Convergence of bi-spatial pullback random attractors and stochastic Liouville
type
equations for nonautonomous stochastic $p$-Laplacian lattice system

Jintao Wang*, Qinghai Peng, Chunqiu Li

Department of Mathematics, Wenzhou University, Wenzhou 325035, China

###### Abstract

We consider convergence properties of the long-term behaviors with respect to
the coefficient of the stochastic term for a nonautonomous stochastic
$p$-Laplacian lattice equation with multiplicative noise. First, the upper
semi-continuity of pullback random $(\ell^{2},\ell^{q})$-attractor is proved
for each $q\in[1,+\infty)$. Then, a convergence result of the time-dependent
invariant sample Borel probability measures is obtained in $\ell^{2}$. Next,
we show that the invariant sample measures satisfy a stochastic Liouville type
equation and a termwise convergence of the stochastic Liouville type equations
is verified. Furthermore, each family of the invariant sample measures is
turned out to be a sample statistical solution, which hence also fulfills a
convergence consequence.

Keywords: Stochastic $p$-Laplacian lattice system; convergence of bi-spatial
attractors; convergence of invariant measures; stochastic Liouville type
theorem; sample statistical solutions.

AMS Subject Classification 2010: 60H10, 37L60, 37L30, 37L40

000 ∗ Corresponding author.
E-mail address<EMAIL_ADDRESS>(J.T. Wang<EMAIL_ADDRESS>(Q.H. Peng<EMAIL_ADDRESS>(C.Q. Li).

##  1 Introduction

This article investigates convergence properties of the long-term dynamical
behaviors (as $\alpha$ changes) of the following nonautonomous stochastic
$p$-Laplacian lattice equation with multiplicative noise,

$\displaystyle{\rm d}u_{i}(t)=$
$\displaystyle\nu(t)\left[|u_{i+1}-u_{i}|^{p-2}(u_{i+1}-u_{i})-|u_{i}-u_{i-1}|^{p-2}(u_{i}-u_{i-1})\right]{\rm
d}t$ $\displaystyle+\left(f_{i}(t,u_{i})-\lambda u_{i}\right){\rm d}t+\alpha
u_{i}{\rm d}W(t),\hskip 11.38092ptt>\tau$ (1.1)

with the initial datum

$u_{i}(\tau)=u_{\tau i}\hskip 11.38092pt\mbox{and}\hskip
11.38092pti\in\mathbb{Z},$ (1.2)

where the coefficient $\nu(t)\in{\mathcal{C}}(\mathbb{R})$ is assumed to
satisfy $0\leqslant\nu(t)\leqslant\nu_{0}<\infty$, $p\geqslant 2$, $f$ is the
nonautonomous nonlinear forcing, $W$ is a two-sided real-valued Wiener process
on the probability space to be determined later and $u_{\tau}:=(u_{\tau
i})_{i\in\mathbb{Z}}\in\ell^{2}$. The lattice equation (1) can be regarded as
the spatial discrete form on infinite lattices of 1-dimensional $p$-Laplacian
equation (see [11])

$\frac{\partial u}{\partial t}=\nu(t)(|u_{x}|^{p-2}u_{x})_{x}+f(t,u)-\lambda
u+\alpha u{\rm d}W(t),\hskip 11.38092ptx\in\mathbb{R}.$ (1.3)

The $p$-Laplacian equation originated from the study of a wide variety of
physical phenomena ([10, 29]) involving non-Newtonian fluids, porous media and
nonlinear elasticity etc. There have been plenty of works about $p$-Laplacian
equations; see [10, 3, 11, 20, 17, 21, 24, 27, 29, 34, 38, 39] and their
references. Lattice dynamical systems are developed by spatial discretizations
of general partial differential equations, where a countable system of
ordinary differential equations is constructed by replacing the spatial
derivatives by differences ([8]). Lattice dynamical systems have been applied
in many important areas and attracted much attention these years; see [1, 2,
3, 10, 11, 13, 12, 15, 16, 22, 23, 27, 36, 37] and their references. In
particular, Wang and Jin recently considered the stochastic $p$-Laplacian
lattice equation with multiplicative noise in [27], in which they obtained the
existence of pullback random $(\ell^{2},\ell^{q})$-attractors for each
$q\in[1,+\infty)$ and constructed the invariant sample measures in $\ell^{2}$
for the nonautonomous random dynamical system (NRDS for short) generated by
(1).

In the present article, based on the results of [27], we mainly consider the
convergence properties for the stochastic $p$-Laplacian lattice equation (1)
in the following four aspects:

  1. 1.

pullback random $(\ell^{2},\ell^{q})$-attractors for all $q\in[1,\infty)$,

  2. 2.

time-dependent invariant sample measures,

  3. 3.

stochastic Liouville type equation, and

  4. 4.

sample statistical solutions.

By convergence of pullback random $(\ell^{2},\ell^{q})$-attractors
${\mathcal{A}}_{\alpha}=\\{{\mathcal{A}}_{\alpha}(\tau,\omega)\\}_{\tau,\omega}\in\mathbb{R}\times\Omega$
(see explanations in Subsection 3.1), we mean the upper semi-continuity of
${\mathcal{A}}_{\alpha}(\tau,\omega)$ in $\ell^{2}\cap\ell^{q}$ as
$\alpha\rightarrow\alpha_{0}$ in $\mathbb{R}$, for each $\tau\in\mathbb{R}$
and $\mathbb{P}$-a.s. $\omega\in\Omega$, where
$(\Omega,{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$ is an
ergodic metric dynamical system defined in Subsection 2.1. The upper semi-
continuity of attractors is a classical topic about attractors. Upper semi-
continuity of random single-space attractors of stochastic partial
differential equations was considered in [19, 20, 28, 40]; the case of random
bi-spatial attractors was proved in [20] Wang studied the upper semi-
continuity of single-space pullback random attractor for the NRDS in [26]. In
[17] the authors obtained the upper semi-continuity of tempered random
attractors for stochastic delay $p$-Laplacian equation on unbounded thin
domains. A result of upper semi-continuity of random bi-spatial pullback
random attractors of NRDS’s was proved in [7] and it was applied to stochastic
parabolic equation on thin domains in [18] when the thin domain collapses onto
a lower dimensional domain.

But up to our knowledge until now, there is still no work about the upper
semi-continuity of pullback random $(\ell^{2},\ell^{q})$-attractors for (1)
for each $q\in[1,\infty)$. In our present work, we make some adjustment of
Theorem 3.4 of [7] to facilitate our estimates and show this convergence
result in $\ell^{2}\cap\ell^{q}$.

For the convergence of invariant sample measures, we recall the theorem of
limiting behaviors for a sequence of invariant sample measures proved by Chen
and Yang in [4]. Originally, invariant sample measures are developed recently
in [4, 29, 41] for (nonautonomous) random dynamical systems from time-
dependent invariant measures ([9, 13, 15, 16, 32, 31, 35, 38]) for
deterministic dynamical systems. This sort of invariant measure is a “weaker”
version of the usual one for random dynamical systems, since the ”sample” here
represents the reliance of the invariant measures on each fixed
$\omega\in\Omega$, while the usual one is in the sense of expectation in
$\Omega$ ([3, 28, 30, 34]). Invariant sample measures has been obtained in
[27] for the stochastic $p$-Laplacian lattice equation (1) in $\ell^{2}$. Then
Theorem 4.1 of [4] provides us an efficient method to obtain a weakly
convergent subsequence of a given sequence of families of invariant sample
measures constructed for (1) with $\alpha=\alpha_{n}$ and
$\alpha_{n}\rightarrow\alpha_{0}$ in $\mathbb{R}$.

Stochastic Liouville type equation we adopted in this article was developed in
[33] and originally established in [4] by introducing Itô’s formula. For
Liouville type equation associated with deterministic differential systems, we
refer the reader to [9, 15, 16]. Liouville type equation was indeed taken as
the basis for the equation for time-dependent statistical solutions ([9]).
Recently, it is extended to random Liouville type equation of (nonautonomous)
random differential systems (see [4, 27, 29, 41]). In [33], the authors
provided a more general definition of the class of test functions satisfying
four pivotal conditions, and hence developed stochastic Liouville type
equation ([4]) into more situations. We will use the framework set up in [33]
to check that the invariant sample measures
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ in
$\ell^{2}$ constructed in [27] satisfy the following stochastic Liouville type
equation, for $s,t\in\mathbb{R}$ with $s\leqslant t$,

$\displaystyle\int_{H}\Psi(u)\mu^{\alpha}_{t,\theta_{t-\tau}\omega}({\rm
d}u)-\int_{H}\Psi(u)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}u)$
$\displaystyle=$
$\displaystyle\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\varsigma+\alpha\int_{s}^{t}\int_{H}(u,\Psi^{\prime}(u))\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\tilde{W}(\varsigma)$
$\displaystyle+\frac{\alpha^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma,$ (1.4)

for each test function $\Psi\in{\mathcal{T}}$ (see definition in Subsection
5.1), where $\tilde{F}$ is defined in (5.1) and
$\tilde{W}(\varsigma)=W(-\tau+\varsigma)-W(-\tau)$. In this part, the
compactness of the $(\ell^{2},\ell^{q})$-attractor in $\ell^{q}$ plays an
important role.

Note that the stochastic Liouville type equation depends on the parameter
$\alpha$. The weak convergence of invariant sample measures
$\mu^{\alpha}_{\tau,\omega}$ also implies convergence of each integral in the
left side of the equation (1.4). This actually has asserted the convergence of
the stochastic Liouville type equation (1.4) in a coarse sense. In fact,
however, the termwise convergence of (1.4) as $\alpha$ varies is more
interesting and deserves more attention. We will thereby consider and prove
the termwise convergence of the stochastic Liouville type equations.

Let
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be
the invariant sample measures constructed in [27] for the NRDS generated by
(1) for each $\alpha\in\mathbb{R}$. In order to prove the termwise convergence
of the stochastic Liouville type equation with respect to $\alpha$, one needs
to further guarantee the convergence in $L^{1}$ and of the stochastic
integrals of the
$\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}$-integrals. The
convergence in $L^{1}$ follows from Lebesgue’s Dominated Convergence Theorem,
and indeed from continuity of the solution $(t,\phi)\mapsto
u_{\alpha}(t,\tau,\omega,\phi)$ and compactness of the pullback random
attractors. Comparatively, the convergence of the stochastic integrals is not
a natural consequence. However, the convergence of other terms in the
(stochastic Liouville type) equation can well assure the convergence of the
stochastic integral terms.

Statistical solutions were defined to describe the time evolution of the
probability distribution functions associated with the fluid flow ([9]). A
significant feature of a statistical solution is its basis, the Liouville type
equation discussed above. In [4], the authors defined a stochastic Liouville
type equation for a stochastic differential equation with respect to the
invariant sample measures. Afterwards, Wang, Zhu and Li not only developed the
stochastic Liouville type equation, but also gave a reasonable definition of
_sample statistical solution_ corresponding to the stochastic Liouville type
equation in [33]. In this article, we follow the definition of sample
statistical solution and prove that each family of invariant sample measures
is a sample statistical solution. And then the convergence result of sample
statistical solutions can be derived from the convergence properties of
invariant sample measures stochastic Liouville type equations with a
supplementary argument for the convergence in ${\mathcal{C}}([s,t])$ with
$s\leqslant t$ by Arzelà-Ascoli theorem.

In the remainder of this article, Section 2 is about the basic settings for
this article and the NDRS induced by (1)-(1.2). Section 3 is devoted to the
convergence of $(\ell^{2},\ell^{q})$-pullback random attractors for each
$q\in[1,+\infty)$. In Section 4, the convergence theorem of the invariant
sample measures is proved. In Section 5, we prove that the invariant sample
measures satisfy the stochastic Liouville type equation and obtain the
termwise convergence of the stochastic Liouville type equations. In Section 6,
we prove that each family of invariant sample measures is a statistical
solution, which fulfills a convergence result as well.

##  2 The induced nonautonomous random dynamical system

We introduce the basic settings of spaces and operators and induce the
nonautonomous random dynamical system for (1)-(1.2).

###  2.1 Basic notations and properties

For metric spaces $X$ and $Y$, we conventionally denote ${\mathcal{C}}(X,Y)$
(${\mathcal{C}}_{\rm b}(X,Y)$) the collection of continuous (and bounded)
functionals from $X$ to $Y$. When $Y=\mathbb{R}$, we simply use
${\mathcal{C}}(X)$ (${\mathcal{C}}_{\rm b}(X)$) to represent
${\mathcal{C}}(X,\mathbb{R})$ (${\mathcal{C}}_{\rm b}(X,\mathbb{R})$).

