚)+𝛿𝜃_𝑡𝜔𝑦^(𝑚)+𝜇𝑥^(𝑚)+𝑒^{−𝛿(𝜃𝑡𝜔)}𝑔⎞⎟⎟⎠,(3.28) where 𝐴_𝑚(𝑡,𝜔)𝑥^(𝑚)_𝑖=^𝑞_𝑗=−𝑞𝜂^(𝑚)_𝑖,𝑗(𝑡,𝜔)𝑥_𝑖+𝑗.(3.29) Follow the same procedure as above, (3.28) has a unique solution 𝐳^(𝑚)(⋅;𝜔,𝐳₀,𝑔,ℎ)∈𝐶([0,𝑇^(𝑚)_max),𝐸), that is, 𝐳^(𝑚)_𝑖(𝑡)=𝐳₀+^𝑡₀𝐅^(𝑚)_𝑖𝐳^(𝑚)(𝑠),𝜔𝑑𝑠,for𝑡∈0,𝑇^(𝑚)_max,(3.30) and by the continuity of 𝐴_𝑚(𝑠,𝜔) in 𝑠, there holds 𝑑𝐳^(𝑚)_𝑖𝑑𝑡=⎛⎜⎜⎝−𝜆𝑥^(𝑚)_𝑖+𝐴_𝑚𝜃_𝑡𝜔𝑥^(𝑚)_𝑖−𝑒^{−𝛿(𝜃𝑡𝜔)}𝑓_𝑖𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖+𝛿𝜃_𝑡𝜔𝑥^(𝑚)_𝑖−𝛼𝑦^(𝑚)_𝑖+𝑒^{−𝛿(𝜃𝑡𝜔)}ℎ_𝑖−𝜎𝑦^(𝑚)_𝑖+𝛿𝜃_𝑡𝜔𝑦^(𝑚)_𝑖+𝜇𝑥^(𝑚)_𝑖+𝑒^{−𝛿(𝜃𝑡𝜔)}𝑔⎞⎟⎟⎠.(3.31) Note that 𝐴_𝑚𝜃_𝑡𝜔𝑥^(𝑚)_𝑖⋅𝑥^(𝑚)_𝑖≤𝜂𝜃_𝑡𝜔⋅|||𝑥^(𝑚)_𝑖𝑥^(𝑚)_𝑖−𝑞+⋯+𝑥^(𝑚)_𝑖𝑥^(𝑚)_𝑖+⋯+𝑥^(𝑚)_𝑖𝑥^(𝑚)_𝑖+𝑞|||,−𝑎𝑒^{2𝛿(𝜃𝑡𝜔)}⋅𝑥^(𝑚)_𝑖²≤𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖⋅𝑓_𝑖𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖≤𝑅𝑒^{𝛿(𝜃𝑡𝜔)}‖‖𝑥^(𝑚)‖‖⋅𝑒^{2𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖²,𝑡∈[0,𝑇],(3.32) multiplying (3.31) by ^{𝜇𝑥(𝑚)𝑖0}_0𝛼𝑦^(𝑚)_𝑖 and sum over 𝑖∈ℤ results in 𝑑𝑑𝑡𝜇‖‖𝑥^(𝑚)‖‖²+𝛼‖‖𝑦^(𝑚)‖‖²≤−min{𝜆,𝜎}+2𝑎+2𝛿𝜃_𝑡𝜔+2(2𝑞+1)𝜂𝜃_𝑡𝜔⋅𝜇‖‖𝑥^(𝑚)‖‖²+𝛼‖‖𝑦^(𝑚)‖‖²+2𝜇𝜆‖ℎ‖²+2𝛼𝜎‖𝑔‖²𝑒^{−2𝛿(𝜃𝑡𝜔)}.(3.33) Applying Gronwall’s inequality to (3.33) we obtain that 𝜇‖‖𝑥^(𝑚)‖‖²+𝛼‖‖𝑦^(𝑚)‖‖²≤𝑒^{2𝑎𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+2(2𝑞+1)𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝜇‖𝑥(0)‖²+𝛼‖𝑦(0)‖²+2𝜇𝜆‖ℎ‖²+2𝛼𝜎‖𝑔‖²𝑒^{2𝑎𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+2(2𝑞+1)𝜂(𝜃𝑠𝜔)]𝑑𝑠}⋅^𝑡₀𝑒^{(min{𝜆,𝜎}−2𝑎)𝑠−2𝛿(𝜃𝑠𝜔)+∫𝑠0[2𝛿(𝜃𝑟𝜔)+2(2𝑞+1)𝜂(𝜃𝑟𝜔)]𝑑𝑟}𝑑𝑠≐𝜅(𝑡),𝑡∈0,𝑇^(𝑚)_max,(3.34) 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), |||𝑥^(𝑚)_𝑖|||²+|||𝑦^(𝑚)_𝑖|||²≤𝜅(𝑡)min{𝜇,𝛼},∀𝑚∈ℕ,𝑡∈[0,𝑇).(3.35) Since |𝐅^(𝑚)_𝑖(𝐳^(𝑚)(𝑡),𝜃_𝑡𝜔)|²≤𝐾²(𝑇,𝜔) for some 𝐾(𝑇,𝜔)>0 and 𝑡∈[0,𝑇), then for any 𝑡,𝜏∈[0,𝑇),𝑚∈ℕ, |||𝐳^(𝑚)_𝑖(𝑡)−𝐳^(𝑚)_𝑖(𝜏)|||=^𝑡_𝜏|||𝐅^(𝑚)_𝑖𝐳^(𝑚)(𝑠),𝜃_𝑠𝜔|||𝑑𝑠≤𝐾(𝑇,𝜔)⋅|𝑡−𝜏|.(3.36) By the Arzela-Acoli Theorem, there exists a convergent subsequence {𝐳^(𝑚𝑘)_𝑖(𝑡),𝑡∈[0,𝑇)} of {𝐳^(𝑚)_𝑖(𝑡),𝑡∈[0,𝑇)} such that 𝐳^(𝑚𝑘)_𝑖(𝑡)⟶𝐳_𝑖(𝑡)as𝑘⟶∞,𝑡∈[0,𝑇),𝑖∈ℤ,(3.37) 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 lim_𝑘→∞^𝑡₀𝜂^(𝑚𝑘)_𝑖,𝑗(𝑠,𝜔)−𝜂_𝑖,𝑗𝜃_𝑠𝜔𝑑𝑠=^𝑡₀lim_𝑘→∞𝜂^(𝑚𝑘)_𝑖,𝑗(𝑠,𝜔)−𝜂_𝑖,𝑗𝜃_𝑠𝜔𝑑𝑠=0,lim_𝑘→∞𝜂^(𝑚𝑘)_𝑖,𝑗(𝑠,𝜔)−𝜂_𝑖,𝑗𝜃_𝑠𝜔=0,fora.e.𝑠∈[0,𝑇],lim_𝑘→∞^𝑡₀𝜂^(𝑚𝑘)_𝑖,𝑗(𝑠,𝜔)𝑥^(𝑚𝑘)_𝑖(𝑠)−𝜂_𝑖,𝑗𝜃_𝑠𝜔𝑥_𝑖(𝑠)𝑑𝑠=0.(3.38) Thus replacing 𝑚 by 𝑚_𝑘 in (3.31) and letting 𝑘→∞ give 𝐳_𝑖(𝑡)=𝐳₀_𝑖+^𝑡₀𝐅_𝑖𝐳_𝑖,𝜃_𝑠𝜔𝑑𝑠for𝑡∈[0,𝑇).(3.39) By the uniquness of the solutions of (3.9), we have 𝐳_𝑖(𝑡)=𝐳_𝑖(𝑡) for 𝑡∈[0,𝑇_max). By (3.34), ‖𝐳(𝑡)‖²_𝐸≤𝜅(𝑡)/min{𝜇,𝛼} for 𝑡∈[0,𝑇_max), which implies that the solution 𝐳(𝑡) of (3.9) exists globally on 𝑡∈[0,𝑇).
(2) Next we prove that for any 𝐳₀∈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 𝐳_1,0,𝐳_2,0∈𝐸 and ℎ₁=(ℎ_1,𝑖)_𝑖∈ℤ, ℎ₂=(ℎ_2,𝑖)_𝑖∈ℤ,𝑔₁=(𝑔_1,𝑖)_𝑖∈ℤ,𝑔₂=(𝑔_2,𝑖)_𝑖∈ℤ∈𝑙². Let 𝐳^(𝑚)₁(𝑡,𝜔),𝐳^(𝑚)₂(𝑡,𝜔) be two solutions of (3.28) with initial data 𝐳_1,0,𝐳_2,0 and ℎ,𝑔 replaced by ℎ₁, ℎ₂, 𝑔₁, 𝑔₂, respectively. Set 𝐝^(𝑚)(𝑡)=𝐳^(𝑚)₁(𝑡)−𝐳^(𝑚)₂(𝑡)=(𝑑^(𝑚)₁(𝑡),𝑑^(𝑚)₂(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), ||||𝐴_𝑚𝜃_𝑡𝜔𝑑^(𝑚)₁,𝑑^(𝑚)₁_𝜌||||≤𝑞𝜂̃𝜃_𝑡𝜔‖‖𝑑^(𝑚)₁‖‖²_𝜌,𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)₁−𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)₂,𝑑^(𝑚)₁_𝜌≥−𝑎𝑒^{𝛿(𝜃𝑡𝜔)}‖‖𝑑^(𝑚)₁‖‖²_𝜌,𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)₁−𝑓𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)₂,𝑑^(𝑚)₁_𝜌≤𝑅𝑒^{𝛿(𝜃𝑡𝜔)}‖‖𝑥^(𝑚)₁‖‖+‖‖𝑥^(𝑚)₂‖‖𝑒^{𝛿(𝜃𝑡𝜔)}‖‖𝑑^(𝑚)₁‖‖²_𝜌,ℎ₁−ℎ₂,𝑑^(𝑚)₁_𝜌≤‖‖ℎ₁−ℎ₂‖‖²_𝜌⋅‖‖𝑑^(𝑚)₁‖‖²_𝜌;𝑔₁−𝑔₂,𝑑^(𝑚)₂_𝜌≤‖‖𝑔₁−𝑔₂‖‖²_𝜌⋅‖‖𝑑^(𝑚)₂‖‖²_𝜌.