Three types of weak pullback attractors for lattice pseudo-parabolic equations driven by locally Lipschitz noise

1.   Introduction

In this article we study the global well-posedness as well as long-time mean random dynamics of the non-autonomous stochastic lattice pseudo-parabolic equation with locally Lipschitz white noise defined on the whole $N$-dimensional integer set $\mathbb{Z}^N$:

along with initial data:

where $N\in\mathbb{N}$, $\tau\in\mathbb{R}$, $\lambda>0$, $g = (g_i)_{i\in\mathbb{Z}^N}$ and $h = (h_{k,i})_{k\in\mathbb{N},i\in\mathbb{Z}^N}$ are two random sequences depending on time, $\delta = (\delta_{k,i})_{k\in\mathbb{N},i\in\mathbb{Z}^N}$ is a given sequence of real numbers, $F_i\in C^1(\mathbb{R},\mathbb{R})$ and $\hat{\sigma} = (\hat{\sigma}_{k,i}) _{k\in\mathbb{N},i\in\mathbb{Z}^N}$ are locally Lipschitz continuous nonlinear functions satisfying some conditions, and the sequence of independent two-sided real-valued Wiener process $(W_{k})_{k\in\mathbb{N}}$ is defined on a complete filtered probability space $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ satisfying the usual conditions. The last stochastic term in (1.1) is interpreted as an It'o stochastic differential.

The pseudo-parabolic equation (see Xu et al. [26,27]) is also called a nonclassical diffusion equation (see Kuttler and Aifantis [11]) in the literature. This equation is used to study solid mechanics, non-Newtonian as well as heat conduction, see e.g., [1]. The blow-up phenomenon of the deterministic pseudo-parabolic equation has been investigated in several interesting papers, see e.g., [26,27,25,32] and the references therein. The long time dynamics by means of random attractors of the stochastic pseudo-parabolic equation was recently discussed in [21,23,29]. A contradictory version for the pseudo-parabolic equation is the classical parabolic (also called reaction-diffusion) equation. The large time dynamics in terms of random attractors of the stochastic classical diffusion equation was investigated in [12,15]. The reader is referred to [5,6] for the study of attractors of other interesting models.

Lattice equations can be regarded as space discretization version of evolution equations that are widely used for image processing, chemical reaction as well as pattern formation see e.g., [7]. The existence of deterministic and random attractors of lattice equations have been extensively examined in [14] and [2,4,9,22,28,31], respectively. In particular, the existence of global attractors of deterministic lattice pseudo-parabolic equation defined on $\mathbb{Z}$ had been investigated in [13,30]. To the best of our knowledge, there are no documented results reported in the literature on the existence of random attractors of stochastic lattice pseudo-parabolic equation (1.1)-(1.2) defined on $\mathbb{Z}^N$ even in a simple case where $N = 1$ and the diffusion coefficient $\hat{\sigma}_{k,i}(u_i(t))$ is linear in $u_i$ or independent of $u_i$. Our main task in the paper is to solve this problem in a more general case, that is, we will prove the existence of random attractors of equation (1.1) on $\mathbb{Z}^N$ when the nonlinear diffusion coefficient $\hat{\sigma}_{k,i}(u_i(t))$ is locally Lipschitz continuous in $u_i(t)$.

Note that those random attractors in the literature aforementioned were studied under the frameworks of pathwise random dynamical systems [15], and hence those random attractors are also called pathwise random attractors. As is well known, a basic but very restrictive condition for investigating the existence of such pathwise random attractors is that we need to transform the stochastic equation into a pathwise one. This transformation can be done for stochastic equations with additive noise or linear multiplicative noise but unavailable for stochastic equations with nonlinear noise. This then introduces many difficulties to prove the existence of pathwise random attractors of stochastic systems with nonlinear noise. For the purpose of dealing with the nonlinear noise in (1.1)-(1.2), in the present article, it is unnecessary to transform the stochastic equation (1.1)-(1.2) into a pathwise one. In other words, we will alternatively investigate the mean random dynamics but not the pathwise random dynamics of (1.1)-(1.2) by using the concept of weak pullback mean random attractors (WPMR attractors for short) of mean random dynamical systems, see Wang [16]. By a WPMR attractor in an abstract bochner space $L^p(\Omega,\mathcal{F},X)$, we here mean a minimal, weakly compact and weakly attracting set in $L^p(\Omega,\mathcal{F},X)$, where $p\in(1,\infty)$ and $X$ is a Banach space. The notation of invariant WPMR attractors can be found in Kloeden and Lorenz [10].

In order to study the existence of WPMR attractors for problem (1.1)-(1.2) in $L^2(\Omega,\mathcal{F},\ell^2)$, we must prove the global existence as well as uniqueness of mean square solutions to (1.1)-(1.2) in $L^2(\Omega,\mathcal{F},\ell^2)$. We remark that, since the nonlinear drift and diffusion functions are locally Lipschitz continuous but not globally Lipschitz continuous, we shall find a way to approximate the two functions in here. Our idea to solve the problem is to utilize the stoping time technique as well as the truncate method, see Theorem 3.1.

Based on the global well-posedness of (1.1)-(1.2) in $L^2(\Omega,\mathcal{F},\ell^2)$ we have established, we define a mean random dynamical system $\Phi$ via the solutions operators to (1.1)-(1.2), and prove that $\Phi$ has a unique WPMR $\mathfrak{D}$-attractor of in $L^2(\Omega,\mathcal{F},\ell^2)$, where $\mathfrak{D}$ is an attracting universe (see (4.6)).