Let $\ell^{p}$, $p\in[1,\infty)$, be the Banach space of all real valued
$p$-power summable bi-infinite sequences with the norm $\|\cdot\|_{p}$ such
that

$\|u\|_{p}^{p}=\sum_{i\in\mathbb{Z}}|u_{i}|^{p},\hskip 22.76228pt\mbox{for
}u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{p}.$

For the case when $p=\infty$, we let $\ell^{\infty}$ be the Banach space of
all bounded bi-infinite sequences with the norm $\|\cdot\|_{\infty}$
satisfying

$\|u\|_{\infty}=\sup_{i\in\mathbb{Z}}|u_{i}|,\hskip 22.76228pt\mbox{for
}u=(u_{i})_{i\in\mathbb{Z}}\in\ell^{\infty}.$

For the case when $p=2$, $\ell^{2}$ is a Hilbert space with the inner product

$(u,v):=\sum_{i\in\mathbb{Z}}u_{i}v_{i}\hskip 11.38092pt\mbox{for all
}u=(u_{i})_{i\in\mathbb{Z}},\,v=(v_{i})_{i\in\mathbb{Z}}\in\ell^{2}$

and the norm simply denoted by $\|\cdot\|$. Generally, we conventionally
denote

$(u,v):=\sum_{i\in\mathbb{Z}}u_{i}v_{i}\hskip 11.38092pt\mbox{for all
}u=(u_{i})_{i\in\mathbb{Z}},\,v=(v_{i})_{i\in\mathbb{Z}}$

as long as the series is convergent.

We have the following embedding relationship between $\ell^{p}$ and
$\ell^{q}$, (see [11, Lemma 2.1] or [27])

$1\leqslant p\leqslant q\leqslant\infty\hskip 11.38092pt\Rightarrow\hskip
11.38092pt\ell^{p}\subset\ell^{q}\mbox{ and }\|u\|_{q}\leqslant\|u\|_{p}\mbox{
for }u\in\ell^{p}.$ (2.1)

For notational convenience in the sequel, given each $u\in\ell^{p}$,
$p\geqslant 1$ and $r>0$, we use $|u|$, $u^{r}$ and ${\rm sgn}u$ to denote the
sequences as

$(|u|)_{i}=|u_{i}|,\hskip 22.76228pt(u^{r})_{i}=u_{i}^{r}\hskip
22.76228pt\mbox{and}\hskip 22.76228pt({\rm sgn}u)_{i}={\rm sgn}u_{i},$

respectively, for each $i\in\mathbb{Z}$, where ${\rm
sgn}:\mathbb{R}\rightarrow\\{\pm 1,0\\}$ is the sign function, i.e., ${\rm
sgn}x=x/|x|$ for $x\neq 0$ and ${\rm sgn}0=0$. According to this, it is easy
to see that $|u|^{r}=(|u_{i}|^{r})_{i\in\mathbb{Z}}$ for $u\in\ell^{p}$. We
also set $0^{0}$ to be $1$ in the sequel. Let $u\in\ell^{p}$, $v\in\ell^{q}$.
By (2.1) we have the following inequality:

$(u,|v|^{r})=\sum_{i\in\mathbb{Z}}u_{i}|v_{i}|^{r}\leqslant\sum_{i\in\mathbb{Z}}|u_{i}|\|v\|_{\infty}^{r}\leqslant\|u\|_{1}\|v\|_{q}^{r}\hskip
11.38092pt\mbox{for }p=1,\,q\geqslant 1,\,r\geqslant 0.$ (2.2)

Define a certain multiplication $\otimes$ for two arbitrary bi-infinite
sequences such that

$(u\otimes v)_{i}:=u_{i}v_{i}\hskip 11.38092pt\mbox{for all
}u=(u_{i})_{i\in\mathbb{Z}},\,v=(v_{i})_{i\in\mathbb{Z}}.$

Define also the self-mappings $B$ and $B^{*}$ on the spaces of bi-infinite
sequences respectively as

$(Bu)_{i}:=u_{i+1}-u_{i},\hskip 22.76228pt(B^{*}u)_{i}:=u_{i-1}-u_{i}\hskip
22.76228pt\mbox{with}\hskip 22.76228ptAu:=B^{*}(|Bu|^{p-2}\otimes(Bu))$
$\mbox{i.e.,}\hskip
22.76228pt(Au)_{i}=|u_{i}-u_{i-1}|^{p-2}(u_{i}-u_{i-1})-|u_{i+1}-u_{i}|^{p-2}(u_{i+1}-u_{i}).$

It is obvious that $(Bu,v)=(u,B^{*}v)$ for all
$u=(u_{i})_{i\in\mathbb{Z}},\,v=(v_{i})_{i\in\mathbb{Z}}\in\ell^{2}$, both $B$
and $B^{*}$ are linear and bounded and

$\|Au\|^{2}\leqslant 4\sum_{i\in\mathbb{Z}}|(Bu)_{i}|^{2p-2}\leqslant
2^{2p-1}\sum_{i\in\mathbb{Z}}\left(|u_{i+1}|^{2p-2}+|v_{i}|^{2p-2}\right)=4^{p}\|u\|_{2p-2}^{2p-2}\leqslant
4^{p}\|u\|^{2p-2},$ (2.3) $(Au,u)=\|Au\|\|u\|=2^{p}\|u\|^{p-1}\|u\|\leqslant
2^{p}\|u\|^{p}.$

Then (1) can be rewritten as

${\rm d}u(t)=[-\nu(t)Au+f(t,u)-\lambda u]{\rm d}t+\alpha u{\rm
d}W(t),\;t>\tau,\hskip 11.38092ptu(\tau)=u_{\tau}\in\ell^{2},$ (2.4)

where $f:\mathbb{R}\times\ell^{2}\rightarrow\ell^{2}$ satisfies that
$(f(t,u))_{i}:=f_{i}(t,u_{i})$, for each $i\in\mathbb{Z}$.

The Wiener process $W$ is determined as follows. Consider the space
${\mathcal{C}}_{0}(\mathbb{R})=\\{\omega\in{\mathcal{C}}(\mathbb{R}):\omega(0)=0\\}$,
with the compact open topology, the Borel $\sigma$-algebra ${\mathcal{F}}$ and
the corresponding Wiener measure $\mathbb{P}$. For sake of seeking for a
solution of (2.4), we need to introduce the Ornstein-Uhlenbeck (OU for short)
equation (see [28]). Let $\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t)$,
$t\in\mathbb{R}$. Then
$({\mathcal{C}}_{0}(\mathbb{R}),{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$
is an ergodic metric dynamical system. Based on the system
$({\mathcal{C}}_{0}(\mathbb{R}),{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$,
we set the stochastic stationary solution

$z(\theta_{t}\omega)=-\int_{-\infty}^{0}{\rm e}^{s}\theta_{t}\omega(s){\rm
d}s,$

as a pathwise solution of the OU equation ${\rm d}z+z{\rm d}t={\rm
d}\omega(t)$. Then there is a $\theta_{t}$-invariant set
$\Omega\subset{\mathcal{C}}_{0}(\mathbb{R})$ such that $\mathbb{P}(\Omega)=1$
and for every $\omega\in\Omega$, $z(\theta_{t}\omega)$ is continuous in $t$.
The random variable $z(\theta_{t}\omega)$ satisfies

$\lim_{t\rightarrow\pm\infty}\frac{z(\theta_{t}\omega)}{t}=\lim_{t\rightarrow\pm\infty}\frac{\int_{0}^{t}z(\theta_{s}\omega){\rm
d}s}{t}=0.$ (2.5)

Hence we only consider the set $\Omega$, and define the Wiener process $W$ on
$(\Omega,{\mathcal{F}},\mathbb{P})$ as

$W(\omega)(t):=\omega(t)\hskip 22.76228pt\mbox{on the probability space
}(\Omega,{\mathcal{F}},\mathbb{P}).$

###  2.2 The nonautonomous random dynamical system

We now present the already-known results in [27] here for the following
discussion. Let $u(t,\tau,\omega,u_{\tau})$ be a solution of (2.4) and set

$v(t)={\rm e}^{-\alpha z(\theta_{t}\omega)}u(t),\hskip
22.76228ptv(\tau)=v_{\tau}:={\rm e}^{-\alpha z(\theta_{\tau}\omega)}u_{\tau}.$
(2.6)

We deduce from (2.4) and (2.6) that $v(t)$ satisfies the following random
system

$\frac{{\rm d}v(t)}{{\rm d}t}=F(t,\omega,v),\hskip 11.38092ptt>\tau,\hskip
22.76228ptv(\tau)=v_{\tau},$ (2.7)

where

$F(t,\omega,v):=-{\rm e}^{\alpha(p-2)z(\theta_{t}\omega)}\nu(t)Av+\left(\alpha
z(\theta_{t}\omega)-\lambda\right)v+{\rm e}^{-\alpha
z(\theta_{t}\omega)}f(t,{\rm e}^{\alpha z(\theta_{t}\omega)}v).$

We present the following assumption on the nonautonomous nonlinear forcing $f$
below.

(F1) The mapping $(t,u_{i})\mapsto f_{i}(t,u_{i})$ is continuous in both
variables and differentiable in $u_{i}$, and there are $\beta>0$, $q\geqslant
1$ and $\psi_{k}(t)=(\psi_{ki}(t))_{i\in\mathbb{Z}}$, $k=1,\,2$, such that

$\displaystyle f_{i}(t,u_{i})u_{i}$
$\displaystyle\leqslant-\beta|u_{i}|^{q}+\psi_{1i}(t),\mbox{ with }\psi_{1}\in
L_{\rm loc}^{1}(\mathbb{R};\ell^{1})$ (2.8) $\displaystyle\mbox{and}\hskip
11.38092pt\frac{\partial f_{i}}{\partial u_{i}}(t,u_{i})$
$\displaystyle\leqslant\psi_{2i}(t),\mbox{ with }\psi_{2}\in L_{\rm
loc}^{\infty}(\mathbb{R};\ell^{\infty}).$

Under the assumption (F1), the equation (2.7) possesses a unique global
solution
$v(\cdot,\tau,\omega,v_{\tau})\in{\mathcal{C}}^{1}([\tau,\infty);\ell^{2})\cap
L^{q}_{\rm loc}(0,\infty;\ell^{q})$ for each
$(\tau,v_{\tau})\in\mathbb{R}\times\ell^{2}$ and $\mathbb{P}$-a.s.
$\omega\in\Omega$, such that $v(\tau,\tau,\omega,v_{\tau})=v_{\tau}$, where
${\mathcal{C}}^{1}$ means differentiability and continuity of the derivative.
Then set

$u(t,\tau,\omega,u_{\tau})={\rm e}^{\alpha
z(\theta_{t}\omega)}v(t,\tau,\omega,{\rm e}^{-\alpha
z(\theta_{\tau}\omega)}u_{\tau}),$ (2.9)

We know that $u(t,\tau,\omega,u_{\tau})$ is exactly the unique global solution
of (2.4) for each $(\tau,u_{\tau})\in\mathbb{R}\times\ell^{2}$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$.

We define a mapping
$\varphi:\mathbb{R}^{+}\times\mathbb{R}\times\Omega\times\ell^{2}\rightarrow\ell^{2}$,
such that for every
$(t,\tau,\omega,u_{\tau})\in\mathbb{R}^{+}\times\mathbb{R}\times\Omega\times\ell^{2}$,

$\varphi(t,\tau,\omega,u_{\tau})=u(t+\tau,\tau,\theta_{-\tau}\omega,u_{\tau})={\rm
e}^{\alpha z(\theta_{t}\omega)}v(t+\tau,\tau,\theta_{-\tau}\omega,v_{\tau}),$
(2.10)

with $v_{\tau}={\rm e}^{-\alpha z(\theta_{\tau}\omega)}u_{\tau}$. It has been
proved in [27] that $\varphi$ defined in (2.10) is actually a continuous NRDS
over $(\Omega,{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$,
i.e., $\varphi$ satisfies the following conditions:

  1. (1)

$\varphi(\cdot,\tau,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times\ell^{2}\rightarrow\ell^{2}$
is
$({\mathcal{B}}(\mathbb{R}^{+})\times{\mathcal{F}}\times{\mathcal{B}}(\ell^{2}),{\mathcal{B}}(\ell^{2}))$-measurable
for every $\tau\in\mathbb{R}$;

  2. (2)

$\varphi(0,\tau,\omega,\cdot)$ is the identity on $\ell^{2}$ for every
$(\tau,\omega)\in\mathbb{R}\times\Omega$;

  3. (3)

$\varphi(t+s,\tau,\omega,\cdot)=\varphi(t,\tau+s,\theta_{s}\omega,\varphi(s,\tau,\omega,\cdot))$
for every $t,s\in\mathbb{R}^{+}$ and $(\tau,\omega)\in\mathbb{R}\times\Omega$;

  4. (4)

$\varphi(\cdot,\tau,\omega,\cdot):\mathbb{R}^{+}\times\ell^{2}\rightarrow\ell^{2}$
is continuous for every $(\tau,\omega)\in\mathbb{R}\times\Omega$ for
$\mathbb{P}$-a.s. $\omega\in\Omega$.

