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., [Reference Li and Wang9]).

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.

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 2.6. 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 [Reference Wang18], 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 [Reference Wang19], 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., [Reference Wang19]) and the proof is omitted.

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 [Reference Li, Wang and Wang10], 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*}

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).

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.

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.

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 [Reference Li, Wang and Wang11], 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}