Another important contribution of the paper is to study the backward weak compactness and attraction of WPMR attractors. More precisely, we will introduce another attracting universe $\mathfrak{B}$ (see (4.7), and prove that $\Phi$ has a unique backward weakly compact WPMR attractor $\mathcal{A}_\mathfrak{B} = \{\mathcal{A}_\mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$ and a unique backward weakly attracting WPMR attractor $\mathcal{U}_\mathfrak{B} = \{\mathcal{U}_\mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$, see Theorem 6.4. An interesting thing is that the three types attractors have the relationship $\mathcal{A}_\mathfrak{D} = \mathcal{A}_\mathfrak{B}\subseteq \mathcal{U}_\mathfrak{B}$ even when the attracting universes $\mathfrak{B}$ and $\mathfrak{D}$ are different ($\mathfrak{B}\subseteq\mathfrak{D}$).

We remark that the backward strong compactness of random attractors has been investigated in [3,20] for stochastic PDEs with linear noise. The usual WPMR attractors of stochastic equations with nonlinear noise was recently studied in [16,17,19,24,18]. In this paper we study backward weak compactness and attraction of WPMR attractors for stochastic lattice pseudo-parabolic equations with nonlinear noise.

This paper is organized as bellow. In the next Section we recall the lattice pseudo-parabolic equations with locally Lipschitz noise. In Section 3 we prove the global well-posedness of (1.1)-(1.2). In Section 4 we define a mean random dynamical system. In Section 5 we derive two types of uniform estimates and esyavlish the existence of absorbing sets of two types. In the last Section we prove the existence of WPMR attractors of three types.

2.   Lattice pseudo-parabolic equations with locally Lipschitz noise

In this section we recall the following non-autonomous stochastic lattice pseudo-parabolic equation with locally Lipschitz noise defined on $\mathbb{Z}^N$:

with initial data:

where $\tau\in\mathbb{R}$, $\lambda>0$, the sequence of independent two-sided real-valued Wiener process $(W_{j})_{j\in\mathbb{N}}$ is defined on the complete filtered probability space $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$, where $\Omega = \{\omega\in C(\mathbb{R},\mathbb{R}):\ \omega(0) = 0\}$ equipped with the compact-open topology, $\mathcal{F} = \mathfrak{B}(\Omega)$ denotes the Borel sigma-algebra of $\Omega$, $\mathbb{P}$ is the Wiener measure acting on $(\Omega,\mathcal{F})$, and $\{\mathcal{F}_t\}_{t\in\mathbb{R}}$ is an increasing right continuous family of sub-sigma-algebras of $\mathcal{F}$ with all $\mathbb{P}$-null sets.

Consider the Banach space

with norm $\|u\|_r = \big(\sum_{i\in\mathbb{Z}^N}|u_i|^r\big)^\frac{1}{r}$. In this paper, the norm and the inner product of $\ell^2$ are denoted by $(\cdot,\cdot)$ and $\|\cdot\|$, respectively. Assume that $F_i$ is locally Lipschitz continuous from $\mathbb{R}$ to $\mathbb{R}$ uniformly for $i\in\mathbb{Z}^N$, that is, for any compact set $I\subseteq\mathbb{R}$, we can find a constant $c_1 = c_1(I)>0$ such that

We also assume that

where $\gamma\in\mathbb{R}$ is a constant. Assume that $\hat{\sigma}_{k,i}$ are locally Lipschitz continuous form $\mathbb{R}$ to $\mathbb{R}$ uniformly for $k\in\mathbb{N}$ and $i\in\mathbb{Z}^N$, that is, for any compact set $I\subseteq\mathbb{R}$, we can find a constant $c_2 = c_2(I)>0$ such that

Assume that for all $s\in \mathbb{R}$ and $k\in\mathbb{N}$,

For $\delta = (\delta_{k,i})_{k\in\mathbb{N},i\in\mathbb{Z}^N}$, $g = (g_i)_{i\in\mathbb{Z}^N}$ and $h_k = (h_{k,i})_{i\in\mathbb{Z}^N}$, we assume, for all $\tau\in\mathbb{R}$ and $T>0$,

Next, we rewrite (2.1)-(2.2) as an abstract one in the sapce $\ell^2$. For all $1\leq j\leq N$, $u = (u_i)_{i\in\mathbb{Z}^N}\in\ell^2$ and $i = (i_1,i_2,\dots,i_N)\in\mathbb{Z}^N$, we define the operators from $\ell^2$ to $\ell^2$ by

and

For all $1\leq j\leq N$, $u = (u_i)_{i\in\mathbb{Z}^N}\in\ell^2$ and $v = (v_i)_{i\in\mathbb{Z}^N}\in\ell^2$, we have

For each $i\in\mathbb{Z}^N$, we let $f_i(s) = F_i(s)-\gamma s$ for all $s\in\mathbb{R}$. Then we see from (2.4) that

Then we define $F,f,\sigma_k:\ell^2\rightarrow \ell^2$ by

Both $F$ and $f$ are well-defined due to (2.3). In addition, $f$: $\ell^2\rightarrow \ell^2$ is also locally Lipschitz continuous, that is, for every $n\in\mathbb{N}$, we can find a constant $c_3(n)>0$ such that for all $u,v\in\ell^2$ with $\|u\|\leq n$ and $\|v\|\leq n$,

By (2.9), we obtain

From (2.6)-(2.7), we infer that for all $u\in\ell^2$,

Thus we find that $\sigma_k$ is also well-defined. By (2.5) and (2.7) we deduce that $\sigma_k$: $\ell^2\rightarrow \ell^2$ is locally Lipschitz continuous. Then for every $n\in\mathbb{N}$, we can find a constant $c_4(n)>0$ such that for any $u,v\in\ell^2$ with $\|u\|\leq n$ and $\|v\|\leq n$,

Let $\beta = \lambda+\gamma$, then we are able to rewrite (2.1)-(2.2) as the following system in $\ell^2$ for $t>\tau$ with $\tau\in\mathbb{R}$:

with initial data:

In this article, the solutions to the stochastic (2.14)-(2.16) are understood as bellow.

