Pullback measure attractors for non-autonomous stochastic lattice systems

Abstract

The aim of this article is to study the asymptotic behaviour of non-autonomous stochastic lattice systems. We first show the existence and uniqueness of a pullback measure attractor. Moreover, when deterministic external forcing terms are periodic in time, we show the pullback measure attractors are periodic. We then study the upper semicontinuity of pullback measure attractors as the noise intensity goes to zero. Pullback asymptotic compact for a family of probability measures with respect to probability distributions of the solutions is demonstrated by using uniform a priori estimates for far-field values of solutions.

1. Introduction

In this article, we study the dynamical behaviour of the non-autonomous stochastic lattice system defined on the integer set $\mathbb Z$: for $\tau \in \mathbb{R}$,

(1.1)\begin{align} \begin{aligned} du_i \left( t \right)& - \nu \left( {u_{i - 1} \left( t \right) - 2u_i \left( t \right) + u_{i + 1} \left( t \right)} \right)dt + \lambda(t) u_i \left( t \right)dt \\ &= \left( {F \left( {t,u_i(t)} \right) + g_i(t) } \right)dt + \varepsilon\sum\limits_{k = 1}^\infty {\left( {h_{i,k}(t) + \sigma _{i,k} \left( {t,u_i \left( t \right)} \right)} \right)} dW_k \left( t \right),\quad t \gt \tau, \end{aligned} \end{align}

with initial data

(1.2)\begin{equation} u_i \left(\tau\right) = \xi _i, \end{equation}

where $u=(u_i)_{i\in \mathbb Z}$ is an unknown sequence, $\xi=(\xi_i)_{i\in \mathbb Z}\in l^2$ is given, $0 \lt \varepsilon \le 1$, ν > 0, $\lambda(t) \gt 0$, $g(t)=(g_i(t))_{i\in \mathbb Z}$ and $h(t)=(h_{i,k}(t))_{i\in \mathbb Z,k\in \mathbb N}$ are given time dependent sequence, $F, \sigma_{i,k}: \mathbb R\times \mathbb{R} \rightarrow \mathbb R$ are nonlinearity satisfying certain structural conditions for every $i\in \mathbb Z$ and $k\in \mathbb N$, and $(W_k)_{k\in \mathbb N}$ is a sequence of independent standard two-side Wiener processes on a complete filtered probability space $ (\Omega ,\mathcal F, \{\mathcal F_t \}_{t\in \mathbb{R}} ,P )$ satisfying the usual condition.

It seems that measure attractors for autonomous stochastic equations was first studied in [15] where Schmalfuss considered the existence of measure attractors of a stochastic Navier–Stokes equation with additive noise and upper semicontinuity of measure attractors as the noise intensity goes to zero. Related equations with nonlinear noise were investigated for the existence results of measure attractors in [12–14]. The relation between measure attractor and random attractor was studied in [7, 16] for stochastic equations with additive noise. In [9], Li and Wang extended the notion of measure attractors to pullback measure attractors in order to capture the dynamic behaviours of non-autonomous stochastic differential equations.

Lattice differential equations arise naturally in a wide variety of applications where the spatial structure has a discrete character. Such systems also arise in numerical simulations when discretizing PDEs（Partial Differential Equations）defined on unbounded domains. When random influences are taken into account, stochastic lattice systems have been extensively investigated. For random attractors, we refer the readers to [1, 3–5, 8, 17] for autonomous case and [2, 21, 22] for non-autonomous case. The invariant measures or periodic measures for stochastic lattice systems have been investigated by [6, 10, 11, 19, 20].

Here we prove the existence, uniqueness, periodicity, and upper semicontinuity of pullback measure attractors for the non-autonomous stochastic lattice differential equations (1.1)–(1.2). Note that the stochastic lattice system (1.1) shares some similar property with stochastic PDEs defined on the entire space $\mathbb{R}$. The main difficulty is how to establish the asymptotic tightness for a family of probability distributions of solutions. The uniform estimates on the tails of solutions are employed to prove the asymptotic tightness.

The rest of this article is organized as follows. In §2, we recall some fundamental results on the existence, uniqueness, and periodicity of a pullback measure attractor for non-autonomous dynamical systems defined on the space of probability measures of Banach spaces. Section 3 is devoted to the existence and uniqueness of solutions to the non-autonomous stochastic lattice system (1.1)–(1.2). In §4, we derive the uniform moment estimates of solutions as $t\rightarrow \infty$. These estimates are necessary for proving the existence of absorbing sets and the pullback asymptotic compactness of the non-autonomous dynamical systems with respect to the Markov semigroup generated by (1.1)–(1.2). In the last two sections, we establish the existence, uniqueness, and periodicity of pullback measure attractors for (1.1)–(1.2) and prove the convergence of pullback measure attractors of system (1.1)–(1.2) as ɛ → 0.

2. Preliminaries

In this section, for the readers convenience, we recall some results regarding pullback measure attractors for non-autonomous dynamical systems on the space of probability measures(see, e.g., [9]).

In what follows, we denote by X a separable Banach space with norm $\|\cdot\|_X$. Let $C_b(X)$ be the space of bounded continuous functions $\varphi:X\rightarrow \mathbb{R}$ endowed with the norm

\begin{equation*} \left\| \varphi \right\|_\infty = \mathop {\sup }\limits_{x \in X} \left| {\varphi\left( x \right)} \right|. \end{equation*}

Denote by $L_b(X)$ the space of bounded Lipschitz functions on X which consists of all functions $\varphi\in C_b(X)$ such that

\begin{equation*} \text{Lip}\left( \varphi \right): = \mathop {\sup }\limits_{x_1 ,x_2 \in X,x_1\neq x_2} \frac{{\left| {\varphi \left( {x_1 } \right)} - {\varphi\left( {x_2 } \right)} \right|}}{{\|x_1-x_2\|_X}} \lt \infty . \end{equation*}

The space $L_b(X)$ is endowed with the norm

\begin{equation*} \left\| \varphi \right\|_L = \left\| \varphi \right\|_\infty + \text{Lip}\left( \varphi \right). \end{equation*}

Let $\mathcal P(X)$ be the set of probability measures on $(X,\mathcal B(X))$, where $\mathcal B(X)$ is the Borel σ-algebra of X. Given $\varphi\in C_b(X)$ and $\mu \in \mathcal P(X)$, we write

\begin{equation*} \left( {\varphi, \mu } \right) = \int_{X} {\varphi \left( x \right)\mu \left( {dx} \right)}. \end{equation*}

Recall that a sequence $\{\mu_n \}_{n=1}^{\infty} \subseteq \mathcal P(X)$ is weakly convergent to $\mu \in \mathcal P(X)$ if for every $\varphi\in C_b(X)$,

\begin{equation*} \lim_{n\to \infty } \left( {\varphi, \mu_n } \right) =\left( {\varphi, \mu } \right). \end{equation*}

Define a metric on $\mathcal P(X)$ by

\begin{equation*} d_{\mathcal P(X)} \left( {\mu _1 ,\mu _2 } \right) = \mathop {\sup }\limits_{ \varphi \in L_b \left( X \right) \atop \left\| \varphi \right\|_L \le 1 } \left| {\left( {\varphi,\mu _1 } \right) - \left( {\varphi,\mu _2 } \right)} \right|, \quad \forall \ \mu_1,\mu_2\in \mathcal P(X). \end{equation*}

Then $(\mathcal P(X), d_{\mathcal P(X)} )$ is a polish space. Moreover, a sequence $\left\{{\mu _n } \right\}_{n = 1}^\infty \subseteq \mathcal P\left( X \right)$ converges to µ in $(\mathcal P(X), d_{\mathcal P(X)} )$ if and only if $\{\mu_n \}_{n=1}^{\infty}$ converges to µ weakly.

Given p > 0, let $\mathcal P_p(X)$ be the subset of $\mathcal P(X)$ defined by

\begin{equation*} \mathcal P_p \left( X \right) = \left\{{\mu \in \mathcal P\left( X \right): \int_X {\|x\|_X^p \mu \left( {dx} \right) \lt \infty } } \right\}. \end{equation*}

Then $(\mathcal P_p(X), d_{\mathcal P(X)} )$ is also a metric space. Without confusion, we denote $(\mathcal P_p(X), d_{\mathcal P(X)} )$ by $(\mathcal P_p(X), d_{\mathcal P_p(X)} )$. Given r > 0, denote by

\begin{equation*} B_ {\mathcal P_p(X)} (r) = \left\{{\mu \in \mathcal P_p \left( X \right) : \left(\int_X \|x\|_X^p \mu \left( {dx} \right)\right)^{\frac{1}{p}} \leq r } \right\}. \end{equation*}

Recall that the Hausdorff semi-metric between subsets of ${\mathcal P_p(X)}$ is given by

\begin{equation*} d_ {\mathcal P_p(X)}(Y,Z) = \mathop {\sup }\limits_{y \in Y} \mathop {\inf }\limits_{z \in Z} d \left( {y,z} \right), \quad Y, Z \subseteq \mathcal P_p(X), \ \ Y, Z \neq \emptyset . \end{equation*}

If ϵ > 0 and $B\subseteq \mathcal P_p \left( X \right)$, then the open ϵ-neighbourhood of B in $\mathcal P_p \left( X \right)$ is defined by

\begin{equation*} \mathcal N_\epsilon(B) = \left\{{\mu \in \mathcal P_p(X):d_ {\mathcal P_p(X)} \left( {\mu , B} \right) \lt \epsilon } \right\}. \end{equation*}

Definition 2.1. A family $S=\{S(t,\tau): t\in \mathbb{R}^+,\,\tau\in \mathbb{R}\}$ of mappings from $\mathcal P_p (X)$ to $\mathcal P_p (X)$ is called a continuous non-autonomous dynamical system on $\mathcal P_p(X)$, if for all $\tau\in \mathbb{R}$ and $t, s\in \mathbb{R}^+$, the following conditions are satisfied:

(a) $S(0,\tau)=I_{\mathcal P_p(X)}$, where $I_{\mathcal P_p(X)}$ is the identity operator on $\mathcal P_p(X)$;

(b) $S(t+s,\tau)=S(t,s+\tau)\circ S(s,\tau)$;

(c) $S(t,\tau): \mathcal P_p(X)\rightarrow \mathcal P_p(X)$ is continuous.