(3.40) It then follows that 𝑑𝑑𝑡𝜇‖‖𝑑^(𝑚)₁‖‖²_𝜌+𝛼‖‖𝑑^(𝑚)₂‖‖²_𝜌≤−min{𝜆,𝜎}+2𝑎+2𝛿𝜃_𝑡𝜔+𝑞𝜂̃𝜃_𝑡𝜔⋅𝜇‖‖𝑑^(𝑚)₁‖‖²_𝜌+𝛼‖‖𝑑^(𝑚)₂‖‖²_𝜌+2𝜇𝜆‖‖ℎ₁−ℎ₂‖‖²_𝜌+2𝛼𝜎‖‖𝑔₁−𝑔₂‖‖²_𝜌𝑒^{−2𝛿(𝜃𝑡𝜔)}.(3.41) For 𝑇>0, applying Gronwall’s inequality to (3.41) on [0,𝑇] implies that 𝜇‖‖𝑑^(𝑚)₁(𝑡)‖‖²_𝜌+𝛼‖‖𝑑^(𝑚)₂(𝑡)‖‖²_𝜌≤𝐶_𝑇𝜇‖‖𝑑^(𝑚)₁(0)‖‖²_𝜌+𝛼‖‖𝑑^(𝑚)₂(0)‖‖²_𝜌+‖‖ℎ₁−ℎ₂‖‖𝑡²_𝜌+‖‖𝑔₁−𝑔₂‖‖²_𝜌(3.42) for some constant 𝐶_𝑇 depending on 𝑇, and thus ‖‖𝐳^(𝑚)₁(𝑡)−𝐳^(𝑚)₂(𝑡)‖‖²_𝐻≤𝐶_𝑇‖‖𝐳^(𝑚)₁(0)−𝐳^(𝑚)₂(0)‖‖²_𝐻+‖‖ℎ₁−ℎ₂‖‖²_𝜌+‖‖𝑔₁−𝑔₂‖‖²_𝜌,(3.43) where 𝐶_𝑇 is a constant depending on 𝑇. Denote by 𝐸=𝑙²̃‌×𝑙²̃‌, where 𝑙²̃‌=𝑙² with the norm ‖⋅‖_𝜌. By (3.43), there exists a mapping Φ^(𝑚)∈𝐶(𝐸×𝐸,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 𝑙²̃‌ is dense in 𝑙²_𝜌, the mapping Φ^(𝑚) can be extended uniquely to a continuous mapping Φ^(𝑚)∶𝐻×𝐻→𝐶([0,𝑇],𝐻).
For given 𝐳₀∈𝐻 and (𝑔,ℎ)∈𝐻, Φ^(𝑚)(𝐳₀,𝑔,ℎ)=𝐳^(𝑚)(⋅;𝜔,𝐳₀,𝑔,ℎ)∈𝐶([0,𝑇],𝐻) for 𝑇>0. There exist sequences {𝐳_0𝑛}⊂𝐸, {(ℎ_𝑛,𝑔_𝑛)}⊂𝐸 such that ‖‖𝐳_0𝑛−𝐳₀‖‖_𝐸⟶0,‖‖ℎ_𝑛−ℎ‖‖_𝜌⟶0,‖‖𝑔_𝑛−𝑔‖‖_𝜌⟶0as𝑛⟶∞.(3.44) Let 𝐳^(𝑚)_𝑛(𝑡)=Φ^(𝑚)(𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛)=Φ^(𝑚)(𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛)=𝐳^(𝑚)(𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛)∈𝐸 be the solution of (3.28), then it satisfies the integral equation 𝐳^(𝑚)_𝑛_𝑖(𝑡)=𝐳_0𝑛_𝑖+^𝑡₀𝐹_𝑖𝐳^(𝑚)_𝑛,𝜃_𝑠𝜔𝑑𝑠.(3.45) By the continuity of Φ^(𝑚), we have for 𝑡∈[0,𝑇), 𝐳^(𝑚)𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛=Φ^(𝑚)𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛⟶^𝑛→∞Φ^(𝑚)𝐳₀,ℎ,𝑔=𝐳^(𝑚)𝑡;𝜔,𝐳₀,ℎ,𝑔∈𝐻.(3.46) Thus for each 𝑖∈ℤ, 𝐳^(𝑚)_𝑛_𝑖(𝑡)⟶𝐳^(𝑚)_𝑖(𝑡)∶=𝐳^(𝑚)_𝑖𝑡;𝜔,𝐳₀,ℎ,𝑔as𝑛⟶∞uniformlyon𝑡∈[0,𝑇).(3.47) Moreover, {(𝐳^(𝑚)_𝑛)_𝑖(𝑡)} is bounded in 𝑛. Let 𝑛→∞, then we have 𝐳^(𝑚)_𝑖(𝑡)=𝐳₀_𝑖+^𝑡₀𝐅_𝑖𝐳^(𝑚),𝜃_𝑠𝜔𝑑𝑠,(3.48) and 𝐳^(𝑚)_𝑖(𝑡) satisfies the differential equation (3.31).
Multiply equation (3.31) by ^{𝜇𝜌𝑖𝑥(𝑚)𝑖0}_{0𝛼𝜌𝑖𝑦}^(𝑚)_𝑖 and sum over 𝑖∈ℤ, we obtain 𝑑𝑑𝑡𝜇‖‖𝑥^(𝑚)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)‖‖²_𝜌≤−min{𝜆,𝜎}+2𝑎+2𝛿𝜃_𝑡𝜔+𝑞𝜂̃𝜃_𝑡𝜔⋅𝜇‖‖𝑥^(𝑚)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)‖‖²_𝜌+2𝜇𝜆‖ℎ‖²_𝜌+2𝛼𝜎‖𝑔‖²_𝜌𝑒^{−2𝛿(𝜃𝑡𝜔)},𝜇‖‖𝑥^(𝑚)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)‖‖²_𝜌≤𝑒^{2𝑎𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝜇‖𝑥(0)‖²_𝜌+𝛼‖𝑦(0)‖²_𝜌+2𝜇𝜆‖ℎ‖²_𝜌+2𝛼𝜎‖𝑔‖²_𝜌𝑒^{2𝑎𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}⋅^𝑡₀𝑒^{(𝜆+𝜎−2𝑎)𝑠−2𝛿(𝜃𝑠𝜔)+∫𝑠0[2𝛿(𝜃𝑟𝜔)+}𝑞̃_{𝜂(𝜃𝑟𝜔)]𝑑𝑟}𝑑𝑠≐𝜅_𝜌(𝑡),𝑡∈0,𝑇_max(3.49) 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,(𝜃_𝑡)_𝑡∈ℝ) with state space 𝐻: 𝜓(𝑡,𝜔)𝐳₀∶=𝐳𝑡;𝜔,𝐳₀for𝐳₀∈𝐻,𝑡≥0,𝜔∈Ω₁.(3.50) Moreover, 𝜑(𝑡,𝜔)𝑢₀,𝑣₀∶=𝑒^{𝛿(𝜃𝑡𝜔)}𝜓(𝑡,𝜔)𝑒^{−𝛿(𝜔)}𝑢₀,𝑣₀for𝑢₀,𝑣₀∈𝐻,𝑡≥0,𝜔∈Ω₁,(3.51) defines a continuous RDS {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω1} over (Ω₁,ℱ₁,ℙ₁,(𝜃_𝑡)_𝑡∈ℝ) associated with (3.1).