##  3 Convergence of pullback random attractors

###  3.1 Preliminaries on pullback random attractors

Let $X$ be a Polish space and $2^{X}$ be the collection of all subsets of $X$.
An $X$-valued time-parametrized random variable $R(\tau,\omega)$ is called a
_nonautonomous random variable_ in $X$. A _nonautonomous random set_ $D$ in
$X$ is a family of nonempty subsets $D(\tau,\omega)\in 2^{X}$ with two
parameters $(\tau,\omega)\in\mathbb{R}\times\Omega$, which is
$(\Omega,{\mathcal{F}},\mathbb{P})$-measurable (${\mathcal{F}}$-measurable for
short) with respect to $\omega\in\Omega$ (see [5, 25]), i.e., the mapping
$\omega\mapsto{\rm d}_{X}(x,D(\tau,\omega))$ is
$({\mathcal{F}},{\mathcal{B}}(\mathbb{R}))$-measurable for each fixed
$(x,\tau)\in X\times\mathbb{R}$, where ${\rm d}_{X}(\cdot,\cdot)$ is the
metric induced by norm of $X$.

In order to present the concept of bi-spatial pullback attractors for an NRDS,
we consider two separable Banach spaces $(X,\|\cdot\|_{X})$ and
$(Y,\|\cdot\|_{Y})$. An NRDS $\varphi$ on $X$ is assumed to take its values in
the terminate space $Y$, i.e.,

$\varphi(t,\tau,\omega,X)\subset Y,\;\mbox{for all
}t,\,\tau\in\mathbb{R}\mbox{ with }t>0\mbox{ and }\mathbb{P}\mbox{-a.s.
}\omega\in\Omega.$

For convergence uniformity, we assume the space pair $(X,Y)$ is _limit-
identical_ in the following sense,

$x_{n}\in X\cap Y,\hskip 11.38092pt\|x_{n}-x_{0}\|_{X}\rightarrow 0\mbox{ and
}\|x_{n}-y_{0}\|_{Y}\rightarrow 0\mbox{ imply }x_{0}=y_{0}\in X\cap Y.$

The norm of $X\cap Y$ is often defined as $\|x\|_{X\cap
Y}=\|x\|_{X}+\|x\|_{Y}$ for all $x\in X\cap Y$, which makes $(X\cap
Y,\|\cdot\|_{X\cap Y})$ a normed linear space. It was proved in [20] that,
$(X\cap Y,\|\cdot\|_{X\cap Y})$ is a Banach space if and only if $(X,Y)$ is a
limit-identical pair. Moreover, when $(X,Y)$ is limit-identical and $A\subset
X\cap Y$, $A$ is compact in $X\cap Y$ if and only if $A$ is compact in $X$ and
$Y$, respectively.

Let
${\mathcal{A}}=\\{{\mathcal{A}}(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$
be a nonautonomous random set in $X\cap Y$ and
$D=\\{D(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be a
nonautonomous random set in $X$. We say that $A$ _pullback attracts_ in (the
topology of) $X$ if

$\lim_{t\rightarrow+\infty}{\rm
dist}_{X}(\varphi(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega)),{\mathcal{A}}(\tau,\omega))=0,$
(3.1)

where ${\rm dist}_{X}(\cdot,\cdot)$ is the _Hausdorff semi-distance_ under the
norm of an arbitrary normed space $X$, i.e., for two nonempty sets $A,B\subset
X$,

${\rm dist}_{X}(A,B):=\sup_{a\in A}\inf_{b\in B}\|a-b\|_{X}.$

Here the pullback attraction can also be taken in (the topology of) $X\cap Y$
instead with $X$ in (3.1) replaced by $X\cap Y$.

We say ${\mathcal{D}}$ is a _universe_ in $X$, if ${\mathcal{D}}$ is an
inclusion-closed collection of nonautonomous random sets, i.e., when
$D\in{\mathcal{D}}$ and a nonautonomous random set $D^{\prime}$ satisfies
$D^{\prime}(\tau,\omega)\subset D(\tau,\omega)$ for each
$(\tau,\omega)\in\mathbb{R}\times\Omega$, then $D^{\prime}\in{\mathcal{D}}$.

In this article, we adopt the much weaker definitions of bi-spatial pullback
(random) attractors presented in [6, 27] as follows, where we do not require
the attractor to be ${\mathcal{F}}$-measurable in the initial phase space. The
existence theorem of pullback random bi-spatial attractors can be found in
[27].

######  Definition 3.1.

Let $\varphi$ be an NRDS on $X$ taking its values in $Y$ and ${\mathcal{D}}$
be a universe in $X$. A set-mapping
${\mathcal{A}}:\mathbb{R}\times\Omega\rightarrow 2^{X\cap Y}$ is said to be a
pullback random $(X,Y)$-attractor for $\varphi$ with respect to
${\mathcal{D}}$ if

  1. (1)

${\mathcal{A}}(\tau,\omega)$ is compact in $Y$ for all $\tau\in\mathbb{R}$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$;

  2. (2)

${\mathcal{A}}$ is invariant under the system $\varphi$, i.e.,

$\varphi(t,\tau,\omega,{\mathcal{A}}(\tau,\omega))={\mathcal{A}}(\tau+t,\theta_{t}\omega),\mbox{
for all }t\geqslant 0\mbox{ and }(\tau,\omega)\in\mathbb{R}\times\Omega;$

  3. (3)

${\mathcal{A}}$ is pullback ${\mathcal{D}}$-attracting in $X\cap Y$, that is,
${\mathcal{A}}$ pullback attracts every $D\in{\mathcal{D}}$ in the topology of
$X\cap Y$,

  4. (4)

there exists a closed set $K\in{\mathcal{D}}$ with ${\mathcal{A}}\subset K$.

  5. (5)

${\mathcal{A}}$ is a nonautonomous random set in $Y$.

Let $\varphi$ be an NRDS on $X$ taking its values in $Y$ and ${\mathcal{D}}$
be a universe. A _pullback ${\mathcal{D}}$-absorbing set_ in $Y$ for the NRDS
$\varphi$ is a nonautonomous random closed set $K$ in $Y$, such that for every
$D\in{\mathcal{D}}$ and all $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s.
$\omega\in\Omega$, there exists $T=T(\tau,\omega,D)>0$ such that for all
$t>T$,

$\varphi(t,\tau-t,\theta_{-t}\omega,D(\tau-t,\theta_{-t}\omega))\subset
K(\tau,\omega).$

For the upper semi-continuity of pullback random attractors, we consider a
family $\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$ of NRDS’s with
$({\mathfrak{A}},\rho)$ a metric space. We can find the following theorem in
[7] with a slight modification (see more in [19, 20, 26]) for the upper semi-
continuity.

######  Theorem 3.2.

Let $(X,Y)$ be a limit-identical pair of separable Banach spaces and
$\varphi_{\alpha}$ be a continuous NRDS on $X$ over
$(\Omega,{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$ taking
values in $Y$ for each $\alpha\in{\mathfrak{A}}$ with ${\mathfrak{A}}$ a
metric space. For each $\alpha\in{\mathfrak{A}}$, let ${\mathcal{D}}_{\alpha}$
be a universe for $\varphi_{\alpha}$ in $X$. Suppose there exists
$\alpha_{0}\in{\mathfrak{A}}$ such that

  1. (1)

for every $t>0$, $\tau\in\mathbb{R}$, $\omega\in\Omega$,
$\alpha_{n}\in{\mathfrak{A}}$ with $\alpha_{n}\rightarrow\alpha_{0}$ and
$x_{n},x\in X$ with $x_{n}\rightarrow x$,

$\lim_{n\rightarrow\infty}\varphi_{\alpha_{n}}(t,\tau,\omega,x_{n})=\varphi_{\alpha_{0}}(t,\tau,\omega,x)\hskip
11.38092pt\mbox{ in }X;$ (3.2)

  2. (2)

for each $\alpha\in{\mathfrak{A}}$, $\varphi_{\alpha}$ has a pullback
${\mathcal{D}}_{\alpha}$-absorbing set $K_{\alpha}\in{\mathcal{D}}_{\alpha}$
in $X$ such that for each $(\tau,\omega)\in\mathbb{R}\times\Omega$ and
sequence $\alpha_{n}\rightarrow\alpha_{0}$ such that

$\bigcap_{n\geqslant 1}\overline{\bigcup_{N\geqslant
n}K_{\alpha_{N}}(\tau,\omega)}^{X}\subset K_{\alpha_{0}}(\tau,\omega)$ (3.3)

and
$K_{\alpha_{0}}=\\{K_{\alpha_{0}}(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}\in{\mathcal{D}}_{\alpha_{0}}$;

  3. (3)

for each $\alpha\in{\mathfrak{A}}$, $\varphi_{\alpha}$ has a pullback random
$(X,Y)$-attractor ${\mathcal{A}}_{\alpha}\in{\mathcal{D}}_{\alpha}$ with
respect to ${\mathcal{D}}_{\alpha}$ and there exists a neighborhood
${\mathfrak{U}}$ of $\alpha_{0}$ in ${\mathfrak{A}}$ such that for every
$(\tau,\omega)\in\mathbb{R}\times\Omega$,

$\bigcup_{\alpha\in{\mathfrak{U}}}{\mathcal{A}}_{\alpha}(\tau,\omega)\hskip
11.38092pt\mbox{is precompact in }X\cap Y.$ (3.4)

Then for every $(\tau,\omega)\in\mathbb{R}\times\Omega$,

$\lim_{\alpha\rightarrow\alpha_{0}}{\rm dist}_{X\cap
Y}({\mathcal{A}}_{\alpha}(\tau,\omega),{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))=0.$
(3.5)

###### Proof.

The proof is similar to that of Proposition 3.2 and Theorem 3.4 of [7]. For
the reader’s convenience, we give a brief verification.

Suppose the contrary. We thus have $(\tau,\omega)\in\mathbb{R}\times\Omega$, a
positive number $a$ and a sequence $\alpha_{n}\rightarrow\alpha_{0}$ in
${\mathfrak{U}}$ such that for all $n\in\mathbb{N}^{+}$, ${\rm dist}_{X\cap
Y}({\mathcal{A}}_{\alpha_{n}}(\tau,\omega),{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))\geqslant
2a$, and hence the existence of a sequence $\\{x_{n}\\}_{n\in\mathbb{N}_{+}}$
with $x_{n}\in{\mathcal{A}}_{\alpha_{n}}(\tau,\omega)$ such that

${\rm dist}_{X\cap Y}(x_{n},{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))\geqslant
a\hskip 11.38092pt\mbox{for all }n\in\mathbb{N}_{+}.$ (3.6)

By (3.4), we can as well assume (pick a subsequence if necessary) that

$\lim_{n\rightarrow\infty}x_{n}=x_{0}\hskip 11.38092pt\mbox{in }X\cap Y.$
(3.7)

Then we claim $x_{0}\in{\mathcal{A}}_{\alpha_{0}}(\tau,\omega)$, which
contradicts (3.6) and completes the proof.

It suffices to show the claim. Take a positive time sequence
$\\{t_{m}\\}_{m\in\mathbb{N}_{+}}$ with $t_{m}\rightarrow+\infty$. Fix $m=1$.
For each $n\in\mathbb{N}^{+}$, by the invariance of
${\mathcal{A}}_{\alpha_{n}}$, there exists
$x_{1,n}\in{\mathcal{A}}_{\alpha_{n}}(\tau-{t_{1}},\theta_{-t_{1}}\omega)$
such that,

$x_{n}=\varphi_{\alpha_{n}}(t_{1},\tau-t_{1},\theta_{-t_{1}}\omega,x_{1,n}).$
(3.8)

Note that $x_{1,n}\in{\mathcal{A}}_{\alpha_{n}}(\tau-
t_{1},\theta_{-t_{1}}\omega)$. By (3.4) again, we can similarly assume that
$x_{1,n}\rightarrow y_{1}$ in $X\cap Y$ as $n\rightarrow\infty$, for some
$y_{1}\in X\cap Y$. Using (3.7), (3.8) and (3.2), we have

$x_{0}=\lim_{n\rightarrow\infty}\varphi_{\alpha_{n}}(t_{1},\tau-
t_{1},\theta_{-t_{1}}\omega,x_{1,n})=\varphi_{\alpha_{0}}(t_{1},\tau-
t_{1},\theta_{-t_{1}}\omega,y_{1})\hskip 11.38092pt\mbox{in }X.$

Recalling that ${\mathcal{A}}_{\alpha}\subset K_{\alpha}$ and
$x_{1,n}\in{\mathcal{A}}_{\alpha_{n}}(\tau-t_{1},\theta_{-t_{1}}\omega)$ for
each $n\in\mathbb{N}^{+}$, we know by (3.3) that

$y_{1}=\lim_{n\rightarrow\infty}x_{1,n}\in\bigcap_{n\geqslant
1}\overline{\bigcup_{N\geqslant n}K_{\alpha_{N}}(\tau,\omega)}^{X}\subset
K_{\alpha_{0}}(\tau-t_{1},\theta_{-t_{1}}\omega).$

By induction, for each $m\in\mathbb{N}^{+}$, we can find a $y_{m}\in
K_{\alpha_{0}}(\tau-t_{m},\theta_{-t_{m}}\omega)$ such that

$x_{0}=\varphi_{\alpha_{0}}(t_{m},\tau-t_{m},\theta_{-t_{m}}\omega,y_{m}).$

Using the attraction property of ${\mathcal{A}}_{\alpha_{0}}$ in
${\mathcal{D}}_{\lambda_{0}}$, we obtain

$\displaystyle{\rm
dist}_{X}(x_{0},{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))={\rm
dist}_{X}(\varphi_{\alpha_{0}}(t_{m},\tau-
t_{m},\theta_{-t_{m}}\omega,y_{m}),{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))$
$\displaystyle\leqslant$ $\displaystyle{\rm
dist}_{X}(\varphi_{\alpha_{0}}(t_{m},\tau-
t_{m},\theta_{-t_{m}}\omega,K_{\alpha_{0}}(\tau-
t_{m},\theta_{-t_{m}}\omega)),{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))\rightarrow
0,$

as $m\rightarrow\infty$, which implies
$x_{0}\in{\mathcal{A}}_{\alpha_{0}}(\tau,\omega)$ by its compactness in $Y$
and proves the claim. ∎

######  Remark 3.3.

If we replace $X\cap Y$ by $Y$ in (3.4) and (3.5), Theorem 3.2 would be a
simple deduction of [7, Theorem 3.4]. In Theorem 3.2, we obtain a stronger
conclusion by a stronger condition.

The condition (3) of Theorem 3.2 can be also inferred by the local uniform
pullback asymptotic compactness of $\varphi_{\alpha}$ (see [19, 20] for the
case without universe).

######  Definition 3.4.

Let $\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$ be a family of NRDS’s
and ${\mathcal{D}}_{\alpha}$ be a universe of $\varphi_{\alpha}$ in $X$ for
each $\alpha\in{\mathfrak{A}}$. We say that
$\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$ is locally uniformly
pullback ${\mathcal{D}}_{\alpha}$-asymptotically compact in $X$ (resp. in
$Y$), if for each $\alpha_{0}\in{\mathfrak{A}}$, there exists a neighborhood
${\mathfrak{U}}$ of $\alpha_{0}$ in ${\mathfrak{A}}$ such that for each
$(\tau,\omega)\in\mathbb{R}\times\Omega$, whenever there are sequences
$\alpha_{n}\in{\mathfrak{U}}$, $0<t_{n}\rightarrow+\infty$ and $x_{n}\in
D_{\alpha_{n}}(\tau-t_{n},\theta_{-t_{n}}\omega)$ with
$D_{\alpha_{n}}\in{\mathcal{D}}_{\alpha_{n}}$, the sequence
$\\{\varphi_{\alpha_{n}}(t_{n},\tau-
t_{n},\theta_{-t_{n}}\omega,x_{n})\\}_{n\in\mathbb{N}^{+}}$ has a convergent
subsequence in $X$ (resp. in $Y$). If $\alpha_{n}\equiv\alpha$ for all
$n\in\mathbb{N}^{+}$, we say $\varphi_{\alpha}$ is pullback
${\mathcal{D}}_{\alpha}$-asymptotically compact in $X$ (resp. in $Y$).

######  Theorem 3.5.

Let $\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$ be a family of NRDS’s
with ${\mathcal{D}}_{\alpha}$ a universe of $\varphi_{\alpha}$ in $X$ for each
$\alpha\in{\mathfrak{A}}$ and $\alpha_{0}\in{\mathfrak{A}}$. Suppose that

  1. (3′)

$\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$ is locally uniformly
pullback ${\mathcal{D}}_{\alpha}$-asymptotically compact in $X$ and $Y$,
respectively and each $\varphi_{\alpha}$ has a pullback random
$(X,Y)$-attractor ${\mathcal{A}}_{\alpha}\in{\mathcal{D}}_{\alpha}$ with
respect to ${\mathcal{D}}_{\alpha}$.

Then there is a neighborhood ${\mathfrak{U}}$ of $\alpha_{0}$ in
${\mathfrak{A}}$ such that for every $(\tau,\omega)\in\mathbb{R}\times\Omega$,
the set $\bigcup_{\alpha\in{\mathfrak{U}}}{\mathcal{A}}_{\alpha}(\tau,\omega)$
is precompact in $X\cap Y$. As a result, the conditions (1), (2) and (3′) can
also ensure the upper semi-continuity (3.5).

###### Proof.

By (3′) and Definition 3.4, we let ${\mathfrak{U}}$ be a neighborhood of
$\alpha_{0}$ in ${\mathfrak{A}}$, such that for arbitrarily fixed
$(\tau,\omega)\in\mathbb{R}\times\Omega$, whenever there are sequences
$\alpha_{n}\in{\mathfrak{U}}$, $0<t_{n}\rightarrow+\infty$ and $x_{n}\in
D_{\alpha_{n}}(\tau-t_{n},\theta_{-t_{n}}\omega)$ with
$D_{\alpha_{n}}\in{\mathcal{D}}_{\alpha_{n}}$, the sequence
$\\{\varphi_{\alpha_{n}}(t_{n},\tau-
t_{n},\theta_{-t_{n}}\omega,x_{n})\\}_{n\in\mathbb{N}^{+}}$ has a convergent
subsequence in $X$ and $Y$, respectively.

In order to show the precompactness of

${\mathscr{A}}(\tau,\omega):=\bigcup_{\alpha\in{\mathfrak{U}}}{\mathcal{A}}_{\alpha}(\tau,\omega)$

in $X$, we pick a sequence $\\{y_{n}\\}$ in ${\mathscr{A}}$. Then by the
invariance of attractors, we can find $\alpha_{n}\in{\mathfrak{U}}$ and
$x^{\prime}_{n}\in\ell^{2}$ such that
$y_{n}\in{\mathcal{A}}_{\alpha_{n}}(\tau,\omega)$,
$x^{\prime}_{n}\in{\mathcal{A}}(\tau-n,\theta_{-n}\omega)$ and

$\varphi_{\alpha_{n}}(n,\tau-n,\theta_{-n}\omega,x^{\prime}_{n})=y_{n}.$

Noting that ${\mathcal{A}}_{\alpha_{n}}\in{\mathcal{D}}_{\alpha_{n}}$, we
actually have already ensured the precompactness of
$\\{y_{n}\\}_{n\in\mathbb{N}^{+}}$ and hence of ${\mathscr{A}}(\tau,\omega)$
in $X$ by the local uniform pullback ${\mathcal{D}}_{\alpha}$-asymptotic
compactness of $\\{\varphi_{\alpha}\\}$ in $X$.

The conclusion in $Y$ can be obtained by a similar argument in $Y$. Since
$(X,Y)$ is a limit-identical pair, ${\mathscr{A}}(\tau,\omega)$ is obviously
precompact in $X\cap Y$. The proof is finished. ∎

###  3.2 Convergence of pullback random bi-spatial attractors

In order to emphasize the coefficient $\alpha$ in (2.4), we always denote the
solutions of (2.4) and (2.7) by $u_{\alpha}$ and $v_{\alpha}$, respectively
and by $\varphi_{\alpha}$ the generated process by $u_{\alpha}$ in the sequel.

Let $\lambda_{0},\,\lambda_{1}\in\mathbb{R}$ be arbitrary such that

$0<\lambda_{1}<\lambda_{0}<2\lambda.$

We recall some other assumptions on $f$ with a slight adjustment for the
following argument:

(F2) The function $\psi_{1}$ in (2.8) satisfies

$\int_{-\infty}^{0}{\rm e}^{\lambda_{1}s}\|\psi_{1}(s)\|_{1}{\rm d}s<+\infty.$

(F3) Let $q\in[1,+\infty)$ be given in (F1). Assume that there are
$\psi_{k}(t)=(\psi_{ki}(t))_{i\in\mathbb{Z}}$, $k=3,4$, such that

$|f_{i}(t,u_{i})|\leqslant\psi_{3i}(t)|u_{i}|^{q-1}+\psi_{4i}(t),$ (3.9)
$\mbox{with}\hskip 11.38092pt\psi_{3}\in L^{q}_{\rm
loc}(\mathbb{R},\ell^{q})\cap L_{\rm loc}^{1}(\mathbb{R},\ell^{1}),\hskip
11.38092pt\psi_{4}\in L_{\rm loc}^{1}(\mathbb{R},\ell^{1}),$

and if moreover $q\in[1,2)$, there are $\kappa_{0}$, $\Lambda$, $t_{0}>0$ with
$q\lambda_{0}>2\kappa_{0}$ such that when $t>t_{0}$,

$\|\psi_{3}(-t)\|_{1}\leqslant\Lambda{\rm e}^{\kappa_{0}t}\hskip
11.38092pt\mbox{and}\hskip 11.38092pt\|\psi_{4}(-t)\|_{1}\leqslant\Lambda{\rm
e}^{\kappa_{0}t}.$

According to the settings in [27], for each $\varphi_{\alpha}$
($\alpha\in\mathbb{R}$), we consider the universe ${\mathcal{D}}_{\alpha}$ to
be all the families
$D_{\alpha}=\\{D_{\alpha}(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$
be a nonautonomous random set of $\ell^{2}$ such that for every
$\tau\in\mathbb{T}$ and $\omega\in\Omega$,

$\lim_{s\rightarrow-\infty}{\rm
e}^{\lambda_{0}s+2\alpha\int_{s}^{0}z(\theta_{r}\omega){\rm d}r-2\alpha
z(\theta_{t}\omega)}\|D_{\alpha}(\tau+s,\theta_{s}\omega)\|^{2}=0,$

where $\|D\|=\sup_{u\in D}\|u\|$ for each subset $D$ of $\ell^{2}$. The
following theorem is a main theorem — Theorem 3.15 of [27].

######  Theorem 3.6.

Let the assumptions (F1), (F2) and (F3) hold for $f$. Then the NRDS
$\varphi_{\alpha}$ has a unique pullback random
$(\ell^{2},\ell^{q})$-attractor
${\mathcal{A}}_{\alpha}\in{\mathcal{D}}_{\alpha}$ with respect to
${\mathcal{D}}_{\alpha}$, which is also the pullback random
$(\ell^{2},\ell^{2})$-attractor with respect to ${\mathcal{D}}_{\alpha}$.

In the following discussion, we are devoted to verify the upper semi-
continuity of ${\mathcal{A}}_{\alpha}$ at each $\alpha\in\mathbb{R}$. By
Theorems 2.2 of [27], we have the following consequence.