Definition 2.1. Given $\tau\in \mathbb{R}$ and a $\mathcal{F}_\tau$-measurable $u_\tau\in L^2(\Omega,\ell^2)$, we say a continuous $\ell^2$-valued $\mathcal{F}_t$-adapted stochastic process $u$ is a solution of (2.14)-(2.16) if $u\in L^2(\Omega, C([\tau,\tau+T], \ell^2 ))$ for all $T>0$, and for all $t\geq\tau$ and almost all $\omega\in\Omega$, we have

in $\ell^2$.

3.   Global existence and uniqueness of solutions to (2.14)-(2.16)

In this section we show the existence and uniqueness of solutions to (2.14)-(2.16) in the sense of Definition 2.1. As mentioned before, we must approximate the locally Lipschitz continuous operators $f$ and $\sigma_k$. For each $n\in\mathbb{N}$, we define a function $\xi_n:\mathbb{R}\rightarrow \mathbb{R}$ by

Then

Let $f^n(u) = \big(f_{i}(\xi_n(u_i))\big)_{i\in\mathbb{Z}^N}$ and $\sigma_k^n(u) = \big(\delta_{k,i}\hat{\sigma}_{k,i}(\xi_n(u_i))\big)_{i\in\mathbb{Z}^N}$ for $k,n\in\mathbb{N}$ and $u\in\ell^2$, By (2.9), we see

By (2.9), (2.10) and (3.2), for every $n\in\mathbb{N}$, we can find a constant $c_5(n)>0$ such that

It follows from (2.5)-(2.7) and (3.2) that there is a $c_6 = c_6(n)>0$ such that

Given $n\in\mathbb{N}$, we now consider the approximate stochastic system in $\ell^{2}$ for $t>\tau$ with $\tau\in\mathbb{R}$:

with initial data:

Following the arguments of showing the well-posedness of stochastic equations in $\mathbb{R}^n$, we can show, under (3.4)-(3.7), that for every $n\in\mathbb{N}$, $\tau\in \mathbb{R}$ and $\mathcal{F}_\tau$-measurable $u_\tau\in L^2(\Omega,\ell^2)$, system (3.8) possesses a unique solution $u_n\in L^2(\Omega, C([\tau,\infty), \ell^2 ))$ in view of Definition 2.1.

In the next theorem, we will establish the existence of solutions to (2.14)-(2.16) in view of Definition 2.1 by considering the limit of $\{u_n\}_{n = 1}^\infty$ of solutions to (3.8)-(3.9) as $n\rightarrow \infty$.

Theorem 3.1. Let (2.3)-(2.7) hold. Then for all $\tau\in \mathbb{R}$ and $\mathcal{F}_\tau$-measurable $u_\tau\in L^2(\Omega,\ell^2)$, system (2.14)-(2.16) has a solution $u$ in the sense of Definition 2.1. In addition, $u$ satisfies

where $M_1>0$ is a positive constant independent of $u_\tau$, $\tau$ and $T$.

Proof. We first prove that the solutions of (3.8)-(3.9) satisfy that for all $t\geq\tau$ and $n\in\mathbb{N}$,

where the stoping time $\varsigma_n$ is defined by

By (3.8)-(3.9), we have

Applying Ito's formula to (3.13), we infer that a.s.,

where we identify $u_{n+1}(s)-u_n(s)$ in the stochastic term with an element in the dual space of $\ell^2$ in view of the Riesz representation theorem. By $\|u_n(s)\|\leq n$ for all $s\in[\tau,\varsigma_n)$ we have

By (3.3) and (3.15) we find

By (3.6) and (3.15) we get

Then we infer from (3.14)-(3.17) that

where $C_0 = (2|\beta|+c_6(n+1))$. From the BDG inequality and (3.6), we find a constant $C_1>0$ such that

where $C_2 = 2c_6(n+1)C^2_1$. It yields from (3.18)-(3.19) that

From the Gronwall lemma and (3.20), we find

Then we find that $u_{n+1}(t\wedge\varsigma_n) = u_n(t\wedge\varsigma_n)$ for all $t\geq\tau$ a.s.. From this and (3.12) we find (3.11).

We then prove the stoping time satisfies:

In order to prove (3.22), we first deduce the following uniform estimates for the solutions $u_n$ to (3.8):

where

and $M_1>0$ is a constant independent of $u_\tau$, $n$, $\tau$ and $T$.

Applying Ito's formula to (3.8), we find

This together with (3.3) and Young's inequality gives

It yields from (3.7) that for $t \in[\tau,\tau+T]$,

From the BDG inequality and (3.26) we find that for $t \in[\tau,\tau+T]$,

where $C_1$ is the same number as given in (3.19). Then we find from (3.25)-(3.27) that for $t \in[\tau,\tau+T]$,

where $C_3 = 2+4|\beta|+8c_\delta\|\varphi_1\|^2_{\ell^\infty}(1+2C^2_1)$ and $C_4$ is given by

From the Gronwall lemma and (3.28) we can deduce that

This implies (3.23). In the following, we prove (3.22). Let $T>0$ be an arbitrary number. By (3.12), we infer that

Then we deduce from Chebychev's inequality and (3.23) that

which implies

Taking $\Omega_T = \bigcap\limits^\infty_{m = 1}\bigcup\limits^\infty_{n = m}\{\varsigma_n<T \}$, we find from the Borel-Cantelli lemma that

Then for every $\omega\in\Omega\setminus\Omega_T$, we find a $n_0 = n_0(\omega)>0$ such that $\varsigma_n(\omega)\geq T$ for all $n\geq n_0$, and thus $\varsigma(\omega)\geq T$ for all $\omega\in\Omega\setminus\Omega_T$. Let $\Omega_0 = \bigcup\limits^\infty_{T = 1}\Omega_T$, then $\mathbb{P}(\Omega_0) = 0$ and $\varsigma(\omega)\geq T$ for all $\omega\in\Omega\setminus\Omega_0$ and $T\in\mathbb{N}$. Then (3.22) yields.