Proof. By Theorem 3.3, 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, for 𝐳₀∈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.
For 𝐳₀∈𝐻 and (ℎ,𝑔)∈𝐻, the solution 𝐳(𝑡;𝜔,𝐳₀,ℎ,𝑔)∈𝐻 for 𝑡∈[0,∞). For any given 𝑁>0, define 𝑇_𝑁∶𝐻→𝐸, (𝑢,𝑣)=((𝑢_𝑖),(𝑣_𝑖))_𝑖∈ℤ→𝑇_𝑁(𝑢,𝑣)=((𝑇_𝑁(𝑢,𝑣))_𝑖)_𝑖∈ℤ by 𝑇_𝑁(𝑢,𝑣)_𝑗=𝑢_𝑗,𝑣_𝑗if||𝑗||≤𝑁,0if||𝑗||>𝑁,(3.52) and write 𝐳_𝑁𝑡;𝜔,𝐳₀,ℎ,𝑔=𝐳𝑡;𝜔,𝑇_𝑁𝐳₀,𝑇_𝑁(ℎ,𝑔).(3.53) Then 𝑇_𝑁 is continuous and for any 𝐳₀∈𝐻,(ℎ,𝑔)∈𝐻, and 𝐳𝑡;𝜔,𝐳₀,ℎ,𝑔=lim_𝑁→∞𝐳𝑡;𝜔,𝑇_𝑁𝐳₀,𝑇_𝑁(ℎ,𝑔).(3.54) Thus 𝐳∶[0,∞)×Ω₁×𝐸×𝐸→𝐻 is (ℬ([0,∞))×ℱ₀×ℬ(𝐸)×ℬ(𝐸),ℬ(𝐻))-measurable. Observe also that (Id,Id,𝑇_𝑁,𝑇_𝑁)∶[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 𝐳_𝑁=𝐳∘(Id,Id,𝑇_𝑁,𝑇_𝑁)∶[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,(𝜃_𝑡)_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 set 𝐵_1𝜌(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 𝑇_𝐷(𝜔)>0 yielding 𝜑(𝑡,𝜃_−𝑡𝜔)𝐷(𝜃_−𝑡𝜔)⊂𝐵_1𝜌(𝜔) for all 𝑡≥𝑇_𝐷(𝜔). In particular, there exists 𝑇_1𝜌(𝜔)>0 such that 𝜑(𝑡,𝜃_−𝑡𝜔)𝐵_1𝜌(𝜃_−𝑡𝜔)⊂𝐵_1𝜌(𝜔) for all𝑡≥𝑇_1𝜌(𝜔).

Proof. (1) For initial condition 𝐳₀∈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 𝜆₁=2min{𝜆,𝜎}−𝜖₁>2𝑞𝐄̃||𝜂(𝜔)||.(3.55) By (H4) and (H5), we have −‖𝑏‖²_𝜌≤∑_𝑖∈ℤ𝜌_𝑖𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖⋅𝑓_𝑖𝑒^{𝛿(𝜃𝑡𝜔)}𝑥^(𝑚)_𝑖<∞,forfixed𝑡≥0,(3.56)𝑑𝑑𝑡𝜇‖‖𝑥^(𝑚)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)‖‖²_𝜌≤−𝜆₁+2𝛿𝜃_𝑡𝜔+2𝑞𝜂̃𝜃_𝑡𝜔⋅𝜇‖‖𝑥^(𝑚)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)‖‖²_𝜌+2𝑒^{−2𝛿(𝜃𝑡𝜔)}𝜇‖𝑏‖²_𝜌+𝜇𝜖₁‖ℎ‖²_𝜌+𝛼𝜎‖𝑔‖²_𝜌.(3.57) Applying Gronwall’s inequality to (3.57), we obtain that for 𝑡>0, 𝜇‖‖𝑥^(𝑚)(𝑡,𝜔)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝑡,𝜔)‖‖²_𝜌≤𝑒^{−𝜆1𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝜇‖𝑥(0)‖²_𝜌+𝛼‖𝑦(0)‖²_𝜌+2𝜇‖𝑏‖²_𝜌+𝜇𝜖₁‖ℎ‖²_𝜌+𝛼𝜎‖𝑔‖²_𝜌𝑒^{−𝜆1𝑡+∫𝑡0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}⋅^𝑡₀𝑒^{𝜆1𝑠−2𝛿(𝜃𝑠𝜔)−∫𝑠0[2𝛿(𝜃𝑟𝜔)+2}𝑞̃_{𝜂(𝜃𝑟𝜔)]𝑑𝑟}𝑑𝑠.(3.58)
(2) For any 𝐳₀∈𝐻 and (ℎ,𝑔)∈𝐻, let {𝐳_0𝑛}⊂𝐸 and {(ℎ_𝑛,𝑔_𝑛)}⊂𝐸 be sequences such that ‖‖𝐳_0𝑛−𝐳₀‖‖_𝐸⟶0,‖‖ℎ_𝑛−ℎ‖‖_𝜌⟶0,‖‖𝑔_𝑛−𝑔‖‖_𝜌⟶0as𝑛⟶∞.(3.59) Then 𝐳^(𝑚)(𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛)→𝐳^(𝑚)(𝑡;𝜔,𝐳₀,ℎ,𝑔) as 𝑛→∞ in 𝐻, and (3.58) holds for 𝐳₀∈𝐻. Therefore, 𝜇‖‖𝑥^(𝑚)𝑡,𝜃_−𝑡𝜔,𝐳₀𝜃_−𝑡𝜔‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)𝑡,𝜃_−𝑡𝜔,𝐳₀𝜃_−𝑡𝜔‖‖²_𝜌≤𝑒^{−𝜆1𝑡+∫𝑡0[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{𝜂(𝜃𝑠−𝑡𝜔)]𝑑𝑠}𝜇‖‖𝑥₀𝜃_−𝑡𝜔‖‖²_𝜌+𝛼‖‖𝑦₀𝜃_−𝑡𝜔‖‖²_𝜌+12𝑟²_1𝜌(𝜔)𝑒^{−2𝛿(𝜔)},(3.60) where 𝑟²_1𝜌(𝜔)=4𝑒^2𝛿(𝜔)𝜇‖𝑏‖²_𝜌+𝜇𝜖₁‖ℎ‖²_𝜌+𝛼𝜎‖𝑔‖²_𝜌⁰_−∞𝑒^{𝜆1𝑠−2𝛿(𝜃𝑠𝜔)+∫0𝑠[2𝛿(𝜃𝑟𝜔)+2}𝑞̃_{𝜂(𝜃𝑟𝜔)]𝑑𝑟}𝑑𝑠<∞.(3.61) For any 𝛽>0, since 𝑒^−𝛽𝑡𝑟_0𝜌𝜃_−𝑡𝜔²𝜇‖𝑏‖²_𝜌+𝜇𝜖₁‖ℎ‖²_𝜌+𝛼𝜎‖𝑔‖²_𝜌⁻¹=4𝑒^{−2𝛽𝑡+𝛿(𝜃−𝑡𝜔)}^−𝑡_−∞𝑒^{𝜆1(𝑠+𝑡)−2𝛿(𝜃𝑠𝜔)+∫−𝑡𝑠[2𝛿(𝜃𝑟𝜔)+2}𝑞̃_{|𝜂(𝜃𝑟𝜔)|]𝑑𝑟}𝑑𝑠⟶^𝑡→+∞0,(3.62) then 𝑟_1𝜌(𝜔) is tempered.
Let 𝐳(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 𝑚_𝑘→∞ 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), 𝐳₀𝜃_−𝑡𝜔=𝑥₀𝜃_−𝑡𝜔,𝑦₀𝜃_−𝑡𝜔=𝑒^{−𝛿(𝜃−𝑡𝜔)}𝑢₀,𝑣₀,𝜇‖‖𝑢𝑡,𝜃_−𝑡𝜔,𝑢₀,𝑣₀‖‖²_𝜌+𝛼‖‖𝑣𝑡,𝜃_−𝑡𝜔,𝑢₀,𝑣₀‖‖²_𝜌≤𝑒^2𝛿(𝜔)𝑒^{−𝜆1𝑡−2𝛿(𝜃−𝑡𝜔)+∫𝑡0[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{𝜂(𝜃𝑠−𝑡𝜔)]𝑑𝑠}sup_{(𝑢,𝑣)∈𝐷𝜃−𝑡𝜔}𝜇‖𝑢‖²_𝜌+𝛼‖𝑣‖²_𝜌+12𝑟²_1𝜌(𝜔).(3.63) Let 𝜖₂=𝜆₁−2𝑞𝐄̃|𝜂(𝜔)|>0. By properties of 𝜂(𝜃_±𝑡𝜔) and 𝐷∈𝒟(𝐻), we have 𝑒^{−[𝜖2/2+2}𝑞̃_{𝐄|𝜂(𝜔)|]𝑡−𝛿(𝜃−𝑡𝜔)+∫}⁰_{−𝑡[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}⟶0as𝑡⟶+∞,(3.64) and hence lim_𝑡→+∞𝑒^2𝛿(𝜔)𝑒^{−𝜆1𝑡−𝛿(𝜃−𝑡𝜔)+∫0−𝑡[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}sup_{(𝑢,𝑣)∈𝐷𝜃−𝑡𝜔}𝜇‖𝑢‖²_𝜌+𝛼‖𝑣‖²_𝜌=0.(3.65) Denote by 𝑟̃_1𝜌=𝑟_1𝜌/√min{𝜇,𝛼}, it follows that 𝐵_1𝜌(𝜔)=(𝑢,𝑣)∈𝐻∶‖(𝑢,𝑣)‖_𝐻≤𝑟̃_1𝜌(𝜔)=ℬ_𝐻0,𝑟̃_1𝜌(𝜔)⊂𝐻(3.66) 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 𝐴₁(𝜔)=_𝜏≥𝑇_{𝐵1𝜌(𝜔)}_𝑡≥𝜏𝜑𝑡,𝜃_−𝑡𝜔𝐵_1𝜌𝜃_−𝑡𝜔⊂𝐻.(3.67)

Proof. According to Theorem 2.2, it remains to prove the asymptotically nullness of {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω1}; that is, for any 𝜀>0, there exists 𝑇(𝜀,𝜔,𝐵_1𝜌)>𝑇_1𝜌(𝜔) and 𝐼(𝜀,𝜔)∈ℕ such that when 𝑡≥𝑇(𝜀,𝜔,𝐵_1𝜌), the solution 𝜑(𝑡,𝜔)(𝑢₀,𝑣₀)=((𝑢_𝑖,𝑣_𝑖)(𝑡;𝜔,𝑢₀,𝑣₀))_𝑖∈ℤ∈𝐻 of (3.1) with (𝑢₀,𝑣₀)∈𝐵_1𝜌(𝜃_−𝑡𝜔) satisfies _|𝑖|≥_{𝐼(𝜀,𝜔)}𝜌_𝑖||𝜑(𝑡,𝜔)𝑢₀,𝑣₀_𝑖||²=_{|𝑖|≥𝐼(𝜀,𝜔)}𝜌_𝑖||𝑢_𝑖,𝑣_𝑖𝑡;𝜔,𝑢₀,𝑣₀||²≤𝜀.(3.68) Choose a smooth increasing function 𝜉∈𝐶¹(ℝ⁺,[0,1]) such that 𝜉(𝑠)=0,0≤𝑠≤1,0≤𝜉(𝑠)≤1,1≤𝑠≤2,||𝜉^(𝑠)||≤𝐶₀(constant)for𝑠∈ℝ⁺,𝜉(𝑠)=1,𝑠≥2.(3.69) Let (𝑢,𝑣)(𝑡;𝜔,𝑢₀,𝑣₀,ℎ,𝑔)=((𝑢_𝑖,𝑣_𝑖)(𝑡;𝜔,𝑢₀,𝑣₀,ℎ,𝑔))_𝑖∈ℤ be a solution of (3.1), then 𝐳𝑡;𝜔,𝐳₀(𝜔),ℎ,𝑔=𝐳_𝑖𝑡;𝜔,𝐳₀(𝜔),ℎ,𝑔_𝑖∈ℤ=𝑒^{−𝛿(𝜃𝑡𝜔)}𝜑(𝑡,𝜔)𝑢₀,𝑣₀(3.70) is a solution of (3.9) with 𝐳₀(𝜔)=𝑒^{−𝛿(𝜔)}(𝑢₀,𝑣₀)∈𝐻.
Let 𝐳_0𝑛=𝑇_𝑛𝐳₀,(ℎ_𝑛,𝑔_𝑛)=𝑇_𝑛(ℎ,𝑔), where 𝑇_𝑛 is as in (3.52). Then 𝐳_0𝑛∈𝐸,(ℎ_𝑛,𝑔_𝑛)∈𝐸 and 𝐳(𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛)→𝐳(𝑡;𝜔,𝐳₀,ℎ,𝑔) in 𝐻. For any 𝑛≥1, let 𝐳^(𝑚)(𝑡)=𝐳^(𝑚)(𝑡;𝜔,𝐳_0𝑛(𝜔),ℎ_𝑛,𝑔_𝑛) be the solution of (3.28), where 𝐳^(𝑚)(0)=𝐳_0𝑛(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)𝑥(𝑚)𝑖0}_{0𝛼𝜌𝑖𝜉(|𝑖|/𝑀)𝑦}^(𝑚)_𝑖 and sum over 𝑖∈ℤ, we obtain 𝑑𝑑𝑡_𝑖∈ℤ𝜉|𝑖|𝑀𝜌_𝑖𝜇𝑥^(𝑚)_𝑖²+𝛼𝑦^(𝑚)_𝑖²≤−𝜆₁+2𝛿𝜃_𝑡𝜔+2𝑞𝜂̃𝜃_𝑡𝜔⋅_𝑖∈ℤ𝜉|𝑖|𝑀𝜌_𝑖𝜇𝑥^(𝑚)_𝑖²+𝛼𝑦^(𝑚)_𝑖²+𝜂𝜃_𝑡𝜔⋅𝑐𝐶₀𝑀‖‖𝑥^(𝑚)‖‖²_𝜌+2𝑒^{−2𝛿(𝜃𝑡𝜔)}_|𝑖_|≥𝑀𝜌_𝑖𝜇𝑏²_𝑖+𝜇𝜖₁ℎ_𝑛²_𝑖+𝛼𝜎𝑔_𝑛²_𝑖.(3.71) Applying Gronwall’s inequality to (3.71) from 𝑇_1𝜌=𝑇_𝐵1𝜌(𝜔) to 𝑡 gives _𝑖∈ℤ𝜉|𝑖|𝑀𝜌_𝑖𝜇𝑥^(𝑚)_𝑖𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²+𝛼𝑦^(𝑚)_𝑖𝑡;𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²≤𝑒^{−𝜆1(𝑡−𝑇1𝜌)+∫𝑡𝑇1𝜌[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}⋅𝜇‖‖𝑥^(𝑚)𝑇_1𝜌‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝑇_1𝜌)‖‖²_𝜌+_{|𝑖|≥𝑀}𝜌_𝑖𝜇𝑏²_𝑖+𝜇𝜖₁ℎ_𝑛²_𝑖+𝛼𝜎𝑔_𝑛²_𝑖^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠−2𝛿(𝜃𝜏𝜔)}𝑑𝜏+𝑐𝐶₀𝑀^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{|𝜂(𝜃𝑠𝜔)|]𝑑𝑠−2𝛿(𝜃𝜏𝜔)}𝜂𝜃_𝜏𝜔𝜇‖‖𝑥^(𝑚)(𝜏)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝜏)‖‖²_𝜌𝑑𝜏.(3.72) Therefore for (𝑢₀,𝑣₀)∈𝐵_1𝜌(𝜃_−𝑡𝜔)∩𝐻, _𝑖∈ℤ𝜉|𝑖|𝑀𝜌_𝑖𝜇𝑥^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²+𝛼𝑦^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²≤𝑒^{−𝜆1(𝑡−𝑇1𝜌)+∫𝑡𝑇1𝜌[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{|𝜂(𝜃𝑠−𝑡𝜔)|]𝑑𝑠}𝜇‖‖𝑥^(𝑚)(𝑇_1𝜌)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝑇_1𝜌)‖‖²_{𝜌(𝑖)}+_{|𝑖|≥𝑀}𝜌_𝑖𝜇𝑏²_𝑖+𝜇𝜖₁ℎ_𝑛²_𝑖+𝛼𝜎𝑔_𝑛²_𝑖^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{|𝜂(𝜃𝑠−𝑡𝜔)|]𝑑𝑠−2𝛿(𝜃𝜏−𝑡𝜔)}𝑑𝜏_{(𝑖𝑖)}+𝑐𝐶₀𝑀^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{|𝜂(𝜃𝑠−𝑡𝜔)|]𝑑𝑠−2𝛿(𝜃𝜏−𝑡𝜔)}𝜂𝜃_𝜏−𝑡𝜔𝜇‖‖𝑥^(𝑚)(𝜏)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝜏)‖‖²_𝜌𝑑𝜏_{(𝑖𝑖𝑖)}.(3.73) We next estimate terms (𝑖),(𝑖𝑖),(𝑖𝑖𝑖) on the right-hand side of (3.73). By (3.61), 𝑒^{−𝜆1(𝑡−𝑇1𝜌)+∫𝑡𝑇1𝜌[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{𝜂(𝜃𝑠−𝑡𝜔)]𝑑𝑠}𝜇‖‖𝑥^(𝑚)𝑇_1𝜌‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)𝑇_1𝜌‖‖²_𝜌⟶0as𝑡⟶+∞,(3.74) which implies that for all 𝜀>0, there exists 𝑇₁(𝜀,𝜔,𝐵_1𝜌)≥𝑇_1𝜌 such that for 𝑡≥𝑇₁(𝜀,𝜔,𝐵_1𝜌), (𝑖)≤𝜀3min{𝜇,𝛼}𝑒^{−2𝛿(𝜔)}.(3.75) By ℎ,𝑔,𝑏∈𝑙²_𝜌 and ^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{𝜂(𝜃𝑠−𝑡𝜔)]𝑑𝑠−2𝛿(𝜃𝜏−𝑡𝜔)}𝑑𝜏≤⁰_−∞𝑒^{𝜆1𝜏−∫0𝜏[𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠−2𝛿(𝜃𝜏𝜔)}𝑑𝜏<∞,(3.76) there exists 𝐼₁(𝜀,𝜔)∈ℕ such that for 𝑀>𝐼₁(𝜀,𝜔), (𝑖𝑖)≤𝜀3min{𝜇,𝛼}𝑒^{−2𝛿(𝜔)}.(3.77)
Note that 𝜖₂=𝜆₁−2𝑞𝐄̃|𝜂(𝜔)|>0, then by (H2), 𝜂(𝜔) is tempered and it follows that there exists a 𝑇^₁>0 such that 𝜂𝜃_𝜏−𝑡𝜔≤𝑒^{(𝜖2/3)(𝑡−𝜏)},∀𝑡−𝜏≥𝑇^₁.(3.78) Therefore ^𝑡_𝑇1𝜌𝑒^{−𝜆1(𝑡−𝜏)+∫𝑡𝜏[2𝛿(𝜃𝑠−𝑡𝜔)+2}𝑞̃_{𝜂(𝜃𝑠−𝑡𝜔)]𝑑𝑠−2𝛿(𝜃𝜏−𝑡𝜔)}𝜂𝜃_𝜏−𝑡𝜔𝜇‖‖𝑥^(𝑚)(𝜏)‖‖²_𝜌+𝛼‖‖𝑦^(𝑚)(𝜏)‖‖²_𝜌𝑑𝜏≤𝜇‖‖𝑥₀‖‖²_𝜌+𝛼‖‖𝑦₀‖‖²_𝜌𝑒^{−(𝜆1−(𝜖2/3))𝑡−4𝛿(𝜃−𝑡𝜔)+∫0−𝑡[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}^𝑡_𝑇1𝜌𝑒^{(−𝜖2/3)𝜏}𝑑𝜏+12𝑟²_1𝜌(𝜔)𝑒^{−2𝛿(𝜔)}^{𝑡−𝑇1𝜌}₀𝑒^{−(𝜆1−(𝜀2/3))𝜏+∫𝜏0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝑑𝜏.(3.79) For 𝑡≫𝑇_1𝜌, by (H2) and (3.6), there exists 0<𝑇^₂=𝑇^₂(𝜔)<𝑡−𝑇_1𝜌 such that 1𝜏^𝜏₀2𝛿𝜃_𝑠𝜔+2𝑞𝜂̃𝜃_𝑠𝜔𝑑𝑠<𝜆₁−2𝜖₂3,for𝜏≥𝑇^₂.(3.80) Let 𝑇₂=max{𝑇^₁,𝑇^₂}<∞, and 𝑡>𝑇₂+𝑇_1𝜌, write ^{𝑡−𝑇1𝜌}₀𝑒^{−(𝜆1−𝜖2/3)𝜏+∫𝜏0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝑑𝜏=^𝑇2₀+^{𝑡−𝑇1𝜌}_𝑇2𝑒^{−(𝜆1−𝜖2/3)𝜏+∫𝜏0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝑑𝜏,(3.81) of which ^𝑇2₀𝑒^{−(𝜆1−𝜖2/3)𝜏+∫𝜏0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝑑𝜏<∞,^{𝑡−𝑇1𝜌}_𝑇2𝑒^{−(𝜆−𝜖2/3)𝜏+∫𝜏0[2𝛿(𝜃𝑠𝜔)+2}𝑞̃_{𝜂(𝜃𝑠𝜔)]𝑑𝑠}𝑑𝜏≤3𝜖₂𝑒^{(−𝜖2/3)𝑇2}.(3.82) Equation (3.79) together with (3.82) implies that there exist 𝑇₃(𝜀,𝜔,𝐵_1𝜌)≥𝑇₂ and 𝐼₂(𝜀,𝜔)∈ℕ such that for 𝑀>𝐼₂(𝜀,𝜔),𝑡≥𝑇₃(𝜀,𝜔,𝐵_1𝜌), (𝑖𝑖𝑖)≤𝜀3min{𝜇,𝛼}𝑒^{−2𝛿(𝜔)}.(3.83) In summary, let 𝑇𝜀,𝜔,𝐵_1𝜌=max𝑇₁𝜀,𝜔,𝐵_1𝜌,𝑇₂𝜀,𝜔,𝐵_1𝜌,𝑇₃𝜀,𝜔,𝐵_1𝜌,𝐼(𝜀,𝜔)=max𝐼₁(𝜀,𝜔),𝐼₂(𝜀,𝜔)(3.84) Then for 𝑡>𝑇(𝜀,𝜔,𝐵_1𝜌) and 𝑀≥𝐼(𝜀,𝜔), we have _|𝑖|_≥2𝑀𝜌_𝑖𝜇𝑥^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²+𝛼𝑦^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²≤𝜀min{𝜇,𝛼}𝑒^{−2𝛿(𝜔)}.(3.85) Since 𝐳^(𝑚𝑘)_𝑖(𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛(𝜃_−𝑡𝜔),ℎ_𝑛,𝑔_𝑛)→𝐳_𝑖(𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛(𝜃_−𝑡𝜔),ℎ_𝑛,𝑔_𝑛) as 𝑚_𝑘→∞, by (3.85), _|𝑖|_≥2𝑀𝜌_𝑖||𝜑(𝑡,𝜔)𝑢₀,𝑣₀_𝑖||²=_{|𝑖|≥2𝑀}𝜌_𝑖𝑒^2𝛿(𝜔)||𝐳_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,𝑔_𝑛,ℎ_𝑛||²≤𝑒^2𝛿(𝜔)min{𝜇,𝛼}_{|𝑖|≥2𝑀}𝜌_𝑖𝜇𝑥^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²+𝛼𝑦^(𝑚)_𝑖𝑡;𝜃_−𝑡𝜔,𝐳_0𝑛,ℎ_𝑛,𝑔_𝑛²≤𝜀.(3.86) Letting 𝑛→∞ in (3.86), we obtain _|𝑖|_≥2𝑀𝜌_𝑖||𝜑(𝑡,𝜔)𝑢₀,𝑣₀_𝑖||²≤𝜀(3.87) That is, {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω1} is asymptotically null on 𝐵_1𝜌(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=𝑙²_𝜌×𝑙²_𝜌. 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

Define Ω₂={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,(𝜃_𝑡)_𝑡∈ℝ) is a metric dynamical system. Let the infinite sequence 𝑒^𝑖 (𝑖∈ℤ) denote the element having 1 at position 𝑖 and 0 for all other components. Write𝑊¹(𝑡,𝜔)=_𝑖∈ℤ𝑎_𝑖𝑤_𝑖(𝑡)𝑒^𝑖,𝑊²(𝑡,𝜔)=_𝑖∈ℤ𝑏_𝑖𝑤_𝑖(𝑡)𝑒^𝑖,(4.1) where {𝑤_𝑖(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:𝑢̇=−𝜆𝑢+𝐴𝜃_𝑡𝜔𝑢−𝑓(𝑢)+𝛼𝑣+ℎ+𝑑𝑊¹(𝑡)𝑑𝑡,𝑣̇‌=−𝜎𝑣+𝜇𝑢+𝑔+𝑑𝑊²(𝑡)𝑑𝑡,𝑖∈𝑍,𝑡>0,(4.2) where 𝑢_𝑖,𝑣_𝑖,ℎ_𝑖,𝑔_𝑖,𝑎_𝑖,𝑏_𝑖∈ℝ; 𝑢=(𝑢_𝑖)_𝑖∈ℤ, 𝑣=(𝑣_𝑖)_𝑖∈ℤ, 𝑓(𝑢)=(𝑓_𝑖(𝑢_𝑖))_𝑖∈ℤ, 𝑔=(𝑔_𝑖)_𝑖∈ℤ,ℎ=(ℎ_𝑖)_𝑖∈ℤ, 𝑎=(𝑎_𝑖)_𝑖∈ℤ∈𝑙², 𝑏=(𝑏_𝑖)_𝑖∈ℤ∈𝑙², 𝑓_𝑖∈𝐶¹(ℝ,ℝ) (𝑖∈ℤ); 𝜆,𝛼,𝜎,𝜇 are positive constants; 𝜂_𝑖,−𝑞(𝜔),…, 𝜂_𝑖,0(𝜔),…,𝜂_𝑖,+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, let𝛿¹𝜃_𝑡𝜔=−𝜆⁰_−∞𝑒^𝜆𝑠𝜃_𝑡𝜔(𝑠)𝑑𝑠,𝛿²𝜃_𝑡𝜔=−𝜎⁰_−∞𝑒^𝜎𝑠𝜃_𝑡𝜔(𝑠)𝑑𝑠,𝑡∈ℝ,𝜔∈Ω₂.(4.3) Then 𝛿¹(𝜃_𝑡𝜔), 𝛿²(𝜃_𝑡𝜔) are both Ornstein-Uhlenbeck processes on (Ω₂,ℱ₂,ℙ₂) and solve the following Ornstein-Uhlenbeck equations (see [1]), respectively,𝑑𝛿¹+𝜆𝛿¹𝑑𝑡=𝑑𝑊¹(𝑡),𝑑𝛿²+𝜎𝛿²𝑑𝑡=𝑑S𝑊²(𝑡),𝑡≥0.(4.4)

Lemma 4.1 (see [1]). There exists a 𝜃_𝑡-invariant set Ω₂∈ℱ₂ of Ω₂ of full ℙ measure such that for 𝜔∈Ω₂,(i)lim_𝑡→±∞‖𝜔(𝑡)‖/𝑡=0; (ii)the random variables ‖𝛿^𝑗(𝜔)‖ are tempered and the mappings (𝑡,𝜔)⟶𝛿^𝑗𝜃_𝑡𝜔∈𝑙²,𝑗=1,2,(4.5) are stationary solutions of Ornstein-Uhlenbeck equations (4.4) in 𝑙² with continuous trajectories;(iii)lim_𝑡→±∞‖‖𝛿^𝑗𝜃_𝑡𝜔‖‖𝑡=lim_𝑡→±∞1𝑡^𝑡₀𝛿^𝑗𝜃_𝑠𝜔𝑑𝑠=0,𝑗=1,2.(4.6)

In the following, we consider the completion of the probability space 𝜔∈Ω₂, still denoted by (Ω₂,ℱ₂,ℙ₂),

Let𝑥̃(𝑡,𝜔)=𝑢(𝑡,𝜔)−𝛿¹𝜃_𝑡𝜔,𝑦̃(𝑡,𝜔)=𝑣(𝑡,𝜔)−𝛿²𝜃_𝑡𝜔,𝜔∈Ω₂,𝑡∈ℝ.(4.7) then system (4.2) becomes the following random system with random coefficients but without white noise:𝑑𝑥̃𝑑𝑡=−𝜆𝑥̃+𝐴𝜃_𝑡𝜔𝑥̃−𝑓𝑥̃+𝛿¹𝜃_𝑡𝜔−𝛼𝑥̃+𝐴𝜃_𝑡𝜔𝛿¹𝜃_𝑡𝜔−𝛼𝛿²𝜃_𝑡𝜔+ℎ,𝑑𝑦̃𝑑𝑡=−𝜎𝑦̃+𝜇𝑥̃+𝜇𝛿¹𝜃_𝑡𝜔+𝑔,𝑖∈ℤ,𝑡>0.(4.8) In addition, we make the following assumptions on functions 𝑓_𝑖,𝑖∈ℤ:(H6)𝑓_𝑖∈𝐶¹(ℝ,ℝ) satisfy𝑠𝑓_𝑖(𝑠)≥𝜇𝑠^2(𝑝+1)−𝑑²_𝑖,||𝑓_𝑖(𝑠)||≤𝑑_𝑓|𝑠||𝑠|^2𝑝+1,∀𝑠∈ℝ,𝑖∈ℤ,(4.9)where 𝜇, 𝑑_𝑖, 𝑑_𝑓 are positive constants, 𝑝∈ℕ, and 𝑑=(𝑑_𝑖)_𝑖∈ℤ∈𝑙².

4.2. Random Dynamical System Generated by Random Lattice System

Denote by 𝐳̃‌=(𝑥̃,𝑦)̃, we have the following.

Theorem 4.2. Let 𝑇>0 and assume that (P0), (H1), (H2), (H4), and (H6) hold. Then for every 𝜔∈Ω₂ and any initial data 𝐳₀̃‌=(𝑥̃₀,𝑦₀̃)∈𝐻, problem (4.8) admits a unique solution 𝐳(̃‌⋅;𝜔,𝐳₀̃‌)∈𝐶([0,𝑇),𝐻) with 𝐳(̃‌0;𝜔,𝐳₀̃‌)=𝐳₀̃‌.

Proof. Similar to the proof of Theorem 3.3.

Theorem 4.3. Assume that (P0), (H1), (H2), (H4), and (H6) hold. Then system (4.8) generates a continuous RDS {𝜓(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω2} over (Ω₂,ℱ₂,ℙ₂,(𝜃_𝑡)_𝑡∈ℝ) with state space 𝐻: 𝜓(𝑡,𝜔)𝐳₀∶=𝑧̃𝑡;𝜔,𝐳₀,for𝐳₀∈𝐻,𝑡≥0,𝜔∈Ω₂.(4.10) Moreover, 𝜑(𝑡,𝜔)𝑢₀,𝑣₀∶=𝜓(𝑡,𝜔)⎛⎜⎜⎝𝑢₀−𝛿¹(𝜔)𝑣₀−𝛿²(𝜔)⎞⎟⎟⎠+⎛⎜⎜⎝𝛿¹𝜃_𝑡𝜔𝛿²𝜃_𝑡𝜔⎞⎟⎟⎠,(4.11) where (𝑢₀,𝑣₀)∈𝐻,𝑡≥0,𝜔∈Ω₂, defines a continuous RDS {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω2} over (Ω₂,ℱ₂,ℙ₂,(𝜃_𝑡)_𝑡∈ℝ) associated with (4.2).

Proof. It follows immediately from similar arguments to the proof of Theorem 3.4.

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 {𝜑(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 set 𝐵_2𝜌(𝜔)∈𝒟(𝐻) of RDS {𝜑(𝑡,𝜔)}_{𝑡≥0,𝜔∈Ω2} such that for any 𝐷∈𝒟(𝐻) and each 𝜔∈Ω₂, there exists 𝑇_𝐷̃‌(𝜔)>0 yielding 𝜑(𝑡,𝜃_−𝑡𝜔)𝐷(𝜃_−𝑡𝜔)⊂𝐵_2𝜌(𝜔), forall𝑡≥𝑇_𝐷̃‌(𝜔). In particular, there exists 𝑇_2𝜌(𝜔)>0 such that 𝜑(𝑡,𝜃_−𝑡𝜔)𝐵_2𝜌(𝜃_−𝑡𝜔)⊂𝐵_2𝜌(𝜔), forall𝑡≥𝑇_2𝜌(𝜔);(b)the RDS {𝜑(𝑡,𝜔₂)}_{𝑡≥0,𝜔∈Ω2} generated by equations (4.2) possesses a unique global random 𝒟 attractor given by𝐴_2𝜌(𝜔)=_𝜏≥_{𝑇2𝜌(𝜔)}_𝑡≥𝜏𝜑𝑡,𝜃_−𝑡𝜔𝐵_2𝜌𝜃_−𝑡𝜔∈𝐻.(4.12)

Proof. Similar to the proofs of Theorems 3.6 and 3.7.