######  Lemma 3.7.

For each $u_{\tau}\in\ell^{2}$, $\alpha\in\mathbb{R}$, $\tau\in\mathbb{R}$,
$T>\tau$, $t\in(\tau,T]$ and a.s. $\omega\in\Omega$, it holds that

$\|v_{\alpha}(t)\|^{2}\leqslant G_{1}(\alpha,\|u_{\tau}\|)\hskip
22.76228pt\mbox{and}\hskip
11.38092pt\int_{\tau}^{t}\|v_{\alpha}(s)\|_{q}^{q}{\rm d}s\leqslant
G_{2}(\alpha,\|u_{\tau}\|)$ (3.10)

where for each $R\in\mathbb{R}^{+}$,

$\displaystyle G_{1}(\alpha,R):=G_{1}(\alpha,\tau,T,\omega,R)$
$\displaystyle=$ $\displaystyle{\rm
e}^{2\alpha\int_{\tau}^{T}|z(\theta_{r}\omega)|{\rm d}r-2\alpha
z(\theta_{\tau}\omega)}R^{2}+2\int_{\tau}^{T}{\rm
e}^{2\alpha\int_{s}^{T}|z(\theta_{r}\omega)|{\rm d}r-2\alpha
z(\theta_{s}\omega)}\|\psi_{1}(s)\|_{1}{\rm d}s,$ $\displaystyle
G_{2}(\alpha,R):=G_{2}(\alpha,\tau,T,\omega,R)=E(\alpha)G_{1}(\alpha,R)$

and

$E(\alpha):=E(\alpha,\tau,T,\omega)=\frac{1}{2\beta}{\rm
e}^{2\lambda(T-\tau)+2|\alpha|\int_{\tau}^{T}|z(\theta_{r}\omega)|{\rm
d}r+|\alpha(q-2)|\max_{s\in[\tau,T]}|z(\theta_{s}\omega)|}.$

Furthermore, $G_{i}(\alpha,R)$ ($i=1,2$) is joint continuous in
$(\alpha,R)\in\mathbb{R}\times\mathbb{R}^{+}$.

###### Proof.

The proof is almost the same with that of Theorems 2.2 of [27]. We only give a
simple sketch of the proof for the reader’s convenience.

Taking the inner product of (2.7) and $v$, and applying Gronwall’s lemma and
the assumption (F1), we can easily obtain (recall the inequality (18) of [27])

$\displaystyle\|v_{\alpha}(t)\|^{2}+2\beta\int_{\tau}^{t}{\rm
e}^{2\lambda(\varsigma-t)+2\alpha\int_{\varsigma}^{t}z(\theta_{r}\omega){\rm
d}r+\alpha(q-2)z(\theta_{\varsigma}\omega)}\|v_{\alpha}(\varsigma)\|_{q}^{q}{\rm
d}\varsigma$ $\displaystyle\leqslant$ $\displaystyle{\rm
e}^{2\lambda(\tau-t)+2\alpha\int_{\tau}^{t}z(\theta_{r}\omega){\rm
d}r}\|v_{\alpha}(\tau)\|^{2}+2\int_{\tau}^{t}{\rm
e}^{2\lambda(s-t)+2\alpha\int_{s}^{t}z(\theta_{r}\omega){\rm d}r-2\alpha
z(\theta_{s}\omega)}\|\psi_{1}(s)\|_{1}{\rm d}s,$ (3.11)

from which the first inequality of (3.10) follows by (2.6) as well.

It is trivial to see that

$\frac{1}{E(\alpha)}\int_{\tau}^{T}\|v_{\alpha}(s)\|^{2}{\rm d}s\leqslant
2\beta\int_{\tau}^{t}{\rm
e}^{2\lambda(\varsigma-t)+2\alpha\int_{\varsigma}^{t}z(\theta_{r}\omega){\rm
d}r+\alpha(q-2)z(\theta_{\varsigma}\omega)}\|v_{\alpha}(\varsigma)\|_{q}^{q}{\rm
d}\varsigma\leqslant G_{1}(\alpha,\|u_{\tau}\|),$

which implies the second inequality of (3.10).

The joint continuity of $G_{i}(\alpha,R)$ ($i=1,2$) in $(\alpha,R)$ is a
simple deduction of the Dominated Convergence Theorem. The proof is complete.
∎

In the following argument, for notational convenience, we set

${\mathcal{Z}}=\left(1\vee\sup_{n\in\mathbb{N}}|\alpha_{n}|\right)\cdot\max_{t\in[\tau,T]}|z(\theta_{t}\omega)|,\hskip
11.38092pt\hskip 11.38092pt{\mathcal{E}}_{r,n}:=\max_{t\in[\tau,T]}\left|{\rm
e}^{\alpha_{n}rz(\theta_{t}\omega)}-{\rm
e}^{\alpha_{0}rz(\theta_{t}\omega)}\right|$ $\mbox{and}\hskip
11.38092ptR_{\delta}:=R_{\delta}(u_{\tau})=\sup\\{\|u\|:\|u-u_{\tau}\|<\delta,\,u\in\ell^{2}\\}\hskip
11.38092pt\mbox{for each }\delta>0,$

where $a\vee b$ means the bigger one between the real numbers $a$ and $b$,
$r\in\mathbb{R}$. It is obvious that for each $r\in\mathbb{R}$ and
$\omega\in\Omega$, ${\mathcal{E}}_{r,n}\rightarrow 0$ as $n\rightarrow\infty$.
By continuity of $G_{1}$ given in Lemma 3.2, we know that for each
$t\in[\tau,T]$, $\omega\in\Omega$ and $u^{\prime}_{\tau}$ with
$\|u^{\prime}_{\tau}-u_{\tau}\|<\delta$,

$\|v_{\alpha_{n}}(t,\tau,\omega,u^{\prime}_{\tau})\|^{2}\leqslant\sup_{n\in\mathbb{N}}G_{1}(\alpha_{n},R_{\delta}):={\mathcal{G}}_{\delta}<\infty.$

All ${\mathcal{Z}}$, ${\mathcal{E}}_{r,n}$ and ${\mathcal{G}}_{\delta}$ are
independent of the time $t\in[\tau,T]$. We fix an arbitrary $\delta>0$ and
consider the initial $u^{\prime}_{\tau}$ with
$\|u^{\prime}_{\tau}-u_{\tau}\|<\delta$. Moreover, we always denote $C$ as an
arbitrary positive constant only depending on $p$ and $q$ and $C$ may be
different from line to line and even in the same line.

Next we check the condition (1) of Theorem 3.2 in the following lemma.

######  Lemma 3.8.

Suppose the assumptions (F1), (F2) and (F3) hold. Let
$\alpha_{0}\in\mathbb{R}$ with a sequence $\\{\alpha_{n}\\}\subset\mathbb{R}$
such that $\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$,
$\tau,\,t,\,T\in\mathbb{R}$ with $\tau<t\leqslant T$, $u_{\tau}\in\ell^{2}$
and $\mathbb{P}$-a.s. $\omega\in\Omega$. Then there exists
$\delta:=\delta(\tau,T)>0$ such that when $\|\phi-u_{\tau}\|<\delta$,

$u_{\alpha_{n}}(t,\tau,\omega,\phi)\rightarrow
u_{\alpha_{0}}(t,\tau,\omega,\phi)$

in $\ell^{2}$ uniformly in $\phi$ with $\|\phi-u_{\tau}\|<\delta$ as
$\alpha_{n}\rightarrow\alpha_{0}$.

###### Proof.

It suffices to prove

$v_{\alpha_{n}}(t,\tau,\omega,{\rm
e}^{-\alpha_{n}z(\theta_{\tau}\omega)}\phi)\rightarrow
v_{\alpha_{0}}(t,\tau,\omega,{\rm e}^{-\alpha_{0}z(\theta_{\tau}\omega)}\phi)$
(3.12)

in $\ell^{2}$ uniformly in $\phi$ with $\|\phi-u_{\tau}\|<\delta$ as
$\alpha_{n}\rightarrow\alpha_{0}$ for some $\delta>0$. This is due to the fact

$\displaystyle\|u_{\alpha_{n}}(t,\tau,\omega,\phi)-u_{\alpha_{0}}(t,\tau,\omega,\phi)\|$
$\displaystyle\leqslant$ $\displaystyle{\rm
e}^{\max\\{|\alpha_{n}|:n\in\mathbb{N}\\}|z(\theta_{t}\omega)|}\|v_{\alpha_{n}}(t)-v_{\alpha_{0}}(t)\|+|{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}-{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}|\|v_{\alpha_{0}}(t)\|,$

which obviously vanishes as $\alpha_{n}\rightarrow\alpha_{0}$.

In order to show (3.12), we take the inner product in $\ell^{2}$ of the
difference between the corresponding equations (2.7) with $\alpha_{n}$ and
$\alpha_{0}$ and $v_{\alpha_{n}}-v_{\alpha_{0}}$. Then we have

$\displaystyle\frac{1}{2}\frac{{\rm d}}{{\rm
d}t}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}=\left(F(t,\omega,v_{\alpha_{n}})-F(t,\omega,v_{\alpha_{0}}),v_{\alpha_{n}}-v_{\alpha_{0}}\right)$
$\displaystyle=$ $\displaystyle-\nu(t)\left({\rm
e}^{\alpha_{n}(p-2)z(\theta_{t}\omega)}Av_{\alpha_{n}}-{\rm
e}^{\alpha_{0}(p-2)z(\theta_{t}\omega)}Av_{\alpha_{0}},v_{\alpha_{n}}-v_{\alpha_{0}}\right)$
$\displaystyle+\left((\alpha_{n}z(\theta_{t}\omega)-\lambda)v_{\alpha_{n}}-(\alpha_{0}z(\theta_{t}\omega)-\lambda)v_{\alpha_{0}},v_{\alpha_{n}}-v_{\alpha_{0}}\right)$
$\displaystyle+\left({\rm e}^{-\alpha_{n}z(\theta_{t}\omega)}f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})-{\rm
e}^{-\alpha_{0}z(\theta_{t}\omega)}f(t,{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}}),v_{\alpha_{n}}-v_{\alpha_{0}}\right)$
$\displaystyle:=$ $\displaystyle\nu(t)J_{1}+J_{2}+J_{3}.$ (3.13)

First, observe by (2.3), (2.2) and the mean value theorem that

$\displaystyle|J_{1}|\leqslant$ $\displaystyle\left|\left(({\rm
e}^{\alpha_{n}(p-2)z(\theta_{t}\omega)}-{\rm
e}^{\alpha_{0}(p-2)z(\theta_{t}\omega)})Av_{\alpha_{n}},v_{\alpha_{n}}-v_{\alpha_{0}}\right)\right|$
$\displaystyle+{\rm
e}^{\alpha_{0}(p-2)z(\theta_{t}\omega)}\left|\left((Av_{\alpha_{n}}-Av_{\alpha_{0}}),v_{\alpha_{n}}-v_{\alpha_{0}}\right)\right|$
$\displaystyle:=$ $\displaystyle J_{11}+J_{12},$ (3.14) $\displaystyle
J_{11}\leqslant$
$\displaystyle{\mathcal{E}}_{p-2,n}\|Av_{\alpha_{n}}\|(\|v_{\alpha_{n}}\|+\|v_{\alpha_{0}}\|)\leqslant
C{\mathcal{G}}_{\delta}^{\frac{p}{2}}{\mathcal{E}}_{p-2,n},$ (3.15)
$\displaystyle J_{12}\leqslant$ $\displaystyle{\rm
e}^{(p-2){\mathcal{Z}}}\sum_{i\in\mathbb{Z}}\left||(Bv_{\alpha_{n}})_{i}|^{p-2}(Bv_{\alpha_{n}})_{i}-|(Bv_{\alpha_{0}})_{i}|^{p-2}(Bv_{\alpha_{0}})_{i}\right||(Bv_{\alpha_{n}}-Bv_{\alpha_{0}})_{i}|$
$\displaystyle\leqslant$ $\displaystyle(p-1){\rm
e}^{(p-2){\mathcal{Z}}}\sum_{i\in\mathbb{Z}}\left(|(Bv_{\alpha_{n}})_{i}|+|(Bv_{\alpha_{0}})_{i}|\right)^{p-2}|(Bv_{\alpha_{n}}-Bv_{\alpha_{0}})_{i}|^{2}$
$\displaystyle\leqslant$ $\displaystyle(p-1){\rm
e}^{(p-2){\mathcal{Z}}}\left(\|Bv_{\alpha_{n}}\|+\|Bv_{\alpha_{0}}\|\right)^{p-2}\|B(v_{\alpha_{n}}-v_{\alpha_{0}})\|^{2}$
$\displaystyle\leqslant$ $\displaystyle C{\rm
e}^{(p-2){\mathcal{Z}}}{\mathcal{G}}_{\delta}^{\frac{p-2}{2}}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}$
(3.16)

and

$\displaystyle J_{2}=$
$\displaystyle(\alpha_{n}-\alpha_{0})z(\theta_{t}\omega)(v_{\alpha_{n}},v_{\alpha_{n}}-v_{\alpha_{0}})+(\alpha_{0}z(\theta_{t}\omega)-\lambda)\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}$
$\displaystyle\leqslant$ $\displaystyle
C{\mathcal{Z}}{\mathcal{G}}_{\delta}|\alpha_{n}-\alpha_{0}|+|{\mathcal{Z}}-\lambda|\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}.$
(3.17)