We finally show the existence of solutions to (2.14). By Steps 1-2, we find a $\Omega_1\subseteq\Omega$ with $\mathbb{P}(\Omega\setminus\Omega_1) = 0$ such that

By (3.29), for every $\omega\in\Omega_1$ and $t\geq\tau$, we find a $n_0 = n_0(t,\omega)\geq1$ such that for all $n\geq n_0$,

Define a mapping $u:[\tau,\infty)\times\Omega\rightarrow \ell^2$ given by

Note that $u_n$ is a continuous $\ell^2$-valued process, by (3.31), we infer that $u$ is also continuous for $t$ in $\ell^2$ a.s.. By (3.31), we find

Since $u_n$ is $\mathcal{F}_t$-adapted, by (3.32) we deduce that $u$ is also $\mathcal{F}_t$-adapted. By (3.32), (3.23) and Fatou's lemma we see

This implies (3.10). By (3.8) we get

in $\ell^2$ for all $t\geq\tau$. By (3.31) we see $u_n(t\wedge \varsigma_n) = u(t\wedge \varsigma_n)$ a.s., which implies a.s.,

Therefore we see from (3.33)-(3.34) that a.s.,

in $\ell^2$ for all $t\geq\tau$. Since $\lim\limits_{n\rightarrow \infty}\varsigma_n = \infty$ a.s.. Then we find from (3.35) that

in $\ell^2$ for all $t\geq\tau$. Thus $u$ is a solution of (2.14) in view of Definition 2.1.

Now, we prove the uniqueness of the solutions to (2.14)-(2.16).

Theorem 3.2. Let (2.3)-(2.7) hold. Then the solution to system (2.14)-(2.16) is unique.

Proof. Let $u_1$ and $u_2$ be two solutions of (2.14). Given $n\in\mathbb{N}$ and $T>0$, we define a stoping time:

By (2.14)-(2.16), we get

From Ito's formula and (3.37), we find that a.s.,

We infer from (2.11) that

By (2.13) and (3.36), we have

Then we find from (3.38)-(3.40) that

This yields

From the BDG inequality and (3.40), the last term in (3.41) satisfies

Then we find from (3.41)-(3.42) that

where $C_7 = 4|\beta|+2c_4(n)+4c_4(n)C^2_1$. From the Gronwall lemma and (3.43) we find

For $u_{1}(\tau) = u_2(\tau)$ in $L^2(\Omega,\ell^2)$, we see from (3.44) that

Then

Note that $T_n = \tau+T$ for large enough $n$ thanks to the continuity of $u_1$ and $u_2$ in $t$. Then we find that

This shows

Since $T$ is an arbitrary number, we have

Thus the uniqueness of the solutions yields.

4.   Mean random dynamical systems, attracting universes and conditions

Now, we rewrite (2.1)-(2.2) as the following stochastic system in $\ell^2$ for $t>\tau$ with $\tau\in\mathbb{R}$:

with initial data:

From Theorems 3.1-3.2 we find that for every $\tau\in \mathbb{R}$ and $\mathcal{F}_\tau$-measurable $u_\tau\in L^2(\Omega,\ell^2)$, system (4.1)-(4.2) has a unique solution $u\in C([\tau,\infty), \ell^2 )$ $\mathbb{P}$-a.s.. Then by the Lebesgue dominated convergence theorem and the uniform estimates similar to (3.10), we can show $u\in C([\tau,\infty), L^2(\Omega, \ell^2))$. Define a mapping

by

Then we find that $\Phi$ is a mean random dynamical system for (4.1)-(4.2) on $L^2(\Omega,\mathcal{F}, \ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in view of [16,Def. 2.1], that is, for all $t,s\in\mathbb{R}^+$ and $\tau\in\mathbb{R}$,

1. $\Phi(t, \tau)$ is a mapping from $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ to $L^2(\Omega,\mathcal{F}_{\tau+t},\ell^2)$;

2. $\Phi(0, \tau)$ is an identity operator on $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$;

3. $\Phi(t+s,\tau) = \Phi(t,s+\tau)\circ\Phi(s,\tau)$.

Notice that $\Phi$ is also said a mean square random dynamical system, see e.g., Kloeden and Lorenz [10].

In order to derive several kinds of estimates of solutions to (4.1)-(4.2), we next define two families $\mathcal{D} = \{\mathcal{D}(\tau) \subseteq L^2(\Omega,\mathcal{F}_\tau,\ell^2) :\tau\in \mathbb{R}\}$ and $\mathcal{B} = \{\mathcal{B}(\tau) \subseteq L^2(\Omega,\mathcal{F}_\tau,\ell^2) :\tau\in \mathbb{R}\}$ of bounded nonempty subsets satisfying the following conditions:

where $\|\mathcal{D}(\tau-t)\|_{L^2(\Omega,\mathcal{F}_{\tau-t}\;\;, \ell^2)} = \sup\limits_{u\in \mathcal{D}(\tau-t)}\|u\|_{L^2(\Omega, \mathcal{F}_{\tau-t}\;\;, \ell^2)}$. Denote by

To prove our main results, we make the following assumptions:

where $\hat{\lambda}: = 2\wedge\lambda$.

5.   Two types of uniform estimates and absorbing sets

In this section we first provide two types of long-time uniform estimates of solutions to problem (4.1)-(4.2).

Lemma 5.1. Let (2.3)-(2.7) and (4.8a)-(4.8b) hold. Then we have the following two types of long-time uniform estimates of solutions to (4.1)-(4.2).

(1) For every $\tau\in\mathbb{R}$ and $\mathcal{D} = \{\mathcal{D}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{D}$, we can find a $T: = T(\tau,\mathcal{D})>0$ such that the solutions of (4.1)-(4.2) satisfy

(2) For every $\tau\in\mathbb{R}$ and $\mathcal{B} = \{\mathcal{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$, we can find a $T: = T(\tau,\mathcal{B})>0$ such that the solutions of (4.1)-(4.2) satisfy

Here $L>0$ is a number depending on $\lambda$ but independent of $\tau$, $\mathcal{D}$ and $\mathcal{B}$.