If, in addition, there exists a positive number T such that for every $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$,

\begin{equation*}S(t, \tau + T ) = S(t, \tau),\end{equation*}

then S is called a continuous periodic non-autonomous dynamical system on $\mathcal P_p(X)$ with period T.

Definition 2.2. A set $D\subseteq {\mathcal P}_p \left( X \right)$ is called a bounded subset if there is r > 0 such that $D\subseteq B_ {{\mathcal P}_p \left( X \right) } (r)$.

In the sequel, we denote by $\mathcal D$ a collection of some families of nonempty subsets of $\mathcal P_p(X)$ parametrized by $\tau \in \mathbb{R}$; that is,

\begin{equation*} \mathcal D = \left\{{D = \left\{{D\left( \tau \right) \subseteq \mathcal P_p \left( X \right): D\left( \tau \right) \ne \emptyset, \,\, \tau \in \mathbb{R}} \right\}:\,\, D\,\,{\text{satisfies some conditions}}} \right\}. \end{equation*}

Definition 2.3. A collection $\mathcal D$ of some families of nonempty subsets of $\mathcal P_p(X)$ is said to be neighbourhood-closed if for each $D = \left\{{D\left( \tau \right):\tau \in \mathbb{R}} \right\} \in \mathcal D$, there exists a positive number ϵ depending on D such that the family

\begin{equation*} \left\{{B\left( \tau \right):B\left( \tau \right)\,{\text{is a nonempty subset of }}\, \mathcal N_\epsilon(D(\tau)), \,\forall \tau\in \mathbb{R} } \right\} \end{equation*}

also belongs to $\mathcal D$.

Note that the neighbourhood closedness of $\mathcal D$ implies for each $D\in \mathcal D$,

(2.1)\begin{equation} \widetilde D = \left\{{\widetilde D\left( \tau \right):\emptyset \ne \widetilde D\left( \tau \right) \subseteq D\left( \tau \right),\,\,\tau \in \mathbb{R}} \right\}\in \mathcal D. \end{equation}

A collection $\mathcal D$ satisfying (2.1) is said to be inclusion-closed in the literature.

Definition 2.4. A family $K = \left\{{K\left( \tau \right):\tau \in \mathbb{R}} \right\}\in \mathcal D$ is called a $\mathcal D$-pullback absorbing set for S if for each $\tau\in \mathbb{R}$ and every $D\in \mathcal D$, there exists $T=T(\tau,D) \gt 0$ such that

\begin{equation*} S \left( t,\tau-t \right)D(\tau-t) \subseteq K(\tau),\quad\text{for all}\quad t \geq T. \end{equation*}

If there exists a positive number T such that $K(\tau+ T ) = K(\tau)$ for every $\tau\in \mathbb{R}$, then K is said to be periodic with period T.

Definition 2.5. The non-autonomous dynamical system S is said to be $\mathcal D$-pullback asymptotically compact in $\mathcal P_p \left( X \right)$ if for each $\tau\in \mathbb{R}$, $\left\{{S \left( {t_n,\tau-t_n } \right)\mu _n } \right\}_{n = 1}^\infty $ has a convergent subsequence in $\mathcal P_p \left( X \right)$ whenever $t_n\rightarrow +\infty$ and $\mu _n\in D(\tau-t_n)$ with $D\in \mathcal D$.

Definition 2.6. A family $\mathcal A=\{\mathcal A(\tau):\,\tau\in \mathbb{R}\}\in \mathcal D$ is called a $\mathcal D$-pullback measure attractor for S if the following conditions are satisfied,

(i) $\mathcal A(\tau)$ is compact in $\mathcal P_p \left( X \right)$ for each $\tau\in \mathbb{R}$;

(ii) $\mathcal A$ is invariant, that is, $S(t,\tau) \mathcal A(\tau)=\mathcal A(\tau+t)$, for all $ \tau\in \mathbb{R}$ and $t\in \mathbb{R}^+$;

(iii) $\mathcal A$ attracts every set in $\mathcal D$, that is, for each $ D = \left\{{D\left( \tau \right):\tau \in \mathbb{R}} \right\} \in \mathcal D,$

\begin{equation*} \mathop {\lim }\limits_{t \to \infty } d \left( {S \left( t,\tau-t \right)D(\tau-t),\mathcal A(\tau)} \right) = 0. \end{equation*}

Definition 2.7. A mapping $\psi :\mathbb{R}\times \mathbb{R}\rightarrow \mathcal P_p \left( X \right)$ is called a complete orbit of S if for every $s\in \mathbb{R}$, $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$, the following holds:

(2.2)\begin{equation} S\left( t ,s+\tau\right)\psi \left( s,\tau \right) = \psi \left( {t+s,\tau} \right). \end{equation}

If, in addition, there exists $D=\{D(\tau):\tau\in \mathbb{R}\}\in \mathcal D$ such that $\psi(t,\tau)$ belongs to $D(\tau+t)$ for every $t\in \mathbb{R}$ and $\tau\in \mathbb{R}$, then ψ is called a $\mathcal D$-complete orbit of S.

Definition 2.8. A mapping $\xi : \mathbb{R}\rightarrow \mathcal P_p \left( X \right)$ is called a complete solution of S if for every $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$, the following holds:

\begin{equation*} S\left( t , \tau \right)\xi \left( \tau \right) = \xi \left( {t + \tau} \right). \end{equation*}

If, in addition, there exists $D=\{D(\tau):\tau\in \mathbb{R}\}\in \mathcal D$ such that $\xi(\tau)$ belongs to $D(\tau)$ for every $\tau\in \mathbb{R}$, then ξ is called a $\mathcal D$-complete solution of S.

Definition 2.9. For each $D=\{D(\tau):\tau\in \mathbb{R}\}\in \mathcal D$ and $\tau \in \mathbb{R}$, the pullback ω-limit set of D at τ is defined by

\begin{equation*} \omega \left( {D,\tau} \right): = \bigcap\limits_{s \geq 0} {\overline {\bigcup\limits_{t \geq s } {S\left( {t,\tau-t} \right)D(\tau-t)} } }, \end{equation*}

that is,

\begin{align*} \omega \left( {D,\tau} \right) & = \left\{{\nu \in \mathcal P_p \left( X \right)}: \, \text{there exist }\, t_n \rightarrow \infty, \ \mu_n \in D(\tau-t_n) \,\text{such that }\,\right. \notag\\ \nu & \left. = \mathop {\lim }\limits_{n \to \infty } S\left( {t_n,\tau-t_n } \right)\mu _n \right\}. \end{align*}

Based on the above notation, from theorem 2.25 and proposition 3.6 in [18], we have the following criterion for the existence, uniqueness, and periodicity of $\mathcal D$-pullback measure attractors.

Proposition 2.10. Let $\mathcal D$ be a neighbourhood-closed collection of families of subsets of $\mathcal P_p \left( X \right)$ and S be a continuous non-autonomous dynamical system on $\mathcal P_p \left( X \right)$. Then S has a unique $\mathcal D$-pullback measure attractor $\mathcal A$ in $\mathcal P_p \left( X \right)$ if and only if S has a closed $\mathcal D$-pullback absorbing set $K\in \mathcal D$ and S is $\mathcal D$-pullback asymptotically compact in $\mathcal P_p \left( X \right)$. The $\mathcal D$-pullback measure attractor $\mathcal A$ is given by, for each $\tau\in \mathbb{R}$,

\begin{align*} \mathcal A\left( \tau \right) = \omega(K,\tau) =\{\psi(0,\tau):\,\psi\,\text{is a}\,\, \mathcal D \text{-complete orbit of}\,\,S\}\\ =\{\xi(\tau):\,\xi\,\text{is a}\,\, \mathcal D \text{-complete solution of}\,\,S\}. \end{align*}

If, in addition, both S and K are T-periodic for some T > 0, then so is the attractor $\mathcal A$, i.e., $\mathcal A(\tau)=\mathcal A(\tau+T)$, for all $\tau\in \mathbb{R}$.

Next, we give an abstract result for the upper semicontinuity of pullback measure attractors of a family of non-autonomous dynamical systems on $\mathcal P_p(X)$.

Suppose Λ is an interval of $\mathbb R$, and for each $\lambda\in\Lambda$, $\Phi_{\lambda}$ is a non-autonomous dynamical system on $\mathcal P_p(X)$. Suppose that for each $\lambda \in \Lambda$, $\Phi_\lambda$ has a $\mathcal D$-pullback measure attractor $\mathcal A_\lambda\in \mathcal D$. Assume there exists $\lambda_{0}\in\Lambda$ such that for $\tau\in \mathbb{R}$ and $t\in\mathbb{R}^+$,

(2.3)\begin{align} \mathop {\lim }\limits_{n \to \infty } \mathop {\sup }\limits_{\mu \in \mathcal A_{\lambda _n } } d_{\mathcal P_p \left( X \right)} \left( {\Phi _{\lambda _n } \left( t,\tau \right)\mu , \Phi _{\lambda _0 } \left( t,\tau \right)\mu } \right) = 0, \end{align}

for any $\lambda_n\rightarrow \lambda_0$.