We divide $J_{3}$ into three parts as follows:

$\displaystyle|J_{3}|\leqslant$ $\displaystyle\left|{\rm
e}^{-\alpha_{n}z(\theta_{t}\omega)}-{\rm
e}^{-\alpha_{0}z(\theta_{t}\omega)}\right|\left|\left(f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}}),v_{\alpha_{n}}-v_{\alpha_{0}}\right)\right|$
$\displaystyle+{\rm e}^{-2\alpha_{0}z(\theta_{t}\omega)}\left|\left(f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})-f(t,{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}}),{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{n}}-{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}}\right)\right|$
$\displaystyle+{\rm e}^{-2\alpha_{0}z(\theta_{t}\omega)}\left(f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})-f(t,{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}}),{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}}-{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}}\right)$ $\displaystyle:=$
$\displaystyle J_{31}+{\rm
e}^{-2\alpha_{0}z(\theta_{t}\omega)}(J_{32}+J_{33}).$ (3.18)

Under the assumption (F3) and Young’s inequality, we have for each
$n\in\mathbb{N}$,

$\displaystyle\sum_{i\in\mathbb{Z}}|f_{i}(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n},i})|$ $\displaystyle\leqslant$
$\displaystyle{\rm
e}^{\alpha_{n}(q-1)z(\theta_{t}\omega)}\sum_{i\in\mathbb{Z}}|\psi_{3i}(t)||v_{\alpha_{n},i}|^{q-1}+\sum_{i\in\mathbb{Z}}|\psi_{4i}(t)|$
$\displaystyle\leqslant$ $\displaystyle{\rm
e}^{(q-1){\mathcal{Z}}}\sum_{i\in\mathbb{Z}}\left(\frac{q-1}{q}|v_{\alpha_{n},i}|^{q}+\frac{1}{q}|\psi_{3i}(t)|^{q}\right)+\|\psi_{4}(t)\|_{1}$
$\displaystyle\leqslant$ $\displaystyle C{\rm
e}^{(q-1){\mathcal{Z}}}\left(\|v_{\alpha_{n}}\|_{q}^{q}+\|\psi_{3}(t)\|_{q}^{q}\right)+\|\psi_{4}(t)\|_{1}<+\infty,$
(3.19)

which means that $f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})\in\ell^{1}$ for all
$n\in\mathbb{N}$ and hence by (2.2), it yields that

$\displaystyle J_{31}\leqslant$ $\displaystyle{\mathcal{E}}_{-1,n}\|f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})\|_{1}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|$
$\displaystyle\leqslant$ $\displaystyle
2{\mathcal{G}}_{\delta}^{\frac{1}{2}}\|f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})\|_{1}{\mathcal{E}}_{-1,n}$
(3.20)

and

$\displaystyle J_{32}\leqslant$ $\displaystyle\left(\|f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})\|_{1}+\|f(t,{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}})\|_{1}\right){\mathcal{E}}_{1,n}\|v_{\alpha_{n}}\|$
$\displaystyle\leqslant$
$\displaystyle{\mathcal{G}}_{\delta}^{\frac{1}{2}}\left(\|f(t,{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}})\|_{1}+\|f(t,{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}})\|_{1}\right){\mathcal{E}}_{1,n}.$
(3.21)

For $J_{33}$, we can use (F1) and obtain that

$\displaystyle J_{33}\leqslant$ $\displaystyle\|\psi_{2}(t)\|_{\infty}\|{\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}v_{\alpha_{n}}-{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}v_{\alpha_{0}}\|^{2}$
$\displaystyle\leqslant$
$\displaystyle\|\psi_{2}(t)\|_{\infty}\left(\left({\rm
e}^{\alpha_{n}z(\theta_{t}\omega)}-{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}\right)\|v_{\alpha_{n}}\|+{\rm
e}^{\alpha_{0}z(\theta_{t}\omega)}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|\right)^{2}$
$\displaystyle\leqslant$ $\displaystyle
2\|\psi_{2}(t)\|_{\infty}\left({\mathcal{G}}_{\delta}{\mathcal{E}}_{1,n}^{2}+{\rm
e}^{2\alpha_{0}z(\theta_{t}\omega)}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}\right).$
(3.22)

Thus by setting $L_{1}$ and $L_{2}(n,t)$ such that

$\frac{1}{2}L_{1}:=C\nu_{0}{\rm
e}^{(p-2){\mathcal{Z}}}{\mathcal{G}}_{\delta}^{\frac{p-2}{2}}+|{\mathcal{Z}}-\lambda|+2\operatorname*{ess\,sup}_{t\in[\tau,T]}\|\psi_{2}(t)\|_{\infty},$

which is independent of $t$ and $n$ and

$\displaystyle\frac{1}{2}L_{2}(n,t):=$ $\displaystyle
C\nu_{0}{\mathcal{G}}_{\delta}^{\frac{p}{2}}{\mathcal{E}}_{p-2,n}+C{\mathcal{Z}}{\mathcal{G}}_{\delta}|\alpha_{n}-\alpha_{0}|+2{\rm
e}^{2{\mathcal{Z}}}{\mathcal{G}}_{\delta}\operatorname*{ess\,sup}_{t\in[\tau,T]}\|\psi_{2}(t)\|_{\infty}{\mathcal{E}}_{1,n}^{2}$
$\displaystyle+2{\mathcal{G}}_{\delta}^{\frac{1}{2}}\left(C{\rm
e}^{(q-1){\mathcal{Z}}}\left(\|v_{\alpha_{n}}\|_{q}^{q}+\|\psi_{3}(t)\|_{q}^{q}\right)+\|\psi_{4}(t)\|_{1}\right)\left({\mathcal{E}}_{-1,n}+{\rm
e}^{2{\mathcal{Z}}}{\mathcal{E}}_{1,n}\right),$ (3.23)

one can infer from (3.13) to (3.22) that

$\frac{{\rm d}}{{\rm d}t}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}\leqslant
L_{1}\|v_{\alpha_{n}}-v_{\alpha_{0}}\|^{2}+L_{2}(n,t).$ (3.24)

Applying Gronwall’s inequality to (3.24), we can obtain

$\displaystyle\|v_{\alpha_{n}}(t)-v_{\alpha_{0}}(t)\|^{2}\leqslant$
$\displaystyle\|v_{\alpha_{n}}(\tau)-v_{\alpha_{0}}(\tau)\|^{2}{\rm
e}^{L_{1}(t-\tau)}+\int_{\tau}^{t}L_{2}(n,s){\rm e}^{L_{1}(t-s)}{\rm d}s$
$\displaystyle\leqslant$ $\displaystyle\left|{\rm
e}^{-\alpha_{n}z(\theta_{\tau}\omega)}-{\rm
e}^{-\alpha_{0}z(\theta_{\tau}\omega)}\right|^{2}\|u_{\tau}\|^{2}{\rm
e}^{L_{1}(T-\tau)}+\int_{\tau}^{T}L_{2}(n,s){\rm e}^{L_{1}(T-s)}{\rm d}s.$
(3.25)

Then combining (3.10) and (F3), we can obtain the uniform boundedness of the
second term (integral) on the right side of (3.2) for all
$n\in\mathbb{N}^{+}$. Moreover, each term in (3.23) contains either
${\mathcal{E}}_{r,n}$ for some $r\in\mathbb{R}$ or $|\alpha_{n}-\alpha_{0}|$,
which does not depend on the time variable $t$. This implies that as
$n\rightarrow\infty$,

$\int_{\tau}^{T}L_{2}(n,s){\rm e}^{L_{1}(t-s)}{\rm d}s\rightarrow 0,$

and hence the right side of (3.2) tends to $0$ as $n\rightarrow\infty$. The
proof is thus finished. ∎

Based on the uniform continuity of $\varphi_{\alpha}$ with respect to $\alpha$
given in Lemma 3.8 and the continuity of $\varphi_{\alpha}$ with respect to
the initial datum, one can trivially show that as $n\rightarrow\infty$,

$\displaystyle\|\varphi_{\alpha_{n}}(t,\tau,\omega,u_{n})-\varphi_{\alpha_{0}}(t,\tau,\omega,u_{0})\|$
$\displaystyle\leqslant$
$\displaystyle\|\varphi_{\alpha_{n}}(t,\tau,\omega,u_{n})-\varphi_{\alpha_{0}}(t,\tau,\omega,u_{n})\|+\|\varphi_{\alpha_{0}}(t,\tau,\omega,u_{n})-\varphi_{\alpha_{0}}(t,\tau,\omega,u_{0})\|\rightarrow
0,$

whenever $\alpha_{n}\rightarrow\alpha_{0}$ and $u_{n}\rightarrow u_{0}$ in
$\ell^{2}$ as $n\rightarrow\infty$. This asserts the condition (1) of Theorem
3.2.

Next we check the condition (2) of Theorem 3.2, for which we need the
following conclusion.

######  Theorem 3.9.

Let $K_{n}$ be a sequence of subsets of $X$ and $K\subset X$. Suppose that for
each subsequence $n_{k}$ of $n$, and $x_{n_{k}}\in K_{n_{k}}$ with
$x_{n_{k}}\rightarrow x_{0}\in X$, it holds that $x_{0}\in K$, Then

$\bigcap_{n\geqslant 1}\overline{\bigcup_{N\geqslant n}K_{N}}^{X}\subset K.$
(3.26)

###### Proof.

Take $x_{0}$ in the left side of (3.26). Then for each $n\geqslant 1$,

$x_{0}\in\overline{\bigcup_{N\geqslant n}K_{N}}^{X}.$

This indicates that for each $k\geqslant 1$, there exists $n_{k}\geqslant 1$
and $x_{n_{k}}\in K_{n_{k}}$ such that

$\|x_{n_{k}}-x_{0}\|<\frac{1}{k}.$

Furthermore, $n_{k}$ can be chosen strictly increasing, i.e., $n_{k+1}>n_{k}$.
Obviously, $x_{n_{k}}\rightarrow x_{0}$ and by the supposion, $x_{0}\in K$.
The proof is finished.∎

Recall the results in [27]. We know that for each $\alpha\in\mathbb{R}$, the
process $\varphi_{\alpha}$ has a pullback ${\mathcal{D}}_{\alpha}$-absorbing
set
$K_{\alpha}=\\{K_{\alpha}(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}\in{\mathcal{D}}_{\alpha}$
with

$K_{\alpha}(\tau,\omega)=\\{u\in\ell^{2}:\|u\|^{2}\leqslant{\rm e}^{2\alpha
z(\omega)}{\mathcal{R}}(\alpha,\tau,\omega)\\},$

where

${\mathcal{R}}(\alpha,\tau,\omega):=1+2\int_{-\infty}^{0}{\rm
e}^{\lambda_{0}s+2\alpha\int_{s}^{0}z(\theta_{r}\omega){\rm d}r-2\alpha
z(\theta_{s}\omega)}\|\psi_{1}(\tau+s)\|_{1}{\rm d}s.$

######  Lemma 3.10.

Let $(\tau,\omega)\in\mathbb{R}\times\Omega$ be arbitrary,
$\\{\alpha_{n}\\}_{n\in\mathbb{N}}$ be a sequence with
$\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$ and
$\\{x_{n}\\}_{n\in\mathbb{N}}$ be a sequence in $\ell^{2}$ with $x_{n}\in
K_{\alpha_{n}}(\tau,\omega)$ for $n\in\mathbb{N}^{+}$ such that
$x_{n}\rightarrow x_{0}\in\ell^{2}$. Then $x_{0}\in
K_{\alpha_{0}}(\tau,\omega)$.

###### Proof.

Indeed, for each fixed $(\tau,\omega)\in\mathbb{R}\times\Omega$, we can let
$g:\ell^{2}\times\mathbb{R}\rightarrow\mathbb{R}$ such that

$g(x,\alpha)={\rm e}^{2\alpha
z(\omega)}{\mathcal{R}}(\alpha,\tau,\omega)-\|x\|^{2}.$

By the Dominated Convergence Theorem, it is easy to see that
${\mathcal{R}}(\alpha,\tau,\omega)$ is continuous in $\alpha$, and hence
$g(x,\alpha))$ is joint continuous in $(x,\alpha)$.

Note that for $x_{n}$ and $\alpha_{n}$, we have

$g(x_{n},\alpha_{n})\geqslant 0.$

Now let $n\rightarrow\infty$. Then by the local inheriting order property of
continuous functions, we know

$g(x_{0},\alpha_{0})\geqslant 0,\hskip 11.38092pt\mbox{i.e.,}\hskip
11.38092pt{\rm
e}^{2\alpha_{0}z(\omega)}{\mathcal{R}}(\alpha,\tau,\omega)-\|x_{0}\|^{2}\geqslant
0.$

This indicates that $x_{0}\in K_{\alpha_{0}}(\tau,\omega)$ and ends the proof.
∎

Theorem 3.9 and Lemma 3.10 have actually guaranteed the condition (2) of
Theorem 3.2. To check the condition (3), we can verify the local uniform
pullback ${\mathcal{D}}_{\alpha}$-asymptotic compactness of
$\\{\varphi_{\alpha}\\}_{\alpha\in\mathbb{R}}$ by Theorem 3.5. For this, we
need again to recall the estimations in [27] as follows.

######  Lemma 3.11.

Let the assumptions (F1), (F2) and (F3) hold for $f$,
$\alpha_{0}\in\mathbb{R}$ with a neighborhood ${\mathcal{I}}$ in $\mathbb{R}$
and ${\rm e}^{\alpha z(\theta_{-t}\omega)}v_{\tau-t}\in
D_{\alpha}(\tau,\omega)$ with $D_{\alpha}\in{\mathcal{D}}_{\alpha}$ for each
$\alpha\in{\mathcal{I}}$. Then for each $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s.
$\omega\in\Omega$, $\alpha\in{\mathcal{I}}$ and $\varepsilon>0$, there exist
$T=T(\varepsilon,\tau,\omega,{\mathcal{I}},\\{D_{\alpha}\\}_{\alpha\in{\mathcal{I}}})>0$
and
$N=N(\varepsilon,\tau,\omega,{\mathcal{I}},\\{D_{\alpha}\\}_{\alpha\in{\mathcal{I}}})\in\mathbb{N}$
such that when $t>T$, it holds that

$\sum_{|i|>N}|v_{\alpha,i}(\tau,\tau-t,\theta_{-\tau}\omega,v_{\alpha,\tau-t})|^{2}<\varepsilon,$
(3.27) $\mbox{and}\hskip
11.38092pt\sum_{|i|>N}|v_{\alpha,i}(\tau,\tau-t,\theta_{-\tau}\omega,v_{\alpha,\tau-t})|^{q}<\varepsilon.$
(3.28)

###### Proof.

For the proofs of (3.27) and (3.28), we can refer to Lemmas 3.7 and 3.10 of
[27]. The only differences between the conclusions (3.27), (3.28) and [27,
Lemmas 3.7 and 3.10] lie in the dependence on the parameter $\alpha$: the
former depends on the neighborhood ${\mathcal{I}}$ of $\alpha_{0}$ uniformly,
while the latter depends on $\alpha$ pointwise.

Following the proofs of Lemmas 3.7 and 3.10 in [27] in the framework of this
article, one can see that every $\pm z(\theta_{s}\omega)$ (resp. $\pm
z(\theta_{\sigma}\omega)$, $\pm z(\theta_{\varsigma}\omega)$ etc.) should
appear with a coefficient $\alpha$. If we take

$\hat{\alpha}:=\sup_{\alpha\in{\mathcal{I}}}|\alpha|$

and let $\hat{\alpha}|z(\theta_{s}\omega)|$ (resp.
$\hat{\alpha}|z(\theta_{\sigma}\omega)|$,
$\hat{\alpha}|z(\theta_{\varsigma}\omega)|$ etc.) replace $\pm\alpha
z(\theta_{s}\omega)$ (resp. $\pm\alpha z(\theta_{\sigma}\omega)$, $\pm\alpha
z(\theta_{\varsigma}\omega)$ etc.), the procedures of the proofs for Lemmas
3.7 and 3.10 in [27] can work smoothly with a slightly modified usage of
(2.5). As a result, these procedures can deduce the uniform conclusions (3.27)
and (3.28). Since the argument is almost the same as Lemmas 3.7 and 3.10 in
[27] but rather tedious, we omit the details here. ∎

The property given in Lemma 3.11 can be called _local uniform pullback
${\mathcal{D}}_{\alpha}$-asymptotic nullity_ of
$\\{\varphi_{\alpha}\\}_{\alpha\in\mathbb{R}}$. The local uniform pullback
${\mathcal{D}}_{\alpha}$-asymptotic nullity of
$\\{\varphi_{\alpha}\\}_{\alpha\in\mathbb{R}}$ can similarly imply the local
uniform pullback ${\mathcal{D}}_{\alpha}$-asymptotic compactness, and hence we
have proved the condition (3′) given in Theorem 3.5.

Summarizing all the discussions above, we have actually proved the first main
theorem.

######  Theorem 3.12.

Let the assumptions (F1), (F2) and (F3) hold for $f$ and
${\mathcal{A}}_{\alpha}$ be the pullback random
$(\ell^{2},\ell^{q})$-attractor with respect to ${\mathcal{D}}_{\alpha}$ for
$\varphi_{\alpha}$ with $q\in[1,\infty)$. Then for every
$(\tau,\omega)\in\mathbb{R}\times\Omega$ and $\alpha_{0}\in\mathbb{R}$, it
holds that

$\lim_{\alpha\rightarrow\alpha_{0}}{\rm
dist}_{\ell^{2}\cap\ell^{q}}({\mathcal{A}}_{\alpha}(\tau,\omega),{\mathcal{A}}_{\alpha_{0}}(\tau,\omega))=0.$

##  4 Convergence of invariant sample measures

###  4.1 Preliminaries on invariant sample measures

We first give the relative definitions and theorems of invariant sample
measures (with two variables) for NRDS’s; see [29, 4].

Let $X$ be a Banach space with norm $\|\cdot\|$ and $\varphi$ be an NRDS over
$(\Omega,{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$. A
family of Borel probability measures
$\\{\mu_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ on $X$ is
called an _invariant sample measures_ for $\varphi$, if for each $t\geqslant
0$, $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$ and all
$U\in{\mathcal{B}}(X)$, it holds that

$\mu_{\tau+t,\theta_{t}\omega}(U)=\mu_{\tau,\omega}((\varphi(t,\tau,\omega,\cdot))^{-1}U).$
(4.1)

The construction of the invariant sample measures usually involves the
_generalized Banach limit_ , which is any linear functional, denoted by
$\displaystyle\operatorname*{LIM}_{t\rightarrow-\infty}$ (we only need to
consider the case $t\rightarrow-\infty$, see more in [32, 29, 35]), defined
for an arbitrary bounded real-valued function on $(-\infty,a]$ for some
$a\in\mathbb{R}$ and satisfying

  1. (1)

$\displaystyle\operatorname*{LIM}_{t\rightarrow-\infty}\zeta(t)\geqslant 0$
for nonnegative functions $\zeta$ on $(-\infty,a]$;

  2. (2)

$\displaystyle\operatorname*{LIM}_{t\rightarrow-\infty}\zeta(t)=\lim_{t\rightarrow-\infty}\zeta(t)$
if the latter limit exists.

Let $({\mathfrak{A}},\rho)$ be a metric space. For each
$\alpha\in{\mathfrak{A}}$, let $\varphi_{\alpha}$ be an NRDS over
$(\Omega,{\mathcal{F}},\mathbb{P},\\{\theta_{t}\\}_{t\in\mathbb{R}})$ on the
state space $X$ and ${\mathcal{D}}_{\alpha}$ be a universe for
$\varphi_{\alpha}$. We introduce [4, Theorem 4.1] to discuss the convergence
property of invariant sample measures for the family
$\\{\varphi_{\alpha}\\}_{\alpha\in{\mathfrak{A}}}$.

######  Theorem 4.1.

Suppose that for each $\alpha\in{\mathfrak{A}}$, $\varphi_{\alpha}$ has a
pullback random ${\mathcal{D}}_{\alpha}$-attractor
${\mathcal{A}}_{\alpha}(\tau,\omega)$ and a family of Borel probability
measures
$\\{\mu_{\tau,\omega}^{\alpha}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$
such that $\mu_{\tau,\omega}^{\alpha}$ is an invariant sample measures and is
supported on ${\mathcal{A}}_{\alpha}(\tau,\omega)$. Let
$\alpha_{0}\in{\mathfrak{A}}$. Suppose that

  1. (1)

the union
$\bigcup_{\alpha\in{\mathfrak{U}}}{\mathcal{A}}_{\alpha}(\tau,\omega)$ is
precompact in $X$ for $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$
and a certain neighborhood ${\mathfrak{U}}$ of $\alpha_{0}$ in
${\mathfrak{A}}$; and

  2. (2)

for each compact set $K\subseteq X$, $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s.
$\omega\in\Omega$ and each sequence $\\{\alpha_{n}\\}_{n=1}^{\infty}$ in
${\mathfrak{A}}$ with $\alpha_{n}\rightarrow\alpha_{0}$ as
$n\rightarrow\infty$, the following equality holds:

$\lim_{t\rightarrow+\infty}\sup_{u\in
K}\|\varphi_{\alpha_{n}}(t,\tau-t,\theta_{-t}\omega,u)-\varphi_{\alpha_{0}}(t,\tau-t,\theta_{-t}\omega,u)\|=0.$
(4.2)

Then for each $\tau\in\mathbb{R}$ and $\mathbb{P}$-a.s. $\omega\in\Omega$,
there exists a subsequence $\\{\alpha_{n_{k}}\\}_{k=1}^{+\infty}$ of
$\\{\alpha_{n}\\}_{n=1}^{+\infty}$ depending on $\tau$ and $\omega$, and an
invariant sample measure $\mu_{\tau,\omega}^{\alpha_{0}}$ of
$\varphi_{\alpha_{0}}$ such that $\mu_{\tau,\omega}^{\alpha_{n_{k}}}$
converges weakly to $\mu_{\tau,\omega}^{\alpha_{0}}$ as $k\rightarrow+\infty$.

###  4.2 Convergence of invariant sample measures

First we recall the existence theorem of invariant sample measures for NRDS’s
in [27]. To this end, we need another assumption on $f$:

(F4) The nonlinear forcing $f$ satisfies (3.9) with

$\psi_{3},\,\psi_{4}\in L_{\rm loc}^{2}(\mathbb{R},\ell^{2}).$ (4.3)

######  Theorem 4.2.

Suppose the assumption (F1), (F2) and (F4) hold for $f$. Let
${\mathcal{A}}_{\alpha}$ be the unique $(\ell^{2},\ell^{2})$-pullback random
${\mathcal{D}}_{\alpha}$-attractor of $\varphi_{\alpha}$ given in Theorem 3.9.
Then for a given generalized Banach limit
$\displaystyle\operatorname*{LIM}_{t\rightarrow-\infty}$ and a mapping
$\xi:\mathbb{R}\times\Omega\rightarrow\ell^{2}$ with
$\\{\xi(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}\in{\mathcal{D}}_{\alpha}$
such that $\xi(\tau,\theta_{\tau}\omega)$ is continuous in $\tau$, there
exists a family of Borel probability measures
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ on
$\ell^{2}$ such that the support of the measure $\mu^{\alpha}_{\tau,\omega}$
is contained in ${\mathcal{A}}_{\alpha}(\tau,\omega)$ and for all
$\Upsilon\in{\mathcal{C}}(\ell^{2})$,

$\displaystyle\operatorname*{LIM}_{\tau\rightarrow-\infty}\frac{1}{t-\tau}\int_{\tau}^{t}\Upsilon(\varphi_{\alpha}(t-s,s,\theta_{s}\omega,\xi(s,\theta_{s}\omega))){\rm
d}s=\int_{{\mathcal{A}}_{\alpha}(t,\theta_{t}\omega)}\Upsilon(u){\rm
d}\mu^{\alpha}_{t,\theta_{t}\omega}(u)$ $\displaystyle=$
$\displaystyle\int_{\ell^{2}}\Upsilon(u){\rm
d}\mu^{\alpha}_{t,\theta_{t}\omega}(u)=\operatorname*{LIM}_{\tau\rightarrow-\infty}\frac{1}{t-\tau}\int_{\tau}^{t}\int_{\ell^{2}}\Upsilon(\varphi_{\alpha}(t-s,s,\theta_{s}\omega,u)){\rm
d}\mu^{\alpha}_{s,\theta_{s}\omega}(u){\rm d}s.$ (4.4)

Additionally, $\mu^{\alpha}_{t,\omega}$ is invariant in the sense that for all
$\tau\in\mathbb{R}$ and $\mathbb{P}$-a.s. $\omega\in\Omega$,

$\int_{{\mathcal{A}}_{\alpha}(\tau+t,\theta_{t}\omega)}\Upsilon(u){\rm
d}\mu^{\alpha}_{\tau+t,\theta_{t}\omega}(u)=\int_{{\mathcal{A}}_{\alpha}(\tau,\omega)}\Upsilon(\varphi_{\alpha}(t,\tau,\omega,u)){\rm
d}\mu^{\alpha}_{\tau,\omega}(u),\hskip 11.38092pt\mbox{for all }t\geqslant 0.$
(4.5)

By using the definition (2.10) for $\varphi_{\alpha}$, one can translate (4.5)
into the version for the solution $u_{\alpha}$ of (2.4) in the following form,

$\int_{{\mathcal{A}}_{\alpha}(\tau+t,\theta_{t}\omega)}\Upsilon(u){\rm
d}\mu^{\alpha}_{\tau+t,\theta_{t}\omega}(u)=\int_{{\mathcal{A}}_{\alpha}(\tau,\omega)}\Upsilon(u_{\alpha}(t+\tau,\tau,\theta_{-\tau}\omega,u)){\rm
d}\mu^{\alpha}_{\tau,\omega}(u),\hskip 11.38092pt\mbox{for all }t\geqslant 0.$
(4.6)

We aim to apply Theorem 4.1 to prove the convergence property of the invariant
sample measures given by Theorem 4.2. By the discussion above, we actually
only need to check the condition (2) in Theorem 4.1. For this purpose, we
recall the continuity given by Lemma 3.8. Combining the definition (2.10), we
only need to ensure the following conclusion.

######  Lemma 4.3.

Let the assumptions (F1), (F2) and (F3) hold for $f$. Then for each compact
set $K\subset\ell^{2}$, $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s.
$\omega\in\Omega$ and $\alpha_{n}\in{\mathfrak{A}}$ with
$\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$, the following
equality holds true:

$\lim_{n\rightarrow+\infty}\sup_{u\in
K}\|u_{\alpha_{n}}(t,\tau-t,\theta_{-\tau}\omega,u)-u_{\alpha_{0}}(t,\tau-t,\theta_{-\tau}\omega,u)\|=0,$
(4.7)

for all $t\geqslant 0$.

###### Proof.

We first fix $u\in\ell^{2}$, $\tau\in\mathbb{R}$, $t\geqslant 0$ and
$\mathbb{P}$-a.s. $\omega\in\Omega$. By Lemma 3.8, we know for each
$\varepsilon>0$, there exists an open neighborhood ${\mathcal{O}}_{u}$ of $u$
in $\ell^{2}$ and $N_{u}\in\mathbb{N}^{+}$ such that for all
$\phi\in{\mathcal{O}}_{u}$ and $n>N_{u}$,

$\|u_{\alpha_{n}}(t,\tau-t,\theta_{-\tau}\omega,\phi)-u_{\alpha_{0}}(t,\tau-t,\theta_{-\tau}\omega,\phi)\|<\varepsilon.$
(4.8)

For the compact set $K$ in $\ell^{2}$, the family
$\\{{\mathcal{O}}_{u}\\}_{u\in K}$ constitutes an open covering of $K$ and
hence has a finite sub-covering $\\{{\mathcal{O}}_{u^{(i)}}\\}_{i=1}^{k}$ with
$u^{(i)}\in K$ for each $i\in[1,k]$ such that

$K\subset\bigcup_{i=1}^{k}{\mathcal{O}}_{u^{(i)}}.$

Let

$N_{K}=\max\\{N_{u^{(1)}},\,N_{u^{(2)}},\,\cdots,\,N_{u^{(k)}}\\}.$

Then one can easily see that when $n>N_{K}$, (4.8) is valid for each $\phi\in
K$, which obviously indicates (4.7). The proof is complete. ∎

Now we observe that the condition (1) in Theorem 4.1 follows from Lemma 3.11
and Theorem 3.5. Furthermore, Lemma 4.3 and (2.10) immediately guarantee the
condition (2) in Theorem 4.1. Consequently, we can thereby obtain the main
theorem in this section.

######  Theorem 4.4.

Suppose the assumptions (F1), (F2), (F3) and (F4) hold for $f$. Let
$\varphi_{\alpha}$ be the NRDS defined as 2.10 generated by (2.4) and
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be
the invariant sample measures of $\varphi_{\alpha}$. Then for each
$\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$ and each sequence
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$, there exists a
subsequence $\\{\alpha_{n_{k}}\\}_{k\in\mathbb{N}^{+}}$ of
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ depending on $\tau$ and $\omega$, and
an invariant sample measure $\mu_{\tau,\omega}^{\alpha_{0}}$ of
$\varphi_{\alpha_{0}}$ such that $\mu_{\tau,\omega}^{\alpha_{n_{k}}}$ (or
$\\{\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{n_{k}}}\\}_{t\in\mathbb{R}}$)
converges weakly to $\mu_{\tau,\omega}^{\alpha_{0}}$ (or
$\\{\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{0}}\\}_{t\in\mathbb{R}}$) as
$k\rightarrow+\infty$ in the sense that

$\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{n_{k}}}\hskip
11.38092pt\mbox{converges weakly to}\hskip
11.38092pt\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{0}}\hskip 11.38092pt\mbox{for
every }t\in\mathbb{R}.$

##  5 Convergence of stochastic Liouville type equation

###  5.1 Stochastic Liouville type equation

In this subsection, we first check that the invariant sample measures
$\\{\mu_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ obtained in
Theorem 4.2 for $\varphi$ fulfill a stochastic Liouville type theorem (see [4,
33]).

We adopt the stochastic Liouville type equation (see [4, 33]) with respect to
the invariant sample measures
$\\{\mu_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$. Let

$V=\ell^{2}\cap\ell^{q},\hskip 11.38092ptH=\ell^{2}\hskip
11.38092pt\mbox{and}\hskip 11.38092ptV^{*}=\ell^{2}+\ell^{q^{\prime}},$

where $q^{\prime}$ is the dual index of $q$ with $q^{\prime}=\frac{q}{q-1}$
for $q=1$ and $q^{\prime}=\infty$ for $q=1$. We set the norm of $V$ as
$\|u\|_{V}=\|u\|+\|u\|_{q}$. Denote the dual pairing between $v^{*}\in V^{*}$
and $v\in V$ by $\langle v^{*},v\rangle$. Let

$\tilde{F}(t,u)=-\nu(t)Au+f(t,u)-\lambda u.$ (5.1)

We know $\tilde{F}:\mathbb{R}\times V\rightarrow V^{*}$. Thus the equation
(2.4) can be written as

${\rm d}u=\tilde{F}(t,u){\rm d}t+\alpha u\circ{\rm d}W.$ (5.2)

Let ${\mathcal{T}}$ denote the class of test functions
$\Psi:H\rightarrow\mathbb{R}$ for (5.2) to be bounded on bounded subsets of
$H$ such that the following conditions hold:

  1. (1)

for each $u\in V$, the Fréchet derivative $\Psi^{\prime}(u)$ taken in $H$
along $V$ exists. More precisely, for each $u\in V$, there exists an element
in $H$ denoted by $\Psi^{\prime}(u)$ such that

$\frac{|\Psi(u+v)-\Psi(v)-(\Psi^{\prime}(u),v)|}{\|v\|}\rightarrow 0\hskip
11.38092pt\mbox{as }\|v\|\rightarrow 0,\;v\in V;$

  2. (2)

for each $u\in V$, $\Psi^{\prime}(u)\in V$ and the mapping
$u\mapsto\Psi^{\prime}(u)$ is continuous and bounded as a functional from $V$
into $V$;

  3. (3)

for each $u\in V$, the second-order Fréchet derivative
$\Psi^{\prime\prime}(u)$ is a bounded bilinear operator from $H\times H$ to
$\mathbb{R}$ and the mapping $u\mapsto\Psi^{\prime\prime}(u)$ is continuous
and bounded from $V$ into $L(H\times H,\mathbb{R})$;

  4. (4)

for every global solution $u(t)$ of (5.2), the following Itô’s Formula holds
for all $t\geqslant s$ and $\mathbb{P}$-a.s. $\omega\in\Omega$,

$\displaystyle\Psi(u(t))-\Psi(u(s))=$
$\displaystyle\int_{s}^{t}\langle\tilde{F}(\varsigma,u(\varsigma)),\Psi^{\prime}(u(\varsigma))\rangle{\rm
d}\varsigma+\alpha\int_{s}^{t}\left(u(\varsigma),\Psi^{\prime}(u(\varsigma))\right){\rm
d}W(\varsigma)$
$\displaystyle+\frac{\alpha^{2}}{2}\int_{s}^{t}\Psi^{\prime\prime}(u(\varsigma))\left(u(\varsigma),u(\varsigma)\right){\rm
d}\varsigma.$ (5.3)

The class ${\mathcal{T}}$ is not empty for (5.2), and one can see examples in
[4, 33].

For the following discussion, we set one more assumption for $f(t,u)$ as
follows,

(F5) there is $\psi_{5}(t)=(\psi_{5i}(t))_{i\in\mathbb{Z}}$, such that

$\left|\frac{\partial f_{i}}{\partial
u_{i}}(t,u_{i})\right|\leqslant\psi_{5i}(t)\mbox{ with }\psi_{5}\in L_{\rm
loc}^{\infty}(\mathbb{R};\ell^{\infty}).$ (5.4)

To prove the final conclusion, we need the following results.

######  Lemma 5.1.

Suppose that the assumptions (F1), (F2) and (F3) hold for $f$. Let
${\mathcal{A}}_{\alpha}$ be the pullback random
$(\ell^{2},\ell^{q})$-attractor of $\varphi_{\alpha}$ with respect to
${\mathcal{D}}_{\alpha}$ given in Theorem 3.6. Then for each
$\alpha\in\mathbb{R}$, $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s.
$\omega\in\Omega$, and $s,t\in\mathbb{R}$ with $s\leqslant t$, the set

${\mathcal{U}}_{\alpha}[s,t]:=\bigcup_{\sigma\in[s,t]}{\mathcal{A}}_{\alpha}(\sigma,\theta_{\sigma-\tau}\omega)$
(5.5)

is compact in $\ell^{2}$.

###### Proof.

We pick a sequence $\\{\phi_{k}\\}_{k\in\mathbb{N}^{+}}$ in
${\mathcal{U}}_{\alpha}[s,t]$ arbitrarily with
$\phi_{k}\in{\mathcal{A}}_{\alpha}(\sigma_{k},\theta_{\sigma_{k}-\tau}\omega)$,
and it suffices to show that this sequence has a convergent subsequence with
its limit in ${\mathcal{U}}_{\alpha}[s,t]$.

Since $\sigma_{k}\in[s,t]$, we can assume that
$\sigma_{k}\rightarrow\sigma_{0}\in[s,t]$ as $k\rightarrow\infty$ (selecting
its subsequence if necessary). Take

$\sigma_{*}=\inf_{k\in\mathbb{N}}\sigma_{k}\in[s,t].$

Then by the invariance of the attractor, we have
$\tilde{\phi}_{k}\in{\mathcal{A}}_{\alpha}(\sigma_{*},\theta_{\sigma_{*}-\tau}\omega)$
for each $n\in\mathbb{N}$ such that

$\phi_{k}=u_{\alpha}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k}).$

Moreover, since
$\tilde{\phi}_{k}\in{\mathcal{A}}_{\alpha}(\sigma_{*},\theta_{\sigma_{*}-\tau}\omega)$,
which is compact in $\ell^{2}$, we can furthermore assume that

$\tilde{\phi}_{k}\rightarrow\phi_{*}\in{\mathcal{A}}_{\alpha}(\sigma_{*},\theta_{\sigma_{*}-\tau}\omega),\hskip
11.38092pt\mbox{as }n\rightarrow\infty.$

Recalling the condition (4) of a continuous NRDS in Section 2 (or [27, Theorem
2.5]), we can infer from $\sigma_{k}\rