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International Journal of Differential Equations
Volume 2011 (2011), Article ID 628459, 23 pages
http://dx.doi.org/10.1155/2011/628459
Research Article

Asymptotic Behavior of Stochastic Partly Dissipative Lattice Systems in Weighted Spaces

Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA

Received 22 June 2011; Accepted 3 September 2011

Academic Editor: I. Chueshov

Copyright © 2011 Xiaoying Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [1]. Continuing studies have been made on various types of SLDS’s with multiplicative or additive noise, see [27].

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:̇𝑢𝑖=𝜆𝑢𝑖+𝑞𝑗=𝑞𝜂𝑖,𝑗𝜃𝑡𝜔𝑢𝑖+𝑗𝑓𝑖𝑢𝑖𝛼𝑣𝑖+𝑖+𝑢𝑖𝑑𝑤(𝑡)𝑑𝑡,̇𝑣𝑖=𝜎𝑣𝑖+𝜇𝑢𝑖+𝑔𝑖+𝑢𝑖𝑑𝑤(𝑡)𝑑𝑡,𝑖,𝑡>0,(1.1)̇𝑢𝑖=𝜆𝑢𝑖+𝑞𝑗=𝑞𝜂𝑖,𝑗𝜃𝑡𝜔𝑢𝑖+𝑗𝑓𝑖𝑢𝑖𝛼𝑣𝑖+𝑖+𝑎𝑖𝑑𝑤𝑖(𝑡)𝑑𝑡,̇𝑣𝑖=𝜎𝑣𝑖+𝜇𝑢𝑖+𝑔𝑖+𝑏𝑖𝑑𝑤𝑖(𝑡)𝑑𝑡,𝑖,𝑡>0,(1.2) where 𝑢𝑖,𝑖,𝑔𝑖,𝑎𝑖,𝑏𝑖𝑓𝑖𝐶1(,); (𝑖), 𝜆,𝛼,𝜎,𝜇>0 are positive constants; 𝐴 is the coupling operator, 𝜂𝑖,𝑞(𝜔),,𝜂𝑖,0(𝜔),,𝜂𝑖,+𝑞(𝜔), 𝑖, 𝑞, are random variables, and 𝑤(𝑡), {𝑤𝑖(𝑡)𝑖} 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 [913]. For stochastic lattice system (1.2) with additive noises, when 𝑞=1, 𝜂𝑖,±1(𝜔)1, 𝜂𝑖,0(𝜔)2 for all 𝑖, 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 𝑙2𝜌×𝑙2𝜌, 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(0,𝑀0] such that 𝑙2𝑙2𝜌. If 𝑖𝜌(𝑖)<, then 𝑙2𝜌 contains any infinite sequences whose components are just bounded and 𝑙2𝑙𝑙2𝜌. Note that when 𝜌(𝑖)1, our results recover the results obtained in [4, 14] while 𝑙2𝜌 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 𝑙2𝜌×𝑙2𝜌 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 to (0,𝑀0]+, where 𝑀0 is a finite positive constant. Define for any 𝑖, 𝜌𝑖=𝜌(𝑖) and𝑙2𝜌=𝑢=𝑢𝑖𝑖𝑖𝜌𝑖||𝑢𝑖||2<,𝑢𝑖,(2.1) then 𝑙2𝜌 is a separable Hilbert space with the inner product 𝑢,𝑣𝜌=𝑖𝜌𝑖𝑢𝑖𝑣𝑖 and norm 𝑢2𝜌=𝑢,𝑢𝜌=𝑖𝜌𝑖|𝑢𝑖|2 for 𝑢=(𝑢𝑖)𝑖, 𝑣=(𝑣𝑖)𝑖𝑙2𝜌. Moreover, define𝐻̇=𝑙2𝜌×𝑙2𝜌(2.2) with inner product𝑢(1),𝑣(1),𝑢(2),𝑣(2)𝐻=𝑢(1),𝑢(2)𝜌+𝑣(1),𝑣(2)𝜌,for𝑢(1),𝑣(1),𝑢(2),𝑣(2)𝐻,(2.3) and norm(𝑢,𝑣)𝐻=𝑢2𝜌+𝑣2𝜌1/2for(𝑢,𝑣)𝐻,(2.4) then 𝐻 is also a separable Hilbert space.

Let (Ω,,) be a probability space and {𝜃𝑡ΩΩ,𝑡} be a family of measure-preserving transformations such that (𝑡,Ω)𝜃𝑡Ω is (()×,)-measurable, 𝜃0=IdΩ and 𝜃𝑡+𝑠=𝜃𝑡𝜃𝑠 for all 𝑠,𝑡. The space (Ω,,,(𝜃𝑡)𝑡) is called a metric dynamical system. In the following, “property (P) holds for a.e. 𝜔Ω with respect to (𝜃𝑡)𝑡" means that there is ΩΩ with (Ω)=1 and 𝜃𝑡Ω=Ω such that (P) holds for all 𝜔Ω.

Recall the following definitions from existing literature.(i)A stochastic process {𝑆(𝑡,𝜔)}𝑡0,𝜔Ω is said to be a continuous RDS over (Ω,,,(𝜃𝑡)𝑡) with state space 𝐻, if 𝑆+×Ω×𝐻𝐻 is ((+)××(𝐻),(𝐻))-measurable, and for each 𝜔Ω, the mapping 𝑆(𝑡,𝜔)𝐻𝐻,𝑢𝑆(𝑡,𝜔)𝑢 is continuous for 𝑡0, 𝑆(0,𝜔)𝑢=𝑢 and 𝑆(𝑡+𝑠,𝜔)=𝑆(𝑡,𝜃𝑠𝜔)𝑆(𝑠,𝜔) for all 𝑢𝐻 and 𝑠,𝑡0.(ii)A set-valued mapping 𝜔𝐷(𝜔)𝐻 (may be written as 𝐷(𝜔) for short) is said to be a random set if the mapping 𝜔dist𝐻(𝑢,𝐷(𝜔)) is measurable for any 𝑢𝐻.(iii)A random set 𝐷(𝜔) is called a closed (compact) random set if 𝐷(𝜔) is closed (compact) for each 𝜔Ω.(iv)A random set 𝐷(𝜔) is said to be bounded if there exist 𝑢0𝐻 and a random variable 𝑟(𝜔)>0 such that 𝐷(𝜔){𝑢𝐻𝑢𝑢0𝐻𝑟(𝜔)} for all 𝜔Ω.(v)A random bounded set 𝐷(𝜔) is said to be tempered if for a.e. 𝜔Ω, lim𝑡𝑒𝛽𝑡sup𝑢𝐻𝑢𝐷𝜃𝑡𝜔=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. 𝜔Ω, there exists 𝑇𝐷(𝜔) such that 𝑆(𝑡,𝜃𝑡𝜔)𝐷(𝜃𝑡𝜔)𝐵(𝜔)forall𝑡𝑇𝐷(𝜔).(vii)A random set 𝐴(𝜔) is said to be a random attracting set if for any 𝐷(𝜔)𝒟(𝐻), we have lim𝑡dist𝐻𝑆𝑡,𝜃𝑡𝜔𝐷𝜃𝑡𝜔,𝐴(𝜔)=0,a.e.𝜔Ω,(2.6) in which dist𝐻 is the Hausdorff semidistance defined via dist𝐻(𝐸,𝐹)=sup𝑢𝐸inf𝑣𝐹𝑢𝑣𝜌 for any 𝐸,𝐹𝑙2𝜌.(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(𝜃𝑡𝜔) for a.e.𝜔Ω and𝑡0.

Definition 2.1 (see [8]). {𝑆(𝑡,𝜔)}𝑡0,𝜔Ω is said to be random asymptotically null in 𝒟(𝐻), if for any 𝐷(𝜔)𝒟(𝐻), a.e. 𝜔Ω, and any 𝜀>0, there exist 𝑇(𝜀,𝜔,𝐷(𝜔))>0 and 𝐼(𝜀,𝜔,𝐷(𝜔)) such that |𝑖|>𝐼(𝜀,𝜔,𝐷(𝜔))𝜌𝑖||𝑆𝑡,𝜃𝑡𝜔𝑢𝜃𝑡𝜔)𝑖||21/2𝜀,𝑡𝑇(𝜀,𝜔,𝐷(𝜔)),𝑢(𝜔)𝐷(𝜔).(2.7)

Theorem 2.2 (see [8]). Let {𝑆(𝑡,𝜔)}𝑡0,𝜔Ω be a continuous RDS over (Ω,,,(𝜃𝑡)𝑡) 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. 𝜔Ω and any 𝐷(𝜔)𝒟(𝐻), there exists 𝑇𝐷(𝜔)>0 yielding 𝑆(𝑡,𝜃𝑡𝜔)𝐷(𝜃𝑡𝜔)𝐵(𝜔) for all 𝑡𝑇𝐷(𝜔);(b){𝑆(𝑡,𝜔)}𝑡0,𝜔Ω is random asymptotically null on 𝐵(𝜔); that is, for a.e. 𝜔Ω and for any 𝜀>0, there exist 𝑇(𝜀,𝜔,𝐵(𝜔))>0 and 𝐼(𝜀,𝜔,𝐵(𝜔)) such thatsup𝑢𝐵(𝜔)|𝑖|>𝐼(𝜀,𝜔,𝐵(𝜔))𝜌𝑖||𝑆𝑡,𝜃𝑡𝜔𝑢𝜃𝑡𝜔𝑖||2𝜀2,𝑡𝑇(𝜀,𝜔,𝐵(𝜔)).(2.8)
Then the RDS {𝑆(𝑡,𝜔)}𝑡0,𝜔Ω possesses a unique global random 𝒟 attractor 𝐴(𝜔) given by 𝐴(𝜔)=𝜏𝑇𝐵(𝜔)𝑡𝜏𝑆𝑡,𝜃𝑡𝜔𝐵𝜃𝑡𝜔.(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=𝑙2𝜌×𝑙2𝜌. 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+ is chosen to satisfy(P0)0<𝜌(𝑖)𝑀0 and 𝜌(𝑖)𝑐𝜌(𝑖±1),forall𝑖 for some positive constants 𝑀0 and 𝑐.

(e.g., 𝜌(𝑥)=1/(1+𝜖2𝑥2)𝑞,𝑞>1/2 [16, 17] and 𝜌(𝑥)=𝑒𝜖|𝑥|,𝑥 where 𝜖>0).

3.1. Mathematical Setting

Define Ω1={𝜔𝐶(,)𝜔(0)=0}=𝐶0(,), and denote by 1 the Borel 𝜎-algebra on Ω1 generated by the compact open topology (see [2, 15]) and 1 the corresponding Wiener measure on 1. Defining (𝜃𝑡)𝑡 on Ω1 via 𝜃𝑡𝜔()=𝜔(+𝑡)𝜔(𝑡) for 𝑡, then (Ω1,1,1,(𝜃𝑡)𝑡) is a metric dynamical system.

Consider the stochastic lattice system (1.1) with random coupled coefficients and multiplicative white noise:𝑑𝑢=𝜆𝑢+𝐴𝜃𝑡𝜔𝑢𝑓(𝑢)𝛼𝑣+𝑑𝑡+𝑢𝑑𝑤(𝑡),𝑑𝑣=(𝜎𝑣+𝜇𝑢+𝑔)+𝑢𝑑𝑤(𝑡),𝑖,𝑡>0,(3.1) where 𝑢=(𝑢𝑖)𝑖, 𝑣=(𝑣𝑖)𝑖; 𝑓(𝑢)=(𝑓𝑖(𝑢𝑖))𝑖 with 𝑓𝑖𝐶1(,) (𝑖), 𝑔=(𝑔𝑖)𝑖, =(𝑖)𝑖; 𝜆,𝛼,𝜎,𝜇 are positive constants; 𝜂𝑖,𝑞(𝜔),,𝜂𝑖,0(𝜔),,𝜂𝑖,+𝑞(𝜔)(𝑞) are random variables on the probability space (Ω1,1,1); 𝐴() is a linear operator on 𝑙2𝜌 defined by(𝐴()𝑢)𝑖=𝑞𝑗=𝑞𝜂𝑖,𝑗()𝑢𝑖+𝑗;(3.2)𝑤(𝑡) is a Brownian motion (Wiener process) on the probability space (Ω1,1,1); denotes the Stratonovich sense of the stochastic term.

For convenience, we first transform (3.1) into a random differential equation without white noise. Let𝛿𝜃𝑡𝜔=0𝑒𝑠𝜃𝑡𝜔(𝑠)𝑑𝑠,𝑡,𝜔Ω1,(3.3) then 𝛿(𝜃𝑡𝜔) is an Ornstein-Uhlenbeck process on (Ω1,1,1,(𝜃𝑡)𝑡) that solves the following Ornstein-Uhlenbeck equation (see [2, 15] for details)𝑑𝛿+𝛿𝑑𝑡=𝑑𝑤(𝑡),𝑡0,(3.4) where 𝑤(𝑡)(𝜔)=𝑤(𝑡,𝜔)=𝜔(𝑡) for 𝜔Ω1, 𝑡, and possesses the following properties.

Lemma 3.1 (see [2, 15]). There exists a 𝜃𝑡-invariant set Ω11 of Ω1 offull 1 measure such that for 𝜔Ω1, one has(i)the random variable |𝛿(𝜔)| is tempered;(ii)the mapping𝛿(𝜃𝑡𝜔)(𝑡,𝜔)𝛿𝜃𝑡𝜔=0𝑒𝑠𝜔(𝑡+𝑠)d𝑠+𝑤(𝑡)(3.5) is a stationary solution of Ornstein-Uhlenbeck equation (3.4) with continuous trajectories;(iii)lim𝑡±||𝛿𝜃𝑡𝜔||𝑡=lim𝑡±1𝑡𝑡0𝛿𝜃𝑠𝜔𝑑𝑠=0.(3.6)

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

Let𝑥(𝑡,𝜔)=𝑒𝛿(𝜃𝑡𝜔)𝑢(𝑡,𝜔),𝑦(𝑡,𝜔)=𝑒𝛿(𝜃𝑡𝜔)𝑣(𝑡,𝜔),𝜔Ω1,(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+𝐴𝜃𝑡𝜔𝑥𝑒𝛿(𝜃𝑡𝜔)𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥+𝛿𝜃𝑡𝜔𝑥𝛼𝑦+𝑒𝛿(𝜃𝑡𝜔),𝑑𝑦𝑑𝑡=𝜎𝑦+𝛿𝜃𝑡𝜔𝑦+𝜇𝑥+𝑒𝛿(𝜃𝑡𝜔)𝑔𝑡>0,(3.8) Letting 𝐳=(𝑥,𝑦), (3.8) are equivalent to𝑑𝐳𝑑𝑡=𝐅𝐳,𝜃𝑡𝜔,𝑡>0,(3.9) where 𝐅𝐳,𝜃𝑡𝜔=𝜆𝑥+𝐴𝜃𝑡𝜔𝑥𝑒𝛿(𝜃𝑡𝜔)𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥+𝛿𝜃𝑡𝜔𝑥𝛼𝑦+𝑒𝛿(𝜃𝑡𝜔)𝜎𝑦+𝛿𝜃𝑡𝜔𝑦+𝜇𝑥+𝑒𝛿(𝜃𝑡𝜔)𝑔.(3.10)

We now make the following standing assumptions on 𝑓𝑖, 𝑔𝑖, 𝑖, and 𝜂𝑖,𝑗,(𝑗=𝑞,,𝑞)𝑖 and study in the following subsections asymptotic behavior of system (3.9).(H1) 𝑔=(𝑔𝑖)𝑖, =(𝑖)𝑖𝑙2𝜌. (H2)Let𝜂(𝜔)=sup||𝜂𝑖,𝑞(𝜔)||,,||𝜂𝑖,0(𝜔)||,,||𝜂𝑖,+𝑞(𝜔)||𝑖0,𝑞.(3.11)𝜂(𝜃𝑡𝜔)(<)  belongs to 𝐿1loc()  with respect to 𝑡  for each 𝜔Ω1. 𝐄(𝜂)=lim𝑡±1𝑡𝑡0𝜂𝜃𝑡𝜔𝑑𝑠<,(3.12)and 𝜂(𝜔)  is tempered, that is, there exists a 𝜃𝑡-invariant set Ω101 of full 1  measure such that for 𝜔Ω10, lim𝑡+𝑒𝛽𝑡sup𝑡||𝜂𝜃𝑡𝜔||=0,𝛽>0.(3.13)In the following, we will consider 𝜔Ω10Ω1 and still write Ω10Ω1 as Ω1.(H3)min{𝜆,𝜎}>̃𝑞𝐄|𝜂(𝜔)|=lim𝑡±(1/𝑡)𝑡0(𝑞+𝑞𝑘=0𝑐𝑘)𝜂(𝜃𝑡𝜔)𝑑𝑠, where ̃𝑞=𝑞+𝑞𝑘=0𝑐𝑘.(H4)There exists a function 𝑅𝐶(+,+) such thatsup𝑖max𝑠[𝑟,𝑟]||𝑓𝑖(𝑠)||𝑅(𝑟),𝑟+.(3.14)(H5)𝑓𝑖𝐶1(,), 𝑓𝑖(0)=0, 𝑠𝑓𝑖(𝑠)𝑏2𝑖, 𝑏=(𝑏𝑖)𝑖𝑙2𝜌, and there exists a constant 𝑎0 such that 𝑓𝑖(𝑠)𝑎, forall𝑠, 𝑖.

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=𝐅𝐳,𝜃𝑡𝜔,𝐳=𝐳𝑖𝑖,𝐅=𝐅𝑖𝑖,(3.15) where 𝜔Ω0, if 𝐳𝐶([0,𝑇),𝐻) satisfies 𝐳𝑖(𝑡)=𝐳𝑖(0)+𝑡0𝐅𝑖𝐳(𝑠),𝜃𝑠𝜔𝑑𝑠,for𝑖,𝑡[0,𝑇).(3.16)

Theorem 3.3. Let 𝑇>0 and (P0), (H1), (H2), (H4), and (H5) hold. Then for any 𝜔Ω1 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 𝐳0𝐸 and (,𝑔)𝐸, then (3.9) admits a unique solution 𝐳(𝑡;𝜔,𝐳0,,𝑔)𝐸 on [0,𝑇) with 𝐳(0;𝜔,𝐳0,𝑔,)=𝐳0. Given 𝐳𝐸, 𝜔Ω1, and (,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=𝑖𝑞𝑗=𝑞𝜂𝑖,𝑗(𝜔)𝑥𝑖+𝑗21/2(2𝑞+1)𝜂(𝜔)𝑥.(3.17) By (H4), 𝑓𝑒𝛿(𝜔)𝑥max𝑅𝑒𝛿(𝜔)𝑥,𝑎𝑒𝛿(𝜔)𝑥,(3.18) and therefore 𝐅(𝐳,𝜔)𝐸𝜆+(2𝑞+1)𝜂(𝜔)+max𝑅𝑒𝛿(𝜔)𝑥,𝑎+||𝛿(𝜔)||+𝜇𝑥+𝛼+𝜎+||𝛿(𝜔)||𝑦+||𝑒𝛿(𝜔)||(+𝑔).(3.19) For any 𝐳(1)=(𝑥(1),𝑦(1)), 𝐳(2)=(𝑥(2),𝑦(2))𝐸, and for some 𝜗(0,1)𝑓𝑒𝛿(𝜔)𝑥(1)𝑓𝑒𝛿(𝜔)𝑥(2)max𝑅𝑒𝛿(𝜔)(1𝜗)𝑥(1)+𝜗𝑥(2),𝑎𝑒𝛿(𝜔)𝑥(1)𝑥(2).(3.20) Also 𝐴(𝜔)𝑥(1)𝐴(𝜔)𝑥(2)=𝑖𝑞𝑗=𝑞𝜂𝑖,𝑗(𝜔)𝑥(1)𝑖+𝑗𝑥(2)𝑖+𝑗1/2(2𝑞+1)𝜂(𝜔)𝑥(1)𝑥(2).(3.21) It then follows that 𝐅𝐳(1),𝜔𝐅𝐳(2),𝜔𝐸𝛼+𝜎+||𝛿(𝜔)||𝑦(1)𝑦(2)+𝜆+(2𝑞+1)𝜂(𝜔)+𝜇+||𝛿(𝜔)||+𝑒𝛿(𝜔)max𝑅𝑒𝛿(𝜔)(1𝜗)𝑣(1)+𝜗𝑣(2),𝑎𝑥(1)𝑥(2).(3.22) For any compact set 𝐷𝐸 with sup𝐳𝐷𝐳𝑟, defining random variable 𝜁𝐷(𝜔)0 via 𝜁𝐷(𝜔)=𝜆+(2𝑞+1)𝜂(𝜔)+𝜇+||𝛿(𝜔)||+max𝑅𝑒𝛿(𝜔)𝑟,𝑎𝑒𝛿(𝜔)𝑟+𝛼+𝜎+||𝛿(𝜔)||𝑟+||𝑒𝛿(𝜔)||(+𝑔),(3.23) we have 𝜏+1𝜏𝜁𝐷𝜃𝑡𝜔𝑑𝑡<,𝜏,(3.24) and for any 𝐳,𝐳(1),𝐳(2)𝐷, 𝐅(𝐳,𝜔)𝐸𝜁𝐷(𝜔),𝐅𝐳(1),𝜔𝐅𝐳(2),𝜔𝐸𝜁𝐷(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,)𝐶([0,𝑇max),𝐸) (0<𝑇max𝑇) satisfying the integral equation 𝐳(𝑡)=𝐳0+𝑡0𝐅(𝐳(𝑠),𝜔)𝑑𝑠,for𝑡0,𝑇max.(3.26)
We will next show that 𝑇max=𝑇. Since the set 𝐶0() of continuous random process in 𝑡 is dense in the set 𝐿1() (see [18, 21]), for each 𝜔Ω1, there exists a sequence {𝜂(𝑚)𝑖,𝑗(𝑡,𝜔)}𝑚=1 of continuous random process in 𝑡 such that lim𝑚𝑇0𝜂(𝑚)𝑖,𝑗(𝑠,𝜔)𝜂𝑖,𝑗(𝑠,𝜔)𝑑𝑠=0,𝑇>0,|||𝜂(𝑚)𝑖,𝑗(𝑡,𝜔)|||||𝜂𝑖,𝑗𝜃𝑡𝜔||||𝜂𝜃𝑡𝜔||,𝑡.(3.27)
Consider the random differential equation with initial data 𝐳0𝐸𝑑𝐳(𝑚)𝑑𝑡=𝜆𝑥(𝑚)+𝐴𝑚(𝑡,𝜔)𝑥(𝑚)𝑒𝛿(𝜃t𝜔)𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)+𝛿𝜃𝑡𝜔𝑥(𝑚)𝛼𝑦(𝑚)+𝑒𝛿(𝜃𝑡𝜔)𝜎𝑦(𝑚)+𝛿𝜃𝑡𝜔𝑦(𝑚)+𝜇𝑥(𝑚)+𝑒𝛿(𝜃𝑡𝜔)𝑔,(3.28) where 𝐴𝑚(𝑡,𝜔)𝑥(𝑚)𝑖=𝑞𝑗=𝑞𝜂(𝑚)𝑖,𝑗(𝑡,𝜔)𝑥𝑖+𝑗.(3.29) Follow the same procedure as above, (3.28) has a unique solution 𝐳(𝑚)(;𝜔,𝐳0,𝑔,)𝐶([0,𝑇(𝑚)max),𝐸), that is, 𝐳(𝑚)𝑖(𝑡)=𝐳0+𝑡0𝐅(𝑚)𝑖𝐳(𝑚)(𝑠),𝜔𝑑𝑠,for𝑡0,𝑇(𝑚)max,(3.30) and by the continuity of 𝐴𝑚(𝑠,𝜔) in 𝑠, there holds 𝑑𝐳(𝑚)𝑖𝑑𝑡=𝜆𝑥(𝑚)𝑖+𝐴𝑚𝜃𝑡𝜔𝑥(𝑚)𝑖𝑒𝛿(𝜃𝑡𝜔)𝑓𝑖𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖+𝛿𝜃𝑡𝜔𝑥(𝑚)𝑖𝛼𝑦(𝑚)𝑖+𝑒𝛿(𝜃𝑡𝜔)𝑖𝜎𝑦(𝑚)𝑖+𝛿𝜃𝑡𝜔𝑦(𝑚)𝑖+𝜇𝑥(𝑚)𝑖+𝑒𝛿(𝜃𝑡𝜔)𝑔.(3.31) Note that 𝐴𝑚𝜃𝑡𝜔𝑥(𝑚)𝑖𝑥(𝑚)𝑖𝜂𝜃𝑡𝜔|||𝑥(𝑚)𝑖𝑥(𝑚)𝑖𝑞++𝑥(𝑚)𝑖𝑥(𝑚)𝑖++𝑥(𝑚)𝑖𝑥(𝑚)𝑖+𝑞|||,𝑎𝑒2𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖2𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖𝑓𝑖𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖𝑅𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑒2𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖2,𝑡[0,𝑇],(3.32) multiplying (3.31) by 𝜇𝑥(𝑚)𝑖00𝛼𝑦(𝑚)𝑖 and sum over 𝑖 results in 𝑑𝑑𝑡𝜇𝑥(𝑚)2+𝛼𝑦(𝑚)2min{𝜆,𝜎}+2𝑎+2𝛿𝜃𝑡𝜔+2(2𝑞+1)𝜂𝜃𝑡𝜔𝜇𝑥(𝑚)2+𝛼𝑦(𝑚)2+2𝜇𝜆2+2𝛼𝜎𝑔2𝑒2𝛿(𝜃𝑡𝜔).(3.33) Applying Gronwall’s inequality to (3.33) we obtain that 𝜇𝑥(𝑚)2+𝛼𝑦(𝑚)2𝑒2𝑎𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+2(2𝑞+1)𝜂(𝜃𝑠𝜔)]𝑑𝑠𝜇𝑥(0)2+𝛼𝑦(0)2+2𝜇𝜆2+2𝛼𝜎𝑔2𝑒2𝑎𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+2(2𝑞+1)𝜂(𝜃𝑠𝜔)]𝑑𝑠𝑡0𝑒(min{𝜆,𝜎}2𝑎)𝑠2𝛿(𝜃𝑠𝜔)+𝑠0[2𝛿(𝜃𝑟𝜔)+2(2𝑞+1)𝜂(𝜃𝑟𝜔)]𝑑𝑟𝑑𝑠𝜅(𝑡),𝑡0,𝑇(𝑚)max,(3.34) where 𝜅(𝑡)𝐶([0,𝑇),) is independent of 𝑚, which implies that the interval of existence of 𝐳(𝑚)(𝑡) is [0,𝑇), and 𝐳(𝑚)(;𝜔,𝐳0,𝑔,)𝐶1([0,𝑇),𝐸).
By (3.34), |||𝑥(𝑚)𝑖|||2+|||𝑦(𝑚)𝑖|||2𝜅(𝑡)min{𝜇,𝛼},𝑚,𝑡[0,𝑇).(3.35) Since |𝐅(𝑚)𝑖(𝐳(𝑚)(𝑡),𝜃𝑡𝜔)|2𝐾2(𝑇,𝜔) for some 𝐾(𝑇,𝜔)>0 and 𝑡[0,𝑇), then for any 𝑡,𝜏[0,𝑇),𝑚, |||𝐳(𝑚)𝑖(𝑡)𝐳(𝑚)𝑖(𝜏)|||=𝑡𝜏|||𝐅(𝑚)𝑖𝐳(𝑚)(𝑠),𝜃𝑠𝜔|||𝑑𝑠𝐾(𝑇,𝜔)|𝑡𝜏|.(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)𝐳𝑖(𝑡)as𝑘,𝑡[0,𝑇),𝑖,(3.37) and 𝐳𝑖(𝑡) is continuous on 𝑡[0,𝑇). Moreover, |𝐳𝑖|2𝜅(𝑡)/min{𝜇,𝛼} for 𝑡[0,𝑇). By (3.27), (3.35), assumption (H2), and the Lebesgue Dominated Convergence Theorem we have lim𝑘𝑡0𝜂(𝑚𝑘)𝑖,𝑗(𝑠,𝜔)𝜂𝑖,𝑗𝜃𝑠𝜔𝑑𝑠=𝑡0lim𝑘𝜂(𝑚𝑘)𝑖,𝑗(𝑠,𝜔)𝜂𝑖,𝑗𝜃𝑠𝜔𝑑𝑠=0,lim𝑘𝜂(𝑚𝑘)𝑖,𝑗(𝑠,𝜔)𝜂𝑖,𝑗𝜃𝑠𝜔=0,fora.e.𝑠[0,𝑇],lim𝑘𝑡0𝜂(𝑚𝑘)𝑖,𝑗(𝑠,𝜔)𝑥(𝑚𝑘)𝑖(𝑠)𝜂𝑖,𝑗𝜃𝑠𝜔𝑥𝑖(𝑠)𝑑𝑠=0.(3.38) Thus replacing 𝑚 by 𝑚𝑘 in (3.31) and letting 𝑘 give 𝐳𝑖(𝑡)=𝐳0𝑖+𝑡0𝐅𝑖𝐳𝑖,𝜃𝑠𝜔𝑑𝑠for𝑡[0,𝑇).(3.39) By the uniquness of the solutions of (3.9), we have 𝐳𝑖(𝑡)=𝐳𝑖(𝑡) for 𝑡[0,𝑇max). By (3.34), 𝐳(𝑡)2𝐸𝜅(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 any 𝐳0𝐻 and (,𝑔)𝐻, (3.9) has a solution 𝐳(𝑡;𝜔,𝐳0,,𝑔) on [0,𝑇) with 𝐳(0;𝜔,𝐳0,,𝑔)=𝐳0. Let 𝐳1,0,𝐳2,0𝐸 and 1=(1,𝑖)𝑖, 2=(2,𝑖)𝑖,𝑔1=(𝑔1,𝑖)𝑖,𝑔2=(𝑔2,𝑖)𝑖𝑙2. Let 𝐳(𝑚)1(𝑡,𝜔),𝐳(𝑚)2(𝑡,𝜔) be two solutions of (3.28) with initial data 𝐳1,0,𝐳2,0 and ,𝑔 replaced by 1, 2, 𝑔1, 𝑔2, respectively. Set 𝐝(𝑚)(𝑡)=𝐳(𝑚)1(𝑡)𝐳(𝑚)2(𝑡)=(𝑑(𝑚)1(𝑡),𝑑(𝑚)2(𝑡))𝐸𝐻. Take inner product ,𝐻 of (𝑑/𝑑𝑡)𝐝(𝑚) with 𝐝(𝑚) and evaluate each term as follows. By (P0), (H1), (H2), and (H4), ||||𝐴𝑚𝜃𝑡𝜔𝑑(𝑚)1,𝑑(𝑚)1𝜌||||̃𝑞𝜂𝜃𝑡𝜔𝑑(𝑚)12𝜌,𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)1𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)2,𝑑(𝑚)1𝜌𝑎𝑒𝛿(𝜃𝑡𝜔)𝑑(𝑚)12𝜌,𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)1𝑓𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)2,𝑑(𝑚)1𝜌𝑅𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)1+𝑥(𝑚)2𝑒𝛿(𝜃𝑡𝜔)𝑑(𝑚)12𝜌,12,𝑑(𝑚)1𝜌122𝜌𝑑(𝑚)12𝜌;𝑔1𝑔2,𝑑(𝑚)2𝜌𝑔1𝑔22𝜌𝑑(𝑚)22𝜌.(3.40) It then follows that 𝑑𝑑𝑡𝜇𝑑(𝑚)12𝜌+𝛼𝑑(𝑚)22𝜌min{𝜆,𝜎}+2𝑎+2𝛿𝜃𝑡𝜔+̃𝑞𝜂𝜃𝑡𝜔𝜇𝑑(𝑚)12𝜌+𝛼𝑑(𝑚)22𝜌+2𝜇𝜆122𝜌+2𝛼𝜎𝑔1𝑔22𝜌𝑒2𝛿(𝜃𝑡𝜔).(3.41) For 𝑇>0, applying Gronwall’s inequality to (3.41) on [0,𝑇] implies that 𝜇𝑑(𝑚)1(𝑡)2𝜌+𝛼𝑑(𝑚)2(𝑡)2𝜌𝐶𝑇𝜇𝑑(𝑚)1(0)2𝜌+𝛼𝑑(𝑚)2(0)2𝜌+12𝑡2𝜌+𝑔1𝑔22𝜌(3.42) for some constant 𝐶𝑇 depending on 𝑇, and thus 𝐳(𝑚)1(𝑡)𝐳(𝑚)2(𝑡)2𝐻𝐶𝑇𝐳(𝑚)1(0)𝐳(𝑚)2(0)2𝐻+122𝜌+𝑔1𝑔22𝜌,(3.43) where 𝐶𝑇 is a constant depending on 𝑇. Denote by 𝐸=̃𝑙2×̃𝑙2, where ̃𝑙2=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,)=𝐳(𝑚)(𝑡;𝜔,𝐳0,𝑔,), where 𝐳(𝑚)(𝑡;𝜔,𝐳0,𝑔,) is the solution of (3.28) on [0,𝑇) with 𝐳(𝑚)(0;𝜔,𝐳0,𝑔,)=𝐳0. Since ̃𝑙2 is dense in 𝑙2𝜌, the mapping Φ(𝑚) can be extended uniquely to a continuous mapping Φ(𝑚)𝐻×𝐻𝐶([0,𝑇],𝐻).
For given 𝐳0𝐻 and (𝑔,)𝐻, Φ(𝑚)(𝐳0,𝑔,)=𝐳(𝑚)(;𝜔,𝐳0,𝑔,)𝐶([0,𝑇],𝐻) for 𝑇>0. There exist sequences {𝐳0𝑛}𝐸, {(𝑛,𝑔𝑛)}𝐸 such that 𝐳0𝑛𝐳0𝐸0,𝑛𝜌0,𝑔𝑛𝑔𝜌0as𝑛.(3.44) Let 𝐳(𝑚)𝑛(𝑡)=Φ(𝑚)(𝐳0𝑛,𝑛,𝑔𝑛)=Φ(𝑚)(𝐳0𝑛,𝑛,𝑔𝑛)=𝐳(𝑚)(𝑡;𝜔,𝐳0𝑛,𝑛,𝑔𝑛)𝐸 be the solution of (3.28), then it satisfies the integral equation 𝐳(𝑚)𝑛𝑖(𝑡)=𝐳0𝑛𝑖+𝑡0𝐹𝑖𝐳(𝑚)𝑛,𝜃𝑠𝜔𝑑𝑠.(3.45) By the continuity of Φ(𝑚), we have for 𝑡[0,𝑇), 𝐳(𝑚)𝑡;𝜔,𝐳0𝑛,𝑛,𝑔𝑛=Φ(𝑚)𝐳0𝑛,𝑛,𝑔𝑛𝑛Φ(𝑚)𝐳0,,𝑔=𝐳(𝑚)𝑡;𝜔,𝐳0,,𝑔𝐻.(3.46) Thus for each 𝑖, 𝐳(𝑚)𝑛𝑖(𝑡)𝐳(𝑚)𝑖(𝑡)=𝐳(𝑚)𝑖𝑡;𝜔,𝐳0,,𝑔as𝑛uniformlyon𝑡[0,𝑇).(3.47) Moreover, {(𝐳(𝑚)𝑛)𝑖(𝑡)} is bounded in 𝑛. Let 𝑛, then we have 𝐳(𝑚)𝑖(𝑡)=𝐳0𝑖+𝑡0𝐅𝑖𝐳(𝑚),𝜃𝑠𝜔𝑑𝑠,(3.48) and 𝐳(𝑚)𝑖(𝑡) satisfies the differential equation (3.31).
Multiply equation (3.31) by 𝜇𝜌𝑖𝑥(𝑚)𝑖00𝛼𝜌𝑖𝑦(𝑚)𝑖 and sum over 𝑖, we obtain 𝑑𝑑𝑡𝜇𝑥(𝑚)2𝜌+𝛼𝑦(𝑚)2𝜌min{𝜆,𝜎}+2𝑎+2𝛿𝜃𝑡𝜔+̃𝑞𝜂𝜃𝑡𝜔𝜇𝑥(𝑚)2𝜌+𝛼𝑦(𝑚)2𝜌+2𝜇𝜆2𝜌+2𝛼𝜎𝑔2𝜌𝑒2𝛿(𝜃𝑡𝜔),𝜇𝑥(𝑚)2𝜌+𝛼𝑦(𝑚)2𝜌𝑒2𝑎𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+̃𝑞𝜂(𝜃𝑠𝜔)]𝑑𝑠𝜇𝑥(0)2𝜌+𝛼𝑦(0)2𝜌+2𝜇𝜆2𝜌+2𝛼𝜎𝑔2𝜌𝑒2𝑎𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+̃𝑞𝜂(𝜃𝑠𝜔)]𝑑𝑠𝑡0𝑒(𝜆+𝜎2𝑎)𝑠2𝛿(𝜃𝑠𝜔)+𝑠0[2𝛿(𝜃𝑟𝜔)+̃𝑞𝜂(𝜃𝑟𝜔)]𝑑𝑟𝑑𝑠𝜅𝜌(𝑡),𝑡0,𝑇max(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 of (3.9) with initial data 𝐳0𝐻, which is the limit function of a subsequence of {𝐳(𝑚)(𝑡;𝜔,𝐳0,𝑔,)} in 𝐻 for 𝑡[0,𝑇). In the latter part of this paper, we may write 𝐳(𝑡;𝜔,𝐳0,,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)}𝑡0,𝜔Ω1 over (Ω1,1,1,(𝜃𝑡)𝑡) with state space 𝐻: 𝜓(𝑡,𝜔)𝐳0=𝐳𝑡;𝜔,𝐳0for𝐳0𝐻,𝑡0,𝜔Ω1.(3.50) Moreover, 𝜑(𝑡,𝜔)𝑢0,𝑣0=𝑒𝛿(𝜃𝑡𝜔)𝜓(𝑡,𝜔)𝑒𝛿(𝜔)𝑢0,𝑣0for𝑢0,𝑣0𝐻,𝑡0,𝜔Ω1,(3.51) defines a continuous RDS {𝜑(𝑡,𝜔)}𝑡0,𝜔Ω1 over (Ω1,1,1,(𝜃𝑡)𝑡) 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,,𝑔) is measurable in (𝑡,𝜔,𝐳0).
In fact, for 𝐳0𝐸 and (,𝑔)𝐸, the solution of (3.9) 𝐳(𝑡;𝜔,𝐳0,,𝑔)𝐸 for 𝑡[0,). In this case, function 𝐅(𝐳,𝑡,𝜔,,𝑔)=𝐅(𝐳,𝑡,𝜔) is continuous in 𝐳,, 𝑔 and measurable in 𝑡,𝜔, which implies that 𝐳[0,)×Ω1×𝐸×𝐸𝐸,(𝑡;𝜔,𝐳0,,𝑔)𝐳(𝑡;𝜔,𝐳0,,𝑔) is (([0,)×1×(𝐸)×(𝐸),(𝐸))-measurable.
For 𝐳0𝐻 and (,𝑔)𝐻, the solution 𝐳(𝑡;𝜔,𝐳0,,𝑔)𝐻 for 𝑡[0,). For any given 𝑁>0, define 𝑇𝑁𝐻𝐸, (𝑢,𝑣)=((𝑢𝑖),(𝑣𝑖))𝑖𝑇𝑁(𝑢,𝑣)=((𝑇𝑁(𝑢,𝑣))𝑖)𝑖 by 𝑇𝑁(𝑢,𝑣)𝑗=𝑢𝑗,𝑣𝑗if||𝑗||𝑁,0if||𝑗||>𝑁,(3.52) and write 𝐳𝑁𝑡;𝜔,𝐳0,,𝑔=𝐳𝑡;𝜔,𝑇𝑁𝐳0,𝑇𝑁(,𝑔).(3.53) Then 𝑇𝑁 is continuous and for any 𝐳0𝐻,(,𝑔)𝐻, and 𝐳𝑡;𝜔,𝐳0,,𝑔=lim𝑁𝐳𝑡;𝜔,𝑇𝑁𝐳0,𝑇𝑁(,𝑔).(3.54) Thus 𝐳[0,)×Ω1×𝐸×𝐸𝐻 is (([0,))×0×(𝐸)×(𝐸),(𝐻))-measurable. Observe also that (Id,Id,𝑇𝑁,𝑇𝑁)[0,)×Ω1×𝐻×𝐻[0,)×Ω0×𝐸×𝐸 is (([0,))×0×(𝐻)×(𝐻),([0,))×0×(𝐸)×(𝐸))-measurable. Hence 𝐳𝑁=𝐳(Id,Id,𝑇𝑁,𝑇𝑁)[0,)×Ω1×𝐻×𝐻𝐻 is (([0,))×0×(𝐻)×(𝐻),(𝐻))-measurable. It then follows from (3.54) that 𝐳[0,)×Ω1×𝐻×𝐻𝐻 is (([0,))×1×(H)×(𝐻),(𝐻))-measurable. Therefore, fix (,𝑔)𝐻 we have that 𝐳(𝑡;𝜔,𝐳0)=𝐳(𝑡;𝜔,𝐳0,,𝑔) is measurable in (𝑡,𝜔,𝐳0). The other statements then follow directly.

Remark 3.5. If (,𝑔)𝐸, system (3.1) defines a continuous RDS {𝜑(𝑡)}t0 over (Ω1,1,1,(𝜃𝑡)𝑡) 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)}𝑡0,𝜔Ω1 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 𝐵1𝜌(𝜔)𝒟(𝐻) of {𝜑(𝑡,𝜔)}𝑡0,𝜔Ω1 such that for any 𝐷(𝜔)𝒟(𝐻) and each 𝜔Ω1, there exists 𝑇𝐷(𝜔)>0 yielding 𝜑(𝑡,𝜃𝑡𝜔)𝐷(𝜃𝑡𝜔)𝐵1𝜌(𝜔) for all 𝑡𝑇𝐷(𝜔). In particular, there exists 𝑇1𝜌(𝜔)>0 such that 𝜑(𝑡,𝜃𝑡𝜔)𝐵1𝜌(𝜃𝑡𝜔)𝐵1𝜌(𝜔) for all𝑡𝑇1𝜌(𝜔).

Proof. (1) For initial condition 𝐳0𝐸 and (,𝑔)𝐸, let 𝐳(𝑚)(𝑡,𝜔)=𝐳(𝑚)(𝑡;𝜔,𝐳0(𝜔),,𝑔) be a solution of (3.28) with 𝐳0(𝜔)=𝑒𝛿(𝜔)𝐳0𝐸, where 𝜔Ω1, then 𝐳(𝑚)(𝑡,𝜔)𝐸 for all 𝑡0. Let 𝜖1>0 be such that 𝜆1=2min{𝜆,𝜎}𝜖1>2̃𝑞𝐄||𝜂(𝜔)||.(3.55) By (H4) and (H5), we have 𝑏2𝜌𝑖𝜌𝑖𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖𝑓𝑖𝑒𝛿(𝜃𝑡𝜔)𝑥(𝑚)𝑖<,forxed𝑡0,(3.56)𝑑𝑑𝑡𝜇𝑥(𝑚)2𝜌+𝛼𝑦(𝑚)2𝜌𝜆1+2𝛿𝜃𝑡𝜔+2̃𝑞𝜂𝜃𝑡𝜔𝜇𝑥(𝑚)2𝜌+𝛼𝑦(𝑚)2𝜌+2𝑒2𝛿(𝜃𝑡𝜔)𝜇𝑏2𝜌+𝜇𝜖12𝜌+𝛼𝜎𝑔2𝜌.(3.57) Applying Gronwall’s inequality to (3.57), we obtain that for 𝑡>0, 𝜇𝑥(𝑚)(𝑡,𝜔)2𝜌+𝛼𝑦(𝑚)(𝑡,𝜔)2𝜌𝑒𝜆1𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+2̃𝑞𝜂(𝜃𝑠𝜔)]𝑑𝑠𝜇𝑥(0)2𝜌+𝛼𝑦(0)2𝜌+2𝜇𝑏2𝜌+𝜇𝜖12𝜌+𝛼𝜎𝑔2𝜌𝑒𝜆1𝑡+𝑡0[2𝛿(𝜃𝑠𝜔)+2̃𝑞𝜂(𝜃𝑠𝜔)]𝑑𝑠𝑡0𝑒𝜆1𝑠2𝛿(𝜃𝑠𝜔)𝑠0[2𝛿(𝜃𝑟𝜔)+2̃𝑞𝜂(𝜃𝑟𝜔)]𝑑𝑟𝑑𝑠.(3.58)
(2) For any 𝐳0𝐻 and (,𝑔)𝐻, let {𝐳0𝑛}𝐸 and {(𝑛,𝑔𝑛)}𝐸 be sequences such that 𝐳0𝑛𝐳0𝐸0,𝑛𝜌0,𝑔𝑛𝑔𝜌0as𝑛.(3.59) Then 𝐳(𝑚)(𝑡;𝜔,𝐳0𝑛,𝑛,𝑔𝑛)𝐳(𝑚)(𝑡;𝜔,𝐳0,,𝑔) as 𝑛 in 𝐻, and (3.58) holds for 𝐳0𝐻. Therefore, 𝜇𝑥(𝑚)𝑡,𝜃𝑡𝜔,𝐳0𝜃𝑡𝜔2𝜌+𝛼𝑦(𝑚)𝑡,𝜃𝑡𝜔,𝐳0𝜃𝑡𝜔2𝜌𝑒𝜆1𝑡+𝑡0[2𝛿(𝜃𝑠𝑡𝜔)+2̃𝑞𝜂(𝜃𝑠𝑡𝜔)]𝑑𝑠𝜇𝑥0𝜃𝑡𝜔2𝜌+𝛼𝑦0𝜃𝑡𝜔2𝜌+12𝑟21𝜌(𝜔)𝑒2𝛿(𝜔),(3.60) where 𝑟21𝜌(𝜔)=4𝑒2𝛿(𝜔)𝜇𝑏2𝜌+𝜇𝜖12𝜌+𝛼𝜎𝑔2𝜌0𝑒𝜆1𝑠2𝛿(𝜃𝑠𝜔)+0𝑠[2𝛿(𝜃𝑟𝜔)+2̃𝑞𝜂(𝜃𝑟𝜔)]𝑑𝑟𝑑𝑠<.(3.61) For any 𝛽>0, since 𝑒𝛽𝑡𝑟0𝜌𝜃𝑡𝜔2𝜇𝑏2𝜌+𝜇𝜖1