We also assume that

(2.4)\begin{align} K=\Big\{K(\tau)=\bigcup\limits_{\lambda\in\Lambda}\mathcal A_{\lambda}(\tau): \tau\in \mathbb{R}\Big\}\in \mathcal{D}. \end{align}

We now present the upper semicontinuity of $\mathcal A_{\lambda}$ as $\lambda\rightarrow \lambda_0$.

Theorem 2.11 Suppose (2.3)–(2.4) hold. Then for $\tau\in \mathbb{R}$,

\begin{align*} \mathop {\lim }\limits_{\lambda\rightarrow \lambda_0 }d_{\mathcal P_p(X)} \big(\mathcal A_{\lambda}(\tau),\mathcal A_{\lambda_0}(\tau)\big)=0. \end{align*}

Proof. Since $K\in \mathcal{D}$ from (2.4), for $\tau\in \mathbb{R}$ and η > 0, there exists a $T=T(\tau,\eta) \gt 0$ such that for all $t\geq T$,

(2.6)\begin{align} d_{\mathcal P_p(X)}(\Phi_{\lambda_0}(t,\tau-t)K(\tau-t),\mathcal A_{\lambda_0}(\tau)) \lt \eta. \end{align}

Now let $\mu_{n}\in\mathcal A_{\lambda_n}(\tau)$, $n\in\mathbb N$. Since the measure attractor $\mathcal A_{\lambda_n}$ is invariant under $\Phi_{\lambda_n}$, there exists a $\nu_{n}\in\mathcal A_{\lambda_n}(\tau-T)$ such that

(2.7)\begin{align} \mu_{n}=\Phi_{\lambda_{n}}(T,\tau-T)\nu_{n}. \end{align}

By (2.3), we obtain

(2.8)\begin{align} \lim\limits_{n\rightarrow\infty}\sup_{\nu_{n}\in\mathcal A_{\lambda_{n}}(\tau-T)} \|\Phi_{\lambda_n}(T,\tau-T)\nu_{n}-\Phi_{\lambda_0}(T,\tau-T)\nu_{n}\|_{X}=0. \end{align}

It follows from (2.6)–(2.8) that for large enough n,

\begin{align*} &d_{\mathcal P_p(X)}\big(\mathcal A_{\lambda_{n}}(\tau),\mathcal A_{\lambda_0}(\tau)\big) =\sup\limits_{\mu_{n}\in\mathcal A_{\lambda_n}(\tau)} d_{\mathcal P_p(X)}\big(\mu_{n},\mathcal A_{\lambda_0}(\tau)\big)\\ &\le\sup\limits_{\mu_{n}\in\mathcal A_{\lambda_n}(\tau)} (d_{\mathcal P_p(X)}(\mu_{n},\Phi_{\lambda_0}(T,\tau-T)\nu_n)+d_{\mathcal P_p(X)}(\Phi_{\lambda_0} (T,\tau-T)\nu_n,\mathcal A_{\lambda_0}(\tau)))\\ &\le\sup\limits_{\nu_{n}\in\mathcal A_{\lambda_n}(\tau-T)}d_{\mathcal P_p(X)} (\Phi_{\lambda_n}(T,\tau-T)\nu_n,\Phi_{\lambda_0}(T,\tau-T)\nu_n)\\ &+\sup\limits_{\nu_{n}\in\mathcal A_{\lambda_n}(\tau-T)} d_{\mathcal P_p(X)}(\Phi_{\lambda_0}(T,\tau-T)\nu_n,\mathcal A_{\lambda_0}(\tau))\\ &\le \sup\limits_{\nu_{n}\in\mathcal A_{\lambda_n}(\tau-T)}d_{\mathcal P_p(X)} (\Phi_{\lambda_n}(T,\tau-T)\nu_n,\Phi_{\lambda_{0}}(T,\tau-T)\nu_n)\\ &+d_{\mathcal P_p(X)}(\Phi_{\lambda_0}(T,\tau-T)K(\tau-T),\mathcal A_{\lambda_0}(\tau))\\ & \lt 2\eta. \end{align*}

This completes the proof.

3. Existence and uniqueness of solutions

In this section, we prove the existence and uniqueness of solutions to system (1.1)–(1.2). We first discuss the assumptions on the nonlinear drift and diffusion terms in (1.1).

Throughout this article, suppose $g, h: \mathbb{R} \to l^2$, $g(t)=(g_i(t))_{i\in \mathbb Z}$ and $h(t)=(h_{i,k} (t))_{i\in \mathbb Z,k\in \mathbb N}$ are both continuous in $t\in \mathbb{R}$, which implies that for every $t\in \mathbb{R}$,

(3.1)\begin{equation} \left\| g(t) \right\|^2 = \sum\limits_{i \in \mathbb Z} {\left| {g_i (t)} \right|^2 } \lt \infty \quad \text{and} \quad \left\| h (t) \right\|^2 = \sum\limits_{i \in \mathbb Z} {\sum\limits_{k \in \mathbb N} {\left| {h_{i,k} (t) } \right|^2 } } \lt \infty , \end{equation}

where $\| \cdot \|$ is the norm of l ². The inner product of l ² will be denoted by $(\cdot, \cdot)$ throughout this article.

Assume that $F :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb R$, $F =F (t, s)$, is continuous in $(t, s)\in \mathbb{R} \times \mathbb{R}$, $\frac{\partial F (t, s)}{\partial s}\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$ and there exists a positive continuous function $\beta_0: \mathbb{R} \to \mathbb{R}$ such that

(3.2)\begin{equation} F\left( {t,0} \right) = 0\quad \text{and}\quad \frac{{\partial F \left( {t,s} \right)}}{{\partial s}} \le -\beta _0 \left( t \right),\,\,\text{for all}\,\,t,s\in \mathbb{R}. \end{equation}

For the diffusion terms in (1.1), we assume $\sigma_{i,k}:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb R$, $\sigma_{i,k}= \sigma_{i,k}(t,s)$, is continuous in $(t,s)\in \mathbb{R} \times \mathbb{R}$ and globally Lipschitz in $s\in \mathbb{R} $ uniformly with respect to $i\in \mathbb Z$; more precisely, for every $k\in \mathbb{N}$, there exists a constant $L_k \gt 0$ such that for all $ t, s,s^* \in \mathbb{R},\, \text{and} \,\, i \in \mathbb Z,$

(3.3)\begin{equation} \left| {\sigma _{i,k} \left( {t,s } \right) - \sigma _{i,k} \left( {t,s^* } \right)} \right| \le L_k\left| {s - s^* } \right|, \end{equation}

where $L=\left( {L _k } \right)_{k \in \mathbb N}\in l^2$. In addition, we assume $\sigma_{i,k}(t,s)$ grows linearly in $ s\in \mathbb{R} $; that is, for each $k\in \mathbb{N}$ and $i \in \mathbb Z$,

(3.4)\begin{equation} \left| {\sigma _{i,k} \left( {t,s } \right)} \right| \le \delta _{i,k}(t) + \beta _k (t) \left| s \right|,\quad\forall\ t, \, s \in \mathbb{R}, \,k\in \mathbb{N}\quad \text{and} \quad i \in \mathbb Z, \end{equation}

where $\delta(\cdot)=\left( {\delta _{i,k}(\cdot) } \right)_{i \in \mathbb Z,k \in \mathbb{N}} : \mathbb{R} \to l^2$ and $\beta(\cdot)= (\beta_k(\cdot))_{k\in \mathbb{N}}: \mathbb{R} \to l^2$ are positive continuous functions.

The following notation will be used throughout the article:

\begin{equation*} \left\| L \right\|^2 = \sum\limits_{k \in \mathbb{N}} {\left| {L_k } \right|^2 } ,\,\,\left\| \beta (t) \right\|^2 = \sum\limits_{k \in \mathbb{N}} {\left| {\beta _k (t) } \right|^2 } ,\,\,\left\| \delta \right\|^2 = \sum\limits_{i\in \mathbb{Z}} {\sum\limits_{k \in \mathbb{N}} {\left| {\delta _{i,k} } \right|^2 } } . \end{equation*}

For convenience, we set for all $i\in \mathbb{Z}$ and $s,t\in \mathbb{R}$,

\begin{equation*} f \left( {t,s} \right) = F \left( {t,s} \right) + \beta _0 \left( t \right)s. \end{equation*}

Then by (3.2) we obtain for all $s,t\in \mathbb{R}$,

(3.5)\begin{equation} f \left( {t,0} \right) = 0\quad \text{and}\quad \frac{{\partial f \left( {t,s} \right)}}{{\partial s}} \le 0. \end{equation}

In addition, for $u=(u_i)_{i\in \mathbb Z} \in l^2$, we write $f(t,u)=(f(t,u_i))_{i\in \mathbb Z}$ and $\sigma _k \left( t,u\right) = \left( {\sigma _{i,k} \left( {t,u_i } \right)} \right)_{i \in \mathbb Z}$. Since $\frac{\partial f (t, s)}{\partial s}\in C(\mathbb{R}\times \mathbb{R},\mathbb{R})$ with $f(t,0) =0$, one can verify that f is a locally Lipschitz mapping from l ² to l ²; that is, for every bounded set E in l ² and I in $\mathbb{R}$, there exists a constant $L_f=L_f(E,I) \gt 0$ such that

(3.6)\begin{equation} \left\| {f\left(t, u_1 \right) - f\left(t, u_2 \right)} \right\|^2 \le L_f \left\| {u_1 - u_2} \right\|^2 ,\quad \text{for all}\quad u_1,u_2\in E\quad \text{and}\quad t\in I. \end{equation}

It follows from (3.5) that for all $t\in \mathbb{R}$ and $u_1,u_2\in l^2,$

(3.7)\begin{equation} \left( {f\left( {t,u_1 } \right) - f\left( {t,u_2 } \right),u_1 - u_2 } \right) \le 0. \end{equation}

Similarly, by (3.3)–(3.4), we have for all $t\in \mathbb{R}$ and $u_1,u_2$,

(3.8)\begin{equation} \sum\limits_{k \in \mathbb{N}} {\left\| {\sigma _k \left( t,u_1 \right) - \sigma _k \left(t,u_2 \right)} \right\|} ^2 \le \left\| L \right\|^2\left\| {u_1 - u_2} \right\|^2 \end{equation}

and

(3.9)\begin{equation} \sum\limits_{k \in \mathbb{N}} {\left\| {\sigma _k \left(t, u_1 \right)} \right\|} ^2 \le 2\left\| \delta(t) \right\|^2 + 2\left\| \beta (t) \right\|^2 \left\| u_1 \right\|^2. \end{equation}

For simplicity, define linear operators $ A,B, :l^2 \to l^2$ by

\begin{equation*} \left( {Au} \right)_i = -u_{i - 1} + 2u_i - u_{{i + 1}} , \quad \left( {Bu} \right)_i = u_{i + 1} - u_i , \quad i \in \mathbb Z,\,\,u=(u_i)_{i\in \mathbb Z} \in l^2 . \end{equation*}

Then, system (1.1)–(1.2) can be put into the following form in l ² for $t \gt \tau$:

(3.10)\begin{align} \begin{aligned} du\left( t \right) &+ \nu Au\left( t \right)dt + (\lambda(t)+\beta_0(t)) u\left( t \right)dt = \left( {f\left( {t,u(t)} \right) + g (t)} \right)dt \\ &\quad +\varepsilon \sum\limits_{k = 1}^\infty {\left( {h_k(t) + \sigma _k \left( {t,u\left( t \right)} \right)} \right)} dW_k \left( t \right), \end{aligned} \end{align}

with initial condition

(3.11)\begin{align} u\left( \tau \right) = \xi, \end{align}

where $h_k(t) = (h_{i,k} (t))_{i\in \mathbb{Z}} \in l^2$ for each $k\in \mathbb{N}$.

We use $L_{\mathcal F_\tau}^2\left( {\Omega,l^2 } \right)$ to denote the space of all ${\mathcal F_\tau}$-measurable, l ²-valued random variables φ with $\mathbb{E}(\|\varphi\|^2) \lt \infty$, where $\mathbb E$ means the mathematical expectation. Similar to [19], under conditions (3.1)–(3.4), we can show that for any $\xi \in L^2_{\mathcal F_\tau}(\Omega,l^2)$, system (3.10)–(3.11) has a unique solution, which is written as u(t). In particular, u(t), $t\geq \tau$, is a continuous l ²-valued $\mathcal F_t$-adapted stochastic process such that

(3.12)\begin{equation} u \in L^2 \left( {\Omega ,C\left( {\left[ {\tau ,\tau + T} \right],l^2 } \right)} \right), \end{equation}

for every T > 0. To highlight the initial time and initial values, we denote by $u(t,\tau,\xi)$ the solution of (3.10)–(3.11) with initial conditions $u (\tau) = \xi\in L^2_{\mathcal F_\tau}(\Omega,l^2) $.

Give a subset E of $\mathcal P_2 \left( l^2 \right)$, define

\begin{equation*} \|E \|_{\mathcal P_2 \left( l^2\right)} =\inf \left\{{r \gt 0: \sup_{\mu\in E} \left ( \int_{l^2} \|z \|^2 \mu (dz)\right ) ^{\frac 12} \le r } \right\}, \end{equation*}

with the convention that inf $\emptyset =\infty$. If E is a bounded subset of $\mathcal P_2 \left( l^2 \right)$, then $ \|E \|_{\mathcal P_2 \left( l^2 \right)} \lt \infty $. Let $\mathcal{D}$ be the collection of families of bounded nonempty subsets of $\mathcal P_2 \left( l^2 \right)$ as given by

\begin{align*} \mathcal D & = \left\{D = \left\{{D\left( \tau \right) \subseteq \mathcal P_2 \left( l^2 \right): \emptyset\ne D\left( \tau \right) \,\, \text{bounded in }\,\,\mathcal P_2 \left( l^2 \right),\,\,\tau \in \mathbb{R}} \right\}:\right. \\& \left. \quad \mathop {\lim }\limits_{\tau \to - \infty } e^{\gamma \tau } \left\| {D\left( \tau \right)} \right\|_{\mathcal P_2 (l^2 )}^2 = 0 \right\}, \end{align*}

where γ > 0 defined later.

Throughout this article, we assume

(3.13)\begin{align} \int_{- \infty }^\tau {e^{\gamma t}(\|g(t)\|^2+\|h(t)\|^2+\|\delta(t)\|^2) }dt \lt \infty ,\quad\forall \tau \in \mathbb{R}. \end{align}

4. Uniform moment estimates

In this section, we derive uniform moment estimates of the solution of problem (3.10)–(3.11) which are necessary for establishing the existence of pullback measure attractors. In the sequel, we use $\mathcal L(\xi) $ to denote the distribution law of a random variable ξ. We assume that

(4.1)\begin{equation} \gamma = \mathop {\inf }\limits_{t \in \mathbb{R}} \left\{{\lambda \left( t \right)+ \beta _0 \left( t \right) -{\frac{1}{2} -2 \left\| {\beta \left( t \right)} \right\|^2 } } \right\} \gt 0. \end{equation}

We first discuss uniform estimates of solutions of problem (3.10)–(3.11) in $L^2(\Omega, l^2)$.

Lemma 4.1. Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Then for every $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D$, there exists $T=T(\tau,D) \gt 0$, independent of ɛ, such that for all $t\geq T$, $\xi \in L^2_{\mathcal{F}_{\tau-t}}(\Omega, l^2)$ with $\mathcal L(\xi)\in D(\tau-t)$, and $0 \lt \varepsilon\leq 1$, the solution u of (3.10)–(3.11) satisfies

(4.2)\begin{align} \begin{aligned} &\mathbb{E}\left( {\left\| {u\left( \tau,\tau-t,\xi \right)} \right\|^2 } \right)+ \int_{\tau-t}^\tau {e^{- \gamma(\tau-s) } \mathbb{E}\left( {\left\| {u\left( s,\tau-t,\xi \right)} \right\|^2 } \right)} ds \le R(\tau), \end{aligned} \end{align}

where

\begin{equation*}R(\tau)=M_1+M_1\int_{-\infty}^\tau {e^{- \gamma(\tau-s) } \left( {\left\| {g\left( s \right)} \right\|^2 + \left\| {h\left( s \right)} \right\|^2 + \left\| \delta(s) \right\|^2 } \right)} ds,\end{equation*}

with $M_1 \gt 0$ being a constant independent of τ, ɛ, and D.

Proof. By (3.10) and Ito’s formula, we have for $t \geq \tau$

(4.3)\begin{align} \begin{aligned} & \mathbb{E}\left( {\left\| {u\left( t \right)} \right\|^2 } \right) + 2\nu \int_{\tau}^t {\mathbb{E}(\left\| {Bu\left( s \right)} \right\|^2)} ds+ 2 \int_{\tau}^t {(\lambda(s)+\beta_0(s))\mathbb{E}(\left\| {u\left( s\right)} \right\|^2)} ds \\ & = \mathbb{E}\left( {\left\| \xi \right\|^2 } \right) + 2 {\int_{\tau}^t {\mathbb{E}\left( {u\left( s \right),f\left( {s,u\left( s\right)} \right)} \right)ds} } \\ &+ 2 {\int_{\tau}^t {\mathbb{E}\left( {u\left( s \right),g(s)} \right)} } ds +\varepsilon^2 {\sum\limits_{k = 1}^\infty {\int_{\tau}^t {\mathbb{E}\left( \left\| {h_k(s) + \sigma _k \left( {s,u\left( s \right)} \right)} \right\|^2\right) ds} } }. \end{aligned} \end{align}

It should be noted that $u\in C([\tau,+\infty);L^{2}(\Omega;l^{2}))$ due to the fact that $u\in L^{2}(\Omega;C([\tau,\tau+T];l^{2}))$ for all T > 0 and the Lebesgue Dominated Theorem. Thus, by (3.7), (3.9), and (4.1), we obtain from (4.3) for $t \gt \tau$

(4.4)\begin{align} \begin{aligned} \frac{d}{dt}\mathbb{E}\left( {\left\| {u\left( t \right)} \right\|^2 } \right) & \le - \varpi(t) {\mathbb{E}\left( {\left\| {u\left( t \right)} \right\|^2 } \right)} + 4(\|g(t)\|^2+\|h(t)\|^2+\left\| \delta(t) \right\|^2) \\ & \le - 2\gamma {\mathbb{E}\left( {\left\| {u\left( t \right)} \right\|^2 } \right)} + 4(\|g(t)\|^2+\|h(t)\|^2+\left\| \delta(t) \right\|^2), \end{aligned} \end{align}

where $\varpi(t)=2(\lambda(t)+\beta _0(t))-1-4\left\| \beta(t) \right\|^2$.

Multiplying (4.4) by $e^{\gamma t}$ and then integrating the resulting inequality on $(\tau-t,\tau)$ with $t\in \mathbb{R}^+$, we obtain

(4.5)\begin{align} \begin{aligned} &\mathbb{E}\left( {\left\| {u\left( \tau,\tau-t,\xi \right)} \right\|^2 } \right) +\gamma\int_{\tau-t}^\tau {e^{- \gamma(\tau-s) } \mathbb{E}\left( {\left\| {u\left( s,\tau-t,\xi \right)} \right\|^2 } \right)} ds \\ & \le \mathbb{E}\left( {\left\| \xi \right\|^2 } \right)e^{- \gamma t } + 4\int_{\tau-t}^\tau {e^{- \gamma(\tau-s) } \left( {\left\| {g\left( s \right)} \right\|^2 + \left\| {h\left( s \right)} \right\|^2 + \left\| \delta(s) \right\|^2 } \right)} ds\\ & \le \mathbb{E}\left( {\left\| \xi \right\|^2 } \right)e^{- \gamma t } + 4\int_{-\infty}^\tau {e^{- \gamma(\tau-s) } \left( {\left\| {g\left( s \right)} \right\|^2 + \left\| {h\left( s \right)} \right\|^2 + \left\| \delta(s) \right\|^2 } \right)} ds. \end{aligned} \end{align}

Since $\mathcal L(\xi) \in D_1\left( {\tau - t } \right)$ we have

\begin{equation*} e^{-\gamma t} \mathbb{E}\left ( \left\| \xi \right\|^2 \right ) \le e^{- \gamma t} \left\| {D \left( {\tau - t } \right)} \right\|^2 \to 0,\quad \text{as } t\to \infty, \end{equation*}

and hence there exists $T = T (\tau, D) \gt 0$ such that for all $t\geq T$,

\begin{equation*} e^{-\gamma t} \mathbb{E}\left ( \left\| \xi \right\|^2 \right ) \le \frac{{4 }}{\gamma }, \end{equation*}

which along with (4.5) concludes the proof.

Next, we derive uniform estimates on the tails of the solutions of (3.10)–(3.11) which are crucial for establishing the $\mathcal{D}$-pullback asymptotic compact in $\mathcal P_2(l^2)$ of the family of probability distributions of the solutions.

Lemma 4.2. Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Then for every η > 0, $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D,$ there exist $T=T(D,\tau,\eta)$ and $N=N(D,\tau,\eta)\in \mathbb{N}$ such that for all $0 \lt \varepsilon\leq 1$, $t\ge T$ and $n\ge N$, the solution u of (3.10)–(3.11) satisfies,

\begin{equation*}\sum_{|i|\ge n} \mathbb{E}(|u_i(\tau,\tau-t,\xi)|^2)\le \eta,\end{equation*}

when $\xi \in L^2_{\mathcal{F}_{\tau-t}}(\Omega, l^2)$ with $\mathcal L(\xi)\in D(\tau-t)$.

Proof. Let $\theta:\mathbb R\to\mathbb R$ be a smooth function such that $0\le\theta(s)\le 1$ for all $s\in\mathbb R$ and

(4.6)\begin{equation} \theta(s)=0,\quad \text{for}\ |s|\le 1,\ \text{and}\quad \theta(s)=1,\ \text{for}\ |s|\ge 2. \end{equation}

Given $n\in\mathbb N$, denote by $\theta_n=(\theta(\frac{i}{n}))_{i\in\mathbb Z}$ and $\theta_nu=(\theta(\frac{i}{n})u_i)_{i\in\mathbb Z}$ for $u=(u_i)_{i\in\mathbb Z}$. By (3.10), we obtain

(4.7)\begin{align} \begin{aligned} d(\theta_n u(t))&=-\theta_n\nu A u(t)dt-(\lambda(t)+\beta_0(t)) \theta_n u(t)dt+\theta_nf( t,u(t))dt\\ &\quad+\theta_ng(t)dt+\varepsilon\sum_{k=1}^\infty(\theta_nh_k(t)+ \theta_n\sigma_k(t, u(t)))dW_k(t),\quad t \gt \tau. \end{aligned} \end{align}

By (4.7), Ito’s formula and taking the expectation we obtain for all $t \geq \tau$ and $\Delta t\geq 0$,

(4.8)\begin{align} \begin{aligned} &\mathbb{E}\left( {\left\| {\theta _n u\left( t+\Delta t \right)} \right\|^2 } \right) = \mathbb{E}\Big(\left\| {\theta _n u(t) } \right\|^2 \big)- 2\nu \int_{t}^{t+\Delta t} {\mathbb{E}\left( {Bu\left( s \right),B\left( {\theta _n^2 u\left( s \right)} \right)} \right)ds}\\ &\quad- 2 \int_{t}^{t+\Delta t} {(\lambda(s)+\beta_0(s))\mathbb{E}\left( \theta _n u\left( s \right),\theta _n u\left( s \right) \right)ds}\\ &\quad+ 2\int_{t}^{t+\Delta t} {\mathbb{E}\left( {\theta _n u\left( s \right),\theta _n f\left(s, {u(s)} \right) } \right)ds}+ 2\int_{t}^{t+\Delta t} {\mathbb{E}\left( {\theta _n u\left( s \right),\theta _n g(s)} \right)ds} \\ &\quad+\varepsilon^2 \sum\limits_{k = 1}^\infty {\int_{t}^{t+\Delta t} {\mathbb{E}\left( {\left\| {\theta _n h_k(s) + \theta _n \sigma _k \left( {s,u\left( s \right)} \right)} \right\|^2 } \right)} } ds. \end{aligned} \end{align}

For the second term on the right-hand side of (4.8), we have

(4.9)\begin{align} \begin{aligned} &- 2\nu \int_t^{t+\triangle t} {\mathbb{E}\left( {Bu\left( s \right),B\left( {\theta _n^2 u\left( s \right)} \right)} \right)} ds \\ &= - 2\nu \int_t^{t+\triangle t} {\mathbb{E}\left( {\sum\limits_{i \in \mathbb{Z}} {\left( {u_{i + 1} - u_i } \right)\left( {\theta ^2 \left( {\frac{{i + 1}}{n}} \right)u_{i + 1} - \theta ^2 \left( {\frac{i}{n}} \right)u_i } \right)} } \right)} ds \\ &\le 4\nu \int_t^{t+\triangle t} {\mathbb{E}\left( {\sum\limits_{i \in \mathbb{Z}} {\left| {\theta \left( {\frac{{i + 1}}{n}} \right) - \theta \left( {\frac{i}{n}} \right)} \right|\left| {u_{i + 1} - u_i } \right|\left| {u_i } \right|} } \right)} ds \\ &\le \frac{c}{n}\int_t^{t+\triangle t} {\mathbb{E}\left( {\sum\limits_{i \in \mathbb{Z}} {\left| {u_{i + 1} - u_i } \right|\left| {u_i } \right|} } \right)} ds \le \frac{{2c}}{n}\int_t^{t+\triangle t} {\mathbb{E}\left( {\left\| {u\left( s \right)} \right\|^2 } \right)} ds, \end{aligned} \end{align}

where c > 0 depends only on θ.

For the fourth term on the right-hand side of (4.8), by (3.5), we obtain

(4.10)\begin{align} \begin{aligned} 2\int_t^{t+\triangle t} {\mathbb{E}\left( {\theta _n u\left(s \right),\theta _nf\left( {s,u(s)} \right)} \right)ds} \le 0. \end{aligned} \end{align}

On the other hand, by Young’s inequality, we get

(4.11)\begin{equation} 2\int_t^{t+\triangle t} {\mathbb{E}\left( {\theta _n u\left( s \right),\theta _n g(s)} \right)d s} \le \int_t^{t+\triangle t}\mathbb{E}\left( {\left\| {\theta _n u\left( \tau \right)} \right\|^2}\right) d\tau + \int_t^{t+\triangle t} \sum\limits_{\left| i \right| \ge n} {g_i^2(s) }ds. \end{equation}

For the last term on the right-hand side of (4.8), by (3.9), we obtain

(4.12)\begin{align} \begin{aligned} \varepsilon^2\sum\limits_{k = 1}^\infty & {\int_t^{t+\triangle t} {\mathbb{E}\left( {\left\| {\theta _n h_k + \theta _n \sigma _k \left( {u\left( \tau \right)} \right)} \right\|^2 } \right)} } d\tau \\ &\le 4 \int_t^{t+\triangle t} {\left\| \beta(s) \right\|^2\mathbb{E}\left( {\left\| {\theta _n u\left( s \right)} \right\|^2 } \right)}ds+4 \int_t^{t+\triangle t} \sum\limits_{\left| i \right| \ge n} {\sum\limits_{k = 1}^\infty {\left( {h_{i,k}^2(s) + \delta _{i,k}^2(s) } \right)}ds }. \end{aligned} \end{align}

We obtain from (4.1) and (4.8)–(4.12) that

(4.13)\begin{align} \begin{aligned} &D^+\mathbb{E}\left( {\left\| {\theta _n u\left( t \right)} \right\|^2 } \right) \le - \left( {2(\lambda(t)+\beta_0(t)) -1 - 4\left\| \beta(t) \right\|^2 } \right)\mathbb{E}\left( {\left\| {\theta _n u\left( t \right)} \right\|^2 } \right)\\ &\quad+\frac{2c}{n}\mathbb{E}(\left\| {u\left( t \right)} \right\|^2) + \sum\limits_{\left| i \right| \ge n} {g_i^2(t) } +4\sum\limits_{\left| i \right| \ge n} {\sum\limits_{k = 1}^\infty {\left( {h_{i,k}^2(t) + \delta _{i,k}^2(t) } \right)} }\\ & \le - \gamma\mathbb{E}\left( {\left\| {\theta _n u\left( t \right)} \right\|^2 } \right) +\frac{2c}{n}\mathbb{E}(\left\| {u\left( t \right)} \right\|^2)+ \sum\limits_{\left| i \right| \ge n} {g_i^2(t) } +4\sum\limits_{\left| i \right| \ge n} {\sum\limits_{k = 1}^\infty {\left( {h_{i,k}^2(t) + \delta _{i,k}^2(t) } \right)} }, \end{aligned} \end{align}

where $D^{+}$ is the upper right Dini derivative. Given $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$, integrating the above over $(\tau-t, \tau )$, we obtain

(4.14)\begin{align} &\mathbb{E}\left( {\left\| {\theta _n u\left( \tau,\tau-t,\xi \right)} \right\|^2 } \right) \le \mathbb{E}\left( {\left\| {\theta _n \xi } \right\|^2 } \right)e^{- \gamma t}\! +\! \frac{{2c}}{n}\int_{\tau - t}^\tau \!{e^{- \gamma \left( {\tau - s} \right)} \mathbb{E}\!\left( \!{\left\| {u\left( s,\tau\!-\!t,\xi \right)} \right\|^2 }\! \right)} ds \notag\\ &\quad+ 4\int_{\tau - t}^\tau {e^{- \gamma \left( {\tau - s} \right)} \left( {{ \sum\limits_{\left| i \right| \ge n} {g_i^2 \left( s \right)} + \sum\limits_{\left| i \right| \ge n} {\sum\limits_{k = 1}^\infty {\left( {h_{i,k}^2 \left( s \right) + \delta _{i,k}^2 \left( s \right)} \right)} } } } \right)} ds. \end{align}

For every η > 0, $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D,$ there exists $T_1=T_1(D,\tau,\eta) \gt 0$ such that

\begin{equation*} \mathbb{E}\left( {\left\| {\theta _n \xi } \right\|^2 } \right)e^{-\gamma t} \le e^{-\gamma \tau}\left\| {D\left( {\tau - t} \right)} \right\|_{\mathcal P_2 \left( {l^2 } \right)}^2 e^{\gamma(\tau-t)} \lt \eta . \end{equation*}

By lemma 4.1 for every η > 0, $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D,$ we find that there exist $T_2=T_2(D,\tau,\eta) \gt 0$ and $N_1=N_1(D,\tau,\eta)\in \mathbb{N}$ such that for all $t\geq T_2$ and $n\geq N_1$,

(4.15)\begin{equation} \frac{{2c}}{n}\int_{\tau - t}^\tau {e^{- \gamma \left( {\tau - s} \right)} \mathbb{E} \left( {\left\| {u\left( {s,\tau - t,\xi } \right)} \right\|^2 } \right)} ds \le \frac{{2c}}{n}R\left( \tau \right) \lt \eta, \end{equation}

where $R(\tau)$ is given by (4.2). By (3.13), we know for every η > 0 and $\tau\in \mathbb{R}$, there exists $N_2=N_2(\tau,\eta)\in \mathbb{N}$ such that for $n \gt N_2$,

(4.16)\begin{align} \begin{aligned} 4\int_{\tau - t}^\tau {e^{- \gamma \left( {\tau - s} \right)} \left( { {\sum\limits_{|i|\geq n} {g_i^2 \left( s \right)} + \sum\limits_{|i|\geq n} {\sum\limits_{k = 1}^\infty {\left( {h_{i,k}^2 \left( s \right) + \delta _{i,k}^2 \left( s \right)} \right)} } } } \right)} ds \lt \eta. \end{aligned} \end{align}

Since $\mathcal{L}(\xi)\in D(\tau-t)$, there exists $T_{2}=T_{2}(D,\tau,\eta) \gt T_{1}$, such that for $t \gt T_{2}$,

\begin{equation*} \mathbb{E}(\|\theta_{n}\xi\|^{2})e^{-\gamma t}\leq\|D(\tau-t)\|^{2}_{\mathcal{P}_{2}(l^{2})}e^{-\gamma t} \lt \eta. \end{equation*}

Combining (4.14), (4.15), and (4.16), we get for every η > 0, $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D,$ there exist $T=\max\{T_1,T_2\}$ and $N=\max\{N_1,N_2\}$ such that for all $0 \lt \varepsilon\leq 1$, $t\ge T$ and $n\ge N$,

\begin{equation*} \mathbb{E}\left( {\left\| {\theta _n u\left( t \right)} \right\|^2 } \right) \le 3\eta, \end{equation*}

when $\xi \in L^2_{\mathcal{F}_{\tau-t}}(\Omega, l^2)$ with $\mathcal L(\xi)\in D(\tau-t)$. This completes the proof.

5. Existence of pullback measure attractors

This section is devoted to the existence, uniqueness and periodicity of $\mathcal D$-pullback measure attractors of (3.10)–(3.11) in $\mathcal P_2(l^2)$.

As usual, if $\phi : l^2\rightarrow \mathbb{R}$ is a bounded Borel function, then for $ r\leq t$ and $\xi \in l^2$, we set

\begin{equation*} \left( {p({t,r}) \phi } \right)\left( \xi \right) = \mathbb{E}\left( {\phi \left( {u \left( {t,r,\xi } \right)} \right)} \right), \end{equation*}

and

\begin{equation*} p\left( {r,\xi ;t,\Gamma } \right) = \left( {p({t,r}) 1_\Gamma } \right) \left( \xi \right) , \end{equation*}

where $\Gamma \in \mathcal B\left( l^2 \right)$ and $1_{\Gamma}$ is the characteristic function of Γ.

The following properties of $\left\{{p({t,r}) } \right\}_{r \le t}$ are standard (see, e.g., [19]) and the proof is omitted.

Lemma 5.1. Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Then:

(i) The family $\left\{{p({t,r}) } \right\}_{r \le t} $ is Feller; that is, for any ${r \le t}$, the function $ p({t,r} )\phi \in C_b(l^2)$ is bounded and continuous if so is ϕ.

(ii) For every $r\in \mathbb{R}$ and $ \xi \in l^2$, the process $ \left\{{u \left( {t,r,\xi } \right)} \right\}_{t \ge r}$ is an l ²-valued Markov process.

We will also investigate the periodicity of pullback measure attractors of system (3.10)–(3.11) for which we assume that all given time-dependent functions are ϖ-periodic in t for some ϖ > 0; that is, for all $t\in \mathbb{R} $ and $k\in \mathbb N$,

(5.1)\begin{align} \begin{aligned} & \lambda \left( {t + \varpi} \right) = \lambda \left( t \right),\quad \beta_0(t+\varpi)=\beta_0(t) \quad g(t+\varpi) =g(t),\\ & f \left( {t + \varpi, \cdot , \cdot } \right) = f \left( {t, \cdot , \cdot } \right), \quad h_k(t+\varpi) =h_k(t), \quad \sigma _{k} \left( {t + \varpi, \cdot , \cdot } \right) = \sigma _{k} \left( {t, \cdot , \cdot } \right). \end{aligned} \end{align}

By the similar argument as that of lemma 4.1 in [10], we get the following lemma.

Lemma 5.2. Suppose (3.1)–(3.4), (3.13), (4.1), and (5.1) hold. Then we have the family $\left\{{p({t,r}) } \right\}_{ r \le t} $ is ϖ-periodic; that is, for all $ t\geq r$,

\begin{equation*} p\left( {r,\xi ;t, \cdot } \right) = p\left( {r+\varpi,\xi ;t+\varpi, \cdot } \right), \quad\forall \xi \in l^2. \end{equation*}

Given $t\geq r$ and $\mu \in \mathcal P(l^2)$, define

(5.2)\begin{equation} p_ * \left( {t,r } \right)\mu \left( \cdot \right) = \int_{l^2} {p\left( {r ,\xi; t, \cdot} \right)} \mu \left( {d\xi} \right). \end{equation}

Then $p_{\ast}(t,r): \mathcal P(l^2)\rightarrow \mathcal P(l^2)$ is the dual operator of $p({t,r})$. By (3.12), we find that for all $t\geq r $, $p_{\ast}(t, r)$ maps $ \mathcal P_2(l^2)$ to $ \mathcal P_2(l^2)$.

We now define a non-autonomous dynamical system $S(t,\tau)$, $t\ge \tau$, for the family of operators $p_{\ast}(t, \tau )$. Given $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$, let $S(t,\tau):\mathcal P_2(l^2)\rightarrow\mathcal P_2(l^2)$ be the map given by

\begin{equation*} S(t,\tau)\mu =p_{\ast}(\tau+t,\tau) \mu , \quad \forall \ \mu \in \mathcal P_2(l^2). \end{equation*}

Lemma 5.3. Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Then $S(t,\tau)$, $t\ge \tau$, is a continuous non-autonomous dynamical system in $\mathcal P_2(l^2)$ generated by (3.10)–(3.11); more precisely, $S(t,\tau): \mathcal P_2(l^2)\rightarrow \mathcal P_2(l^2)$ satisfies the following conditions

(a) $S(0,\tau)=I_{\mathcal P_2(l^2)}$, for all $\tau\in \mathbb{R}$;

(b) $S(s+t,\tau)=S(t,\tau+s)\circ S(s,\tau)$, for any $\tau \in \mathbb{R}$ and $t,s\in \mathbb{R}^+$;

(c) $S(t,\tau): \mathcal P_2(l^2)\rightarrow \mathcal P_2(l^2)$ is continuous, for every $\tau\in \mathbb{R}$ and $t\in \mathbb{R}^+$.

Proof. Note that (a) follows from the definition of S, and (b) follows the Markov property of the solutions of (3.10)–(3.11).