Proof. Note that the proof of (1) is just a special case of (2) for $s = \tau$, we will only prove (1).

Applying Ito's formula to (4.1)-(4.2), we obtain

This along with (4.8a) implies

It follows from (2.12) and (4.8a) that

By (5.3)-(5.4), we get

where $\hat{\lambda}: = 2\wedge\lambda$ and $C_8 = \frac{2}{\lambda}+ 2$.

For each $\tau\in\mathbb{R}$ and $\mathcal{B} = \{\mathcal{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$, multiplying (5.5) by $e^{\hat{\lambda}t}$ and integrating the result over $(s-t,s)$ for $s\leq\tau$, we see

By $u_{s-t}\in\mathcal{B}(s-t)$ for $s\leq\tau$ and $\mathcal{B}\in\mathfrak{B}$, we get, as $t\rightarrow +\infty$,

This along with (5.5) and (4.8b) implies there is a $T: = T(\tau,\mathcal{D})>0$ such that for all $t\geq T$,

From this we get (5.1). The proof is completed.

As a direct consequence of Lemma 5.1, we have the existence of two types of absorbing sets for the mean random dynamical system $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$.

Lemma 5.2. Let (2.3)-(2.7) and (4.8a)-(4.8b) hold. Then $\Phi$ has two types of absorbing sets in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$.

(1) $\Phi$ has a weakly compact $\mathfrak{D}$-pullback absorbing set $\mathcal{K}_\mathfrak{D} = \{\mathcal{K}_ \mathfrak{D}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{D}$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$, that is, for every $\tau\in\mathbb{R}$ and $\mathcal{D}\in\mathfrak{D}$, there exists $T = T(\tau,\mathcal{D}) > 0$ such that $\Phi(t, \tau-t)\mathcal{D}(\tau-t) \subseteq \mathcal{K}_\mathfrak{D}(\tau )$ for all $t\geq T$, where $\mathcal{K}_\mathfrak{D}(\tau) = \{u\in L^2(\Omega,\mathcal{F}_\tau,\ell^2): \mathbb{E}(\|u\|^2) \leq R_\mathfrak{D}(\tau)\}$.

(2) $\Phi$ has a weakly compact $\rm{backward}$ $\mathfrak{B}$-pullback absorbing set $\mathcal{K}_\mathfrak{B} = \{\mathcal{K}_ \mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$, that is, for every $\tau\in\mathbb{R}$ and $\mathcal{B}\in\mathfrak{B}$, there exists $T = T(\tau,\mathcal{B}) > 0$ such that $\bigcup_{s\leq\tau}\Phi(t, s-t)\mathcal{B}(s-t) \subseteq \mathcal{K}_\mathfrak{B}(\tau )$ for all $t\geq T$, where $\mathcal{K}_\mathfrak{B}(\tau) = \{u\in L^2(\Omega,\mathcal{F}_\tau,\ell^2): \mathbb{E}(\|u\|^2) \leq R_\mathfrak{B}(\tau)\}$.

Proof. The proof of (2) is just a specifical case of (2). Then we only focuss on the proof of (2).

First, by (4.8b) we know that $\mathcal{K}_\mathfrak{B}(\tau)$ is a bounded closed convex set in $L^2(\Omega,\mathcal{F}_\tau, \ell^2)$, and so it is a weakly compact subset in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$. It yields from (2) of Lemma 5.1 that for every $(\tau,\mathcal{B})\in \mathbb{R}\times\mathfrak{B}$, there exists $T: = T(\tau,\mathcal{B})>0$ such that

This implies that for all $t \geq T$,

Note that for all $s\leq\tau$ and $t\geq0$,

This together with (4.8b) implies, as $t\rightarrow +\infty$,

Therefore $\mathcal{K}_{\mathfrak{B}}\in\mathfrak{B}$ is a weakly compact backward $\mathfrak{B}$-pullback absorbing set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$. This concludes the proof.

6.   Three types of weak pullback mean random attractors of (2.1)-(2.2)

Before introduce our main results, we first give some preparations. Given $\tau\in\mathbb{R}$, we note that the weak topology of $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ has a neighborhood base at a point $\psi_0\in L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ given by the collection:

Here the set $\mathcal{N}_{\phi_{1}^{*}, \ldots, \phi_{n}^{*}}^{\varepsilon}\left(\psi_{0}\right)$ is given by, for $\varepsilon>0$ and $\phi_1^\ast,\dots, \phi_n^\ast\in(L^2(\Omega,\mathcal{F}_\tau,\ell^2))^\ast$,

If $B$ is a weakly open set containing a point $\psi_0\in L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, then we say $B$ is a weak neighborhood of the point $\psi_0$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$. If $\mathcal{B}$ is a weakly open set containing a set $B\subseteq L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, then we say $\mathcal{B}$ is a weak neighborhood of the set $B$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$.

After all preparations established in above sections, we now present the main results on the existence and uniqueness of three types of weak pullback mean random attractors (WPMRAs) for the mean random dynamical system $\Phi$ as follows. Here the three types WPMRAs are understood in the following sense.

Definition 6.1. (The usual WPMRA, see [16,Def. 2.4]) A family sets $\mathcal{A} = \{\mathcal{A}(\tau) : \tau\in\mathbb{R}\}\in\mathfrak{D}$ is called a WPMRA for $\Phi$ on $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ if the following conditions are satisfied.

(ⅰ) $\mathcal{A}(\tau)$ is a weakly compact subset of $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ for every $\tau\in \mathbb{R}$.

(ⅱ) $\mathcal{A}$ is a $\mathfrak{D}$-pullback weakly attracting set of $\Phi$, that is, for each $\tau\in\mathbb{R}$, $\mathcal{D}\in\mathfrak{D}$ and each weak neighborhood $\mathcal{N}^w(\mathcal{A}(\tau ))$ of $\mathcal{A}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, there is a time $T = T(\tau,\mathcal{D},\mathcal{N}^w(\mathcal{A}(\tau ))) > 0$ such that

(ⅲ) $\mathcal{A}$ is the minimal element of $\mathfrak{D}$ satisfying (ⅰ) as well as (ⅱ).