We now prove (c). Suppose $\mu_n\rightarrow \mu$ in $\mathcal P_2(l^2)$. We will show $S(t,\tau)\mu_n\rightarrow S(t,\tau)\mu$ in $\mathcal P_2(l^2)$ for every $\tau\in \mathbb{R}$ and $t\in \mathbb{R}^+$. Let $\varphi\in C_b(l^2)$. By lemma 5.1, we have $p(\tau +t, \tau) \varphi\in C_b(l^2)$ for all $\tau\in \mathbb{R}$ and $t\in \mathbb{R}^+$, and hence

(5.3)\begin{align} \begin{aligned} &\mathop {\lim }\limits_{n \to \infty } \left(\varphi, {S \left( t,\tau \right)\mu _n } \right)= \mathop {\lim }\limits_{n \to \infty } \left(\varphi, {p_ * \left( \tau+t,\tau \right)\mu _n } \right)\\ &= \mathop {\lim }\limits_{n \to \infty } \left( { p(\tau+t,\tau) \varphi }, \mu_n \right) = \left( {p(\tau+t,\tau) \varphi }, \mu \right)\\ &= \left( \varphi, {p_ * \left( \tau+t,\tau \right)\mu } \right) = \left(\varphi, {S \left( t,\tau \right)\mu } \right), \end{aligned} \end{align}

as desired.

By lemma 4.1, we obtain a $\mathcal D$-pullback absorbing set for S as stated below.

Lemma 5.4. Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Given $\tau\in \mathbb{R}$, denote by

(5.4)\begin{equation} K(\tau) = B_{ {\mathcal{P}}_2 (l^2)} \left( {L^{\frac 12} _1(\tau) } \right), \end{equation}

where

(5.5)\begin{align} \begin{aligned} L_1(\tau)= M_1+M_1\int_{-\infty}^\tau {e^{- \gamma(\tau-s) } \left( {\left\| {g\left( s \right)} \right\|^2 + \left\| {h\left( s \right)} \right\|^2 + \left\| \delta(s) \right\|^2 } \right)} ds, \end{aligned} \end{align}

and $M_1 \gt 0$ is the same constant as in lemma 4.1, independent of τ and ɛ. Then $K=\{K(\tau): \ \tau\in \mathbb{R} \} \in \mathcal D$ is a closed $\mathcal D$-pullback absorbing set of S.

Proof. By (5.4) and lemma 4.1, we see that for every $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D$, there exists $T=T(\tau,D) \gt 0$, independent of ɛ, such that for all $t\geq T$ and $0 \lt \varepsilon\leq 1$, S satisfies

(5.6)\begin{equation} S\left( t,\tau-t \right)D(\tau-t) \subseteq K(\tau). \end{equation}

We now prove $K=\{K(\tau): \ \tau\in \mathbb{R} \} \in \mathcal D$. By (5.4), (5.5), and (3.13), we have

\begin{align*} e^{\gamma \tau}\| K(\tau) \|^2_{{\mathcal{P}}_2(l^2)} &= e^{\gamma \tau} L_1(\tau)\\ & = {e^{\gamma \tau} M_1} + M_1\int_{-\infty}^\tau {e^{ \gamma s } \left( {\left\| {g\left( s \right)} \right\|^2 + \left\| {h\left( s \right)} \right\|^2 + \left\| \delta(s) \right\|^2 } \right)} ds \to 0,\\ & \times \text{as } \tau\to -\infty, \end{align*}

and hence $K=\{K(\tau): \ \tau\in \mathbb{R} \} \in \mathcal D$, which along with (5.6) concludes the proof.

We now present the $\mathcal D$-pullback asymptotically compact of S associated with (3.10)–(3.11).

Lemma 5.5. If (3.1)–(3.4), (3.13), and (4.1) hold, then S is $\mathcal D$-pullback asymptotically compact in $\mathcal P_2 \left( l^2 \right)$; that is, for every $\tau\in \mathbb{R}$, $\left\{{S \left( {t_n,\tau-t_n} \right)\mu _n } \right\}_{n = 1}^\infty $ has a convergent subsequence in $\mathcal P_2 \left( l^2 \right)$ whenever $t_n\rightarrow +\infty$ and $\mu _n\in D(\tau-t_n)$ with $D\in \mathcal D$.

Proof. To complete the proof, by Prohorov theorem, we need to prove that for each $\tau\in \mathbb{R}$, the sequence $\{\mathcal L({u(\tau,\tau-t_n,\xi_n) })\}_{n=1}^\infty$ is tight. It follows from lemma 4.1 that for $\tau\in \mathbb{R}$ and $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D$ there exists a $N_1=N_1(\tau,D)\in \mathbb{N}$ such that for all $\xi_n \in L^2_{\mathcal{F}_{\tau-t_n}}(\Omega, l^2)$ with $\mathcal L(\xi_n)\in D(\tau-t_n)$ and $n \gt N_1$,

(5.7)\begin{equation} \mathbb{E}\left( {\left\| {u(\tau,\tau-t_n,\xi_n) } \right\|^2 } \right) \le M, \end{equation}

where M > 0 is a constant depending on τ, but independent of ɛ and D. By Chebyshev’s inequality, we obtain from (5.7) that for all $\xi_n \in L^2_{\mathcal{F}_{\tau-t_n}}(\Omega, l^2)$ with $\mathcal L(\xi_n)\in D(\tau-t_n)$ and $n \gt N_1$,

\begin{equation*} P\left( {\left\| {u(\tau,\tau-t_n,\xi_n) } \right\|^2 \gt R_1 } \right) \le \frac{{M }}{{R_1^2 }}\rightarrow 0\quad\text{as}\quad R_1\rightarrow \infty. \end{equation*}

Hence for every $\tau\in \mathbb{R}$, η > 0 and $m\in \mathbb{N}$, there exists $R_2=R_2(\tau,\eta,m) \gt 0$ such that for all $\xi_n \in L^2_{\mathcal{F}_{\tau-t_n}}(\Omega, l^2)$ with $\mathcal L(\xi_n)\in D(\tau-t_n)$ and $n \gt N_1$,

(5.8)\begin{equation} P\left\{{\left\| {u(\tau,\tau-t_n,\xi_n) } \right\|^2 \gt R_2 } \right\} \lt \frac{\eta}{2^{m+1}}. \end{equation}

By lemma 4.2, we infer that for each $\tau\in \mathbb{R}$, $D=\{D(t):t\in \mathbb{R}\}\in \mathcal D$, η > 0 and $m\in \mathbb{N}$, there exist an integer $n_m=n_m(\tau,D,\eta,m)$ and $H_m=H_m(\tau,D,\eta,m) \gt N_1$ such that for all $\xi_n \in L^2_{\mathcal{F}_{\tau-t_n}}(\Omega, l^2)$ with $\mathcal L(\xi_n)\in D(\tau-t_n)$ and $n\geq H_m$,

\begin{equation*} \mathbb{E}\left( {\sum\limits_{\left| i \right| \gt n_m } {\left| {u_i \left(\tau,\tau-t_n,\xi_n \right)} \right|^2 } } \right) \lt \frac{\eta }{{2^{2m + 1} }}, \end{equation*}

and hence for all $\xi_n \in L^2_{\mathcal{F}_{\tau-t_n}}(\Omega, l^2)$ with $\mathcal L(\xi_n)\in D(\tau-t_n)$ and $n\geq H_m$,

(5.9)\begin{align} & P\left( {\left\{{ \sum\limits_{\left| i \right| \gt n_m } {\left| {u_i \left(\tau,\tau-t_n,\xi_n \right)} \right|^2 } \gt \frac{1}{{2^m }}} \right\}} \right) \le 2^m \mathbb{E}\left( { \sum\limits_{\left| i \right| \gt n_m } {\left| {u_i \left(\tau,\tau-t_n,\xi_n \right)} \right|^2 } } \right)\\& \quad \lt \frac{\eta }{{2^{m + 1} }}. \end{align}

Given $m\in \mathbb{N}$, set

(5.10)\begin{align} Y_{1,m } &= \left\{{v \in l^2:\left\| {v } \right\|^2 \le R_2 } \right\}, \end{align}

(5.11)\begin{align} Y_{2,m} &= \left\{{v \in l^2: \sum\limits_{\left| i \right| \gt n_m } {\left| {v_i } \right|^2 } \leq \frac{1}{{2^m }}} \right\}, \end{align}

and

(5.12)\begin{equation} Y_m = Y_{1,m} \cap Y_{2,m }. \end{equation}

By (5.10), we see that the set $\left\{{(v_i)_{|i|\leq n_m}:v \in Y_m } \right\}$ is bounded in the finite-dimensional space $\mathbb{R}^{2n_m+1}$ and hence precompact. Consequently, $\left\{{(v_i)_{|i|\leq n_m}:v \in Y_m } \right\}$ has a finite open cover of balls with radius $\frac{1 }{\sqrt{2}^{m}}$, which along with (5.11) implies that the set $\left\{{v:v \in Y_m } \right\}$ has a finite open cover of balls with radius $\frac{1 }{\sqrt{2}^{m-1}}$ in l ². For each $\tau\in \mathbb{R}$ and $m\in \mathbb{N}$, there exists a compact set $K_m=K_m(\tau)$ such that for all $n\leq H_m$, $P\left( \{u\left( {\tau,\tau-t_n ,\xi _n } \right) \in K_m \} \right) \gt 1 - \frac{\eta }{{2^{m } }}.$ Then by (5.8) and (5.9), there exists a set $\mathcal Y_m=Y_m\cup K_m $, which has a finite open cover of balls with radius $\frac{1 }{\sqrt{2}^{m-1}}$ in l ², such that for all $n\in \mathbb{N}$, $P\left( \{u\left( {\tau,\tau-t_n ,\xi _n } \right) \in \mathcal Y_m\} \right) \gt 1-\frac{\eta }{{2^{m } }}.$ Set $\mathcal Y=\bigcap\limits_{m = 1}^\infty {\mathcal Y_m }$. Then $\mathcal Y$ is a closed and totally bounded subset of l ², and hence is compact. For all $n\in \mathbb{N}$,

\begin{equation*}P\left( \{u\left( {\tau,\tau-t_n ,\xi _n } \right) \in \mathcal Y\} \right) \gt 1-\sum\limits_{m = 1}^\infty {\frac{\eta}{{2^{m } }}}=1-\eta,\end{equation*}

as desired.

Next, we establish the existence, uniqueness, and periodicity of $\mathcal D$-pullback measure attractors for (3.10)–(3.11) on $\mathcal P_2 (l^2)$.

Theorem 5.6 If (3.1)–(3.4), (3.13), and (4.1) hold, then for every $0 \lt \varepsilon \leq 1$, S associated with (3.10)–(3.11) has a unique $\mathcal D$-pullback measure attractor $\mathcal{A} = \{\mathcal A (\tau): \tau\in \mathbb{R} \} \in {\mathcal D}$ in $\mathcal P_2(l^2)$, which is given by, for each $\tau\in \mathbb{R}$,

\begin{align*} \begin{aligned} \mathcal A\left( \tau \right) &= \omega(K,\tau) =\{\psi(0,\tau):\,\psi\,\text{is a}\,\, \mathcal D \text{-complete orbit of}\,\,S\} \\ & =\{\xi (\tau):\,\xi\,\text{is a}\,\, \mathcal D \text{-complete solution of}\,\,S\}, \end{aligned} \end{align*}

where $K=\{K(\tau):\,\tau\in \mathbb{R}\}$ is the $\mathcal{D}$-pullback absorbing set of S as given by lemma 5.4.

Proof. It follows from lemma 5.3 that S is a continuous non-autonomous dynamical system on $\mathcal P_2(l^2)$. Notice that S has a closed $\mathcal D$-pullback absorbing set K in $\mathcal P_2(l^2)$ by lemma 5.4 and is $\mathcal D$-pullback asymptotically compact in $\mathcal P_2(l^2)$ by lemma 5.5. Hence the existence and uniqueness of the $\mathcal D$-pullback measure attractor for S follows from proposition 2.10 immediately.

We now consider the periodicity of the measure attractor $\mathcal A$. By (5.4) and (5.5), we find that K is ϖ-periodic. In addition, it follows from lemma 5.2 and (5.2), the non-autonomous dynamical system S associated with system (3.10)–(3.11) is also ϖ-periodic. Thus, from proposition 2.10, the periodicity of the measure attractor $\mathcal A$ follows.

Theorem 5.7 If (3.1)–(3.4), (3.13), (4.1), and (5.1) hold, then for every $0 \lt \varepsilon \leq 1$, S associated with (3.10)–(3.11) has a unique ϖ-periodic $\mathcal D$-pullback measure attractor $\mathcal A$ in $\mathcal P_2(l^2)$.

6. Upper semicontinuity of pullback measure attractors

In this section, we prove the upper semicontinuity of $\mathcal D$-pullback measure attractors for the non-autonomous stochastic lattice systems as the noise intensity ɛ tends to zero.

We apply theorem 2.11 to the non-autonomous stochastic lattice systems (3.10)–(3.11) with $\varepsilon\in [0,1]$. Note that all results in the previous sections are valid for ɛ = 0 in which case the proof is actually simpler. From now on, we write the solution of system (3.10)–(3.11) as $u^\varepsilon(t,\tau,\xi)$ at initial time τ with initial value $\xi \in L^2_{\mathcal{F}_\tau}(\Omega, l^2)$ to highlight the dependence of solutions on the parameter ɛ. Given $\varepsilon \in [0,1]$, let $p^\varepsilon(t,\tau) $ be the transition operator of $u^\varepsilon (t, \tau,\xi)$ and $p^\varepsilon_{\ast} (t,\tau)$ be the duality operator of p ^ɛ. Given $t\in \mathbb{R}^+$ and $\tau\in \mathbb{R}$, let $S^\varepsilon(t,\tau):\mathcal P_2(l^2)\rightarrow\mathcal P_2(l^2)$ be the map given by

\begin{equation*} S^\varepsilon(t,\tau)\mu =p_{\ast}^\varepsilon(\tau+t,\tau) \mu , \quad \forall \ \mu \in \mathcal P_2(l^2). \end{equation*}

Let $\mathcal A_\varepsilon$ be the $\mathcal{D}$-pullback measure attractor of S ^ɛ.

Next, we establish the convergence of solutions of problem (3.10)–(3.11) when $\varepsilon\rightarrow 0$.

Lemma 6.1. Suppose (3.1)–(3.4) hold. Then given $\tau\in \mathbb{R}$ and a positive constant $\mathcal K(\tau)$, if $\xi \in L^2_{\mathcal{F}_{\tau}}(\Omega, l^2)$ with $\mathbb{E}( {\| \xi \|^2 } ) \le \mathcal K^2 ( \tau)$, then we have for $t\in \mathbb{R}^+$,

\begin{equation*} \mathop{\lim }\limits_{\varepsilon \to 0}\mathop{\sup }\limits_{\mu\in B_{\mathcal P_2(l^2)}(\mathcal K(\tau))} d_{{\mathcal P_2(l^2)}} \left( {S^\varepsilon(t,\tau)\mu, S^0(t,\tau)\mu} \right)=0. \end{equation*}

Proof. By the similar argument as that of lemma 6.2 in [11], we obtain for $\tau\in \mathbb{R}$, $\mathcal K(\tau)$ and $t\in \mathbb{R}^+$,

(6.1)\begin{align} \begin{aligned} \mathop {\sup }\limits_{\mathbb{E} ({{\left\| {{\xi }} \right\|}^2}) \le \mathcal K^2(\tau)} \mathbb{E}\left( \left\| u^{\varepsilon} (\tau+t,\tau,\xi) -u^{0} (\tau+t,\tau,\xi) \right\|^2 \right)\le \chi(\varepsilon), \end{aligned} \end{align}

where $\chi(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. Note that for all $t\in \mathbb{R}^+$ we have

\begin{align*} &\mathop {\sup }\limits_{\mathbb{E} ({{\left\| {{\xi }} \right\|}^2}) \le \mathcal K^2(\tau)}\ \ \mathop {\sup }\limits_{ \varphi \in L_b \left( l^2 \right) \atop \left\| \varphi \right\|_L \le 1 } \left| {\mathbb{E} \left ( \varphi\left( {u^\varepsilon \left( {\tau+t,\tau,\xi } \right)} \right) \right ) - \mathbb{E} \left ( \varphi\left( {u^0 \left( {\tau+t,\tau,\mathcal \xi } \right)} \right ) \right ) } \right|\notag\\ &\le \mathop {\sup }\limits_{\mathbb{E} ({{\left\| {{\xi ^\varepsilon }} \right\|}^2}) \le \mathcal K^2(\tau)}\ \ \mathop {\sup }\limits_{ \varphi \in L_b \left( l^2 \right) \atop \left\| \varphi \right\|_L \le 1 } {\mathbb{E} \left ( \left | \varphi\left( {u^\varepsilon \left( {\tau+t,\tau,\xi } \right)} \right ) - \varphi\left( {u^0 \left( {\tau+t,\tau,\mathcal \xi} \right)} \right ) \right | \right ) }\notag\\ &\le \mathop {\sup }\limits_{\mathbb{E} ({{\left\| {{\xi }} \right\|}^2}) \le \mathcal K^2(\tau)} {\mathbb{E} \left ( \| {u^\varepsilon \left( {\tau+t,\tau,\xi } \right)} - {u^0 \left( {\tau+t,\tau,\mathcal \xi } \right) } \|\right ) }\notag\\ &\le \left ( \mathop {\sup }\limits_{\mathbb{E} ({{\left\| {{\xi ^\varepsilon }} \right\|}^2}) \le \mathcal K^2(\tau)} {\mathbb{E} \left ( \| {u^\varepsilon \left( {\tau+t,\tau,\xi } \right)} - {u^0 \left( {\tau+t,\tau,\xi } \right) } \|^2 \right ) } \right )^{\frac 12} \end{align*}

which along with (6.1) implies that for all $t\in \mathbb{R}^+$,

\begin{equation*} \lim_{\varepsilon \to 0} \ \ \mathop {\sup }\limits_{\mathbb{E}({{\left\| {{\xi }} \right\|}^2}) \le \mathcal K^2(\tau)} \ \ \mathop {\sup }\limits_{ \varphi \in L_b \left(l^2 \right) \atop \left\| \varphi \right\|_L \le 1 } \left| {\mathbb{E} \left ( \varphi\left( {u^\varepsilon \left( {\tau+t,\tau,\xi } \right)} \right) \right ) - \mathbb{E} \left ( \varphi\left( {u^0 \left( {\tau+t,\tau,\xi } \right)} \right ) \right ) } \right| =0. \end{equation*}

This completes the proof.

By lemma 5.4, one can verify that

(6.2)\begin{align} K=\Big\{K(\tau)=\bigcup\limits_{\varepsilon\in[0,1]}\mathcal A_{\varepsilon}(\tau):\tau\in \mathbb{R}\Big\}\in \mathcal{D}. \end{align}

Then the main result of this section are given below.

Theorem 6.2 Suppose (3.1)–(3.4), (3.13), and (4.1) hold. Then for $\tau\in \mathbb{R}$,

(6.3)\begin{equation} \mathop {\lim }\limits_{\varepsilon \to 0} d_{\mathcal P_2(l^2)} \left( {\mathcal A_\varepsilon(\tau), \mathcal A_{0 }(\tau)} \right) = 0. \end{equation}

Proof. Based on (6.2) and lemma 6.1, we obtain (6.3) immediately from theorem 2.11.