Definition 6.2. (The backward weakly compact WPMRA) A family sets $\mathcal{A} = \{\mathcal{A}(\tau) : \tau\in\mathbb{R}\}\in\mathfrak{B}$ is called a backward weakly compact WPMRA for $\Phi$ on $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ if the following conditions are satisfied.

(ⅰ) The set $\overline{\bigcup_{s\leq\tau}\mathcal{A}(s )}^w$ is weakly compact in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, where the closure is taken in the sense of the weak topology of $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$.

(ⅱ) $\mathcal{A}$ is a $\mathfrak{B}$-pullback weakly attracting set of $\Phi$ in the similar sense of Definition 6.1.

(ⅲ) $\mathcal{A}$ is the minimal element of $\mathfrak{B}$ satisfying (ⅰ) as well as (ⅱ).

Definition 6.3. (The backward weakly attracting WPMRA) A family sets $\mathcal{U} = \{\mathcal{U}(\tau) : \tau\in\mathbb{R}\}\in\mathfrak{B}$ is called a backward weakly attracting WPMRA for $\Phi$ on $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ if the following conditions are satisfied.

(ⅰ) $\mathcal{U}(\tau)$ is a weakly compact subset of $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$.

(ⅱ) $\mathcal{U}$ is a backward $\mathfrak{B}$-pullback weakly attracting set of $\Phi$ in the sense that, for each $\tau\in\mathbb{R}$, $\mathcal{B}\in\mathfrak{B}$ and each weak neighborhood $\mathcal{N}^w(\mathcal{U}(\tau ))$ of $\mathcal{U}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, there is a time $T = T(\tau,\mathcal{B},\mathcal{N}^w(\mathcal{U}(\tau ))) > 0$ such that

(ⅲ) $\mathcal{U}$ is the minimal element of $\mathfrak{B}$ satisfying (ⅰ) as well as (ⅱ).

Note that the notation of backward weakly compact WPMRA and backward weakly attracting WPMRA are strong than the usual WPMRA. The following theorem is concerned with the main results of the paper.

Theorem 6.4. Let (2.3)-(2.7) and (4.8a)-(4.8b) be satisfied. Then $\Phi$ possesses three kinds of WPMRAs.

(1) $\Phi$ has a unique $\rm{usual}$ WPMRA $\mathcal{A}_\mathfrak{D} = \{\mathcal{A}_\mathfrak{D}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{D}$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in the sense of Definition 6.1, which is given by

(2) $\Phi$ has a unique $\rm{backward}$ weakly compact WPMRA $\mathcal{A}_\mathfrak{B} = \{\mathcal{A}_\mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in the sense of Definition 6.2, which is given by

(3) $\Phi$ has a unique $\rm{backward}$ weakly attracting WPMRA $\mathcal{U}_\mathfrak{B} = \{\mathcal{U}_\mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in the sense of Definition 6.3, which is given by

(4) The relation of $\mathcal{A}_\mathfrak{D}$, $\mathcal{A}_\mathfrak{B}$ and $\mathcal{U}_\mathfrak{B}$ is $\mathcal{A}_\mathfrak{D} = \mathcal{A}_\mathfrak{B}\subseteq \mathcal{U}_\mathfrak{B}$.

Proof. Proof of (1). By (1) of Lemma 5.2 and the abstract results in [16,Theorem 2.7] we complete the proof of (1) immediately.

Proof of (2). By (2) of Lemma 5.2 and the abstract results in [16,Theorem 2.7] we find that $\mathcal{A}_\mathfrak{B} = \{\mathcal{A}_\mathfrak{B}(\tau):\tau\in \mathbb{R}\}\in\mathfrak{B}$ is a WPMRA for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in the sense of Definition 6.1. To show that $\mathcal{A}_\mathfrak{B}\in\mathfrak{B}$ is a backward weakly compact WPMRA for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$ over $(\Omega,\mathcal{F}, \{\mathcal{F}_t\}_{t\in\mathbb{R}},\mathbb{P})$ in the sense of Definition 6.2, we only need to show that $\overline{\bigcup_{s\leq\tau}\mathcal{A}_\mathfrak{B}(s )}^w$ is weakly compact in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$. By (2) of Lemma 5.2 we know that $\mathcal{K}_{\mathfrak{B}}\in\mathfrak{B}$ and $\mathcal{K}_{\mathfrak{B}}$ is a backward $\mathfrak{B}$-pullback absorbing set for $\Phi$. Then for each $\tau\in\mathbb{R}$, there exists $T_1 = T_1(\tau,\mathcal{K}_{\mathfrak{B}}) > 0$ such that for all $r\geq T_1$,

which along with the structure of $\mathcal{A}_\mathfrak{B}$ and the weak compactness of $\mathcal{K}_{\mathfrak{B}}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ yields

By the weak compactness of $\mathcal{K}_{\mathfrak{B}}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$ we further find that $\overline{\bigcup_{s\leq\tau}\mathcal{A}_\mathfrak{B}(s )}^w$ is weakly compact in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$. This completes the proof of (2).

Proof of (3). Similar to (6.4) we can prove that $\mathcal{U}_\mathfrak{B}(\tau )\subseteq \mathcal{K}_\mathfrak{B}(\tau )$, and hence $\mathcal{U}_\mathfrak{B}(\tau )$ is a weakly compact set in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$. By $\mathcal{K}_{\mathfrak{B}}\in\mathfrak{B}$, $\mathcal{U}_\mathfrak{B}\subseteq \mathcal{K}_\mathfrak{B}$ and the definition of $\mathfrak{B}$ we find $\mathcal{U}_\mathfrak{B}\in\mathfrak{B}$. It is easy to check that $\mathcal{A}_\mathfrak{B}(\tau)\subseteq\mathcal{U}_\mathfrak{B}(\tau)$. This along with the nonemptyness of $\mathcal{A}_\mathfrak{B}$ implies the nonemptyness of $\mathcal{U}_\mathfrak{B}$.

Next, we show that $\mathcal{U}_\mathfrak{B}$ is a backward $\mathfrak{B}$-pullback weakly attracting set in $L^2(\Omega,\mathcal{F},\ell^2)$. First, we need to prove that for every $\tau\in\mathbb{R}$ and every weak neighborhood $\mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau ))$ of $\mathcal{U}_\mathfrak{B}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, there exists $T_2 = T_2(\tau,\mathcal{K}_\mathfrak{B},N^w(\mathcal{U}_\mathfrak{B}(\tau ))) \geq T_1$ such that for all $t\geq T_2$,

If (6.5) is incorrect, then we can find $\tau_0\in\mathbb{R}$, $t_n\rightarrow +\infty$, $s_n\leq\tau_0$, $\psi_n\in\mathcal{K}_\mathfrak{B}(s_n-t_n)$ and a weak neighborhood $\mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau_0 ))$ of $\mathcal{U}_\mathfrak{B}(\tau_0)$ in $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ such that

Since $t_n\rightarrow +\infty$, we know, there exists $N_1 = N_1(\tau_0,\mathcal{K}_\mathfrak{B}) > 0$ such that $t_n\geq T_2$ for all $n\geq N_1$. And hence by (6.3), we find $\Phi(t_n, s_n-t_n)\psi_n\in\mathcal{K}_\mathfrak{B}(\tau_0)$ for all $n\geq N_1$. This along with the weak compactness of $\mathcal{K}_\mathfrak{B}(\tau_0)$ in $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ implies that there exist $\psi_0\in L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ and a subsequence which we do not relabel satisfying

Since $\mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau_0 ))$ is weakly open, we know $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)\setminus \mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau_0 ))$ is weakly closed. In this way we deduce from (6.6) and (6.7) that

In addition, it follows from (6.7) that for every $\varepsilon>0$ and $\phi_1^\ast,\dots, \phi_m^\ast\in(L^2(\Omega,\mathcal{F}_{\tau_0}, \ell^2))^\ast$, there exists $N_2 = N_2(\varepsilon,\psi_0, \phi_1^\ast,\dots, \phi_m^\ast)\in\mathbb{N}$ such that for all $n\geq N_2$,

As an immediate consequence of (6.9) we find $\psi_0\in\mathcal{U}_\mathfrak{B}(\tau_0) \subseteq\mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau_0 ))$. This is just a contradiction of (6.8). In fact, the details on the proof of $\psi_0\in\mathcal{U}_\mathfrak{B}(\tau_0)$ are given bellow. Let $\mathcal{N}^w(\psi_0)$ be an arbitrary weak neighborhood of $\psi_0$ with respect to the weak topology of $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$. Since the collection in (6.1) is a neighborhood base at $\psi_0$ in $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$, we can find that there are $\varepsilon>0$ and $\phi_1^\ast,\dots, \phi_m^\ast\in(L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2))^\ast$ so that $\mathcal{N}^ \varepsilon_{\phi_1^\ast,\dots, \phi_m^\ast}(\psi_0)\subseteq\mathcal{N}^w(\psi_0)$. This along with (6.9) implies

Given $r\geq0$, by $t_n\rightarrow +\infty$, we know, there exists $N\in\mathbb{N}$ so that $t_n\geq r$ for all $n\geq N$. Then by (6.10) we get, for all $n\geq N$,

This implies that $\mathcal{N}^w(\psi_0)\bigcap\Big( \bigcap_{t\geq r}\bigcap_{s\leq\tau_0}\Phi(t,s-t)\mathcal{K}_\mathfrak{B}(s-t)\Big)$ is a nonempty set. This shows that $\psi_0$ is a weak limit point of the set $\bigcup_{t\geq r}\bigcup_{s\leq\tau_0}\Phi(t,s-t)\mathcal{K}_\mathfrak{B} (s-t)$ in $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$, and hence

On the other hand, by (2) of Lemma 5.2 we know that for every $s\leq\tau$ and $\mathcal{B}\in\mathfrak{B}$, there exists $T_3 = T_3(s,\mathcal{K}_\mathfrak{B},N^w(\mathcal{U}_\mathfrak{B}(\tau )),\mathcal{B}) \geq T_2$ such that $\Phi(t, s-T_2-t)\mathcal{B}(s-T_2-t) \subseteq \mathcal{K}_\mathfrak{B}(s-T_2)$ for all $t\geq T_3$. This along with (6.5) yields, for all $t\geq T_3$,

This shows that $\bigcup_{s\leq\tau}\Phi(t, s-t)\mathcal{B}(s-t) \subseteq \mathcal{N}^w(\mathcal{U}_\mathfrak{B}(\tau ))$ for all $t\geq T_3+T_2$. Then $\mathcal{U}_\mathfrak{B}$ is a backward $\mathfrak{B}$-pullback weakly attracting set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$.

According to Definition 6.3, we now only need to prove that $\mathcal{U}_\mathfrak{B}$ is the minimal element of $\mathfrak{B}$ with properties (ⅰ) and (ⅱ) in Definition 6.3. Let $\mathcal{B}\in\mathfrak{B}$ be a weakly compact backward $\mathfrak{B}$-pullback weakly attracting set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$, we next show $\mathcal{U}(\tau)\subseteq\mathcal{B}(\tau)$ for all $\tau\in\mathbb{R}$. If this is not true, then there are $\tau_0\in\mathbb{R}$ and $\psi_0\in L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ satisfying $\psi_0\in\mathcal{U}(\tau_0)\setminus\mathcal{B}(\tau_0)$. And thereby $\psi_0\in L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2) \setminus\mathcal{B}(\tau_0)$. Notice that $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2) \setminus\mathcal{B}(\tau_0)$ is weakly open, we know, there exists a weak neighborhood of $\mathcal{N}^w(\psi_0 )$ at $\varphi_0$ in $L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ such that $\mathcal{N}^w(\psi_0 )\subseteq L^p(\Omega,\mathcal{F}_{\tau_0},\ell^2) \setminus\mathcal{B}(\tau_0)$. In light of the neighborhood base at $\psi_0\in L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2)$ given by (6.1), we find, there exist $\varepsilon>0$ and $\phi_1^\ast,\dots, \phi_m^\ast\in(L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2))^\ast$ such that $\mathcal{N}^ \varepsilon_{\phi_1^\ast,\dots, \phi_m^\ast}(\psi_0)\subseteq\mathcal{N}^w(\psi_0)$. As a result, we find

By $\psi_0\in\mathcal{U}_\mathfrak{B}(\tau_0)$ we find, there are $s_n\leq\tau_0$, $t_n\rightarrow +\infty$ and $\psi_n\in\mathcal{B}(s_n-t_n)$ such that

Let

be the neighborhood of $\mathcal{B}(\tau)$. Since $t_n\rightarrow +\infty$ and $\mathcal{B}\in\mathfrak{B}$ is a backward $\mathfrak{B}$-pullback weakly attracting set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$, there exists $N_3 = N_3(\tau_0,\varepsilon,\phi_1^\ast,\dots, \phi_m^\ast, \mathcal{K}_\mathfrak{B},\mathcal{B})\in\mathbb{N}$ such that for all $n\geq N_3$,

By $s_n\leq\tau_0$, $\psi_n\in\mathcal{D}(s_n-t_n)$ and (6.13) we find $\Phi(t_n, s_n-t_n)\psi_n \in\mathcal{N}^{\frac{\varepsilon}{2}} _{\phi_1^\ast,\dots, \phi_m^\ast}(\mathcal{B}(\tau_0))$ for all $n\geq N_3$. Then, there exists $\widehat{\psi} \in\mathcal{B}(\tau_0)$ such that $\Phi(t_n, s_n-t_n)\psi_n \in\mathcal{N}^{\frac{\varepsilon}{2}} _{\phi_1^\ast,\dots, \phi_m^\ast}(\widehat{\psi})$ for all $n\geq N_3$. This along with (6.12) implies $|\phi_i^\ast(\widehat{\psi})-\phi_i^\ast(\psi_0)| <\varepsilon$ for all $i = 1,\dots,m$. Then $\widehat{\psi}\in\mathcal{N}^{\varepsilon} _{\phi_1^\ast,\dots, \phi_m^\ast}(\psi_0)$, and hence by (6.11) we find $\widehat{\psi}\in L^2(\Omega,\mathcal{F}_{\tau_0},\ell^2) \setminus\mathcal{B}(\tau_0)$. This is a contradiction with $\widehat{\psi} \in\mathcal{B}(\tau_0)$, and therefore we have $\mathcal{U}(\tau)\subseteq\mathcal{B}(\tau)$ for all $\tau\in\mathbb{R}$. This completes the proof of (3).

Proof of (4). By Lemma 5.2 we find $\mathcal{K}_\mathfrak{D}(\tau)\subseteq \mathcal{K}_\mathfrak{B}(\tau)$, and hence $\mathcal{A}_\mathfrak{D}(\tau)\subseteq \mathcal{A}_\mathfrak{B}(\tau)\subseteq \mathcal{U}_\mathfrak{B}(\tau)$. Since $\mathcal{A}_\mathfrak{D}$ is a $\mathfrak{D}$-pullback weakly attracting set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$, we find, for each $\tau\in\mathbb{R}$, $\mathcal{D}\in\mathfrak{B}\subseteq\mathfrak{D}$ and each weak neighborhood $\mathcal{N}^w(\mathcal{A}_\mathfrak{D}(\tau ))$ of $\mathcal{A}_\mathfrak{D}(\tau)$ in $L^2(\Omega,\mathcal{F}_\tau,\ell^2)$, there exists $T = T(\tau,\mathcal{D},\mathcal{N}^w(\mathcal{A}_\mathfrak{D}(\tau ))) > 0$ such that for all $t\geq T$,

By $\mathcal{A}_\mathfrak{D}\subseteq \mathcal{A}_\mathfrak{B}$ and $\mathcal{A}_\mathfrak{B}\in\mathfrak{B}$ we also have $\mathcal{A}_\mathfrak{D}\in\mathfrak{B}$. Then by (6.14) we know that $\mathcal{A}_\mathfrak{D}$ is also a weakly compact $\mathfrak{B}$-pullback weakly attracting set for $\Phi$ in $L^2(\Omega,\mathcal{F},\ell^2)$. Note that $\mathcal{A}_\mathfrak{B}\in\mathfrak{B}$ is a WPMRA in the usual sense. Then by the minimality of $\mathcal{A}_\mathfrak{B}$ we have $\mathcal{A}_\mathfrak{B}\subseteq\mathcal{A}_\mathfrak{D}$. The completes the proof of (4).

Remark 6.5. It maybe interesting to study weak pullback mean random attractors of stochastic equations driven by fractional noise, see [8].

Acknowledgments

Lianbing She was supported by the Science and Technology Foundation of Guizhou Province ([2020]1Y007), the Natural Science Foundation of Education of Guizhou Province (KY[2019]139, KY[2019]143)) and School level Foundation of Liupanshui Normal University (LPSSYKJTD201907). Nan Liu was supported by China Postdoctoral Science Foundation under grant numbers 2019TQ0041 and 2019M660553. Xin Li was supported by the general project of scientific research project of the Beijing education committee of China (KM202111232008) Renhai Wang was supported by China Postdoctoral Science Foundation under grant numbers 2020TQ0053 and 2020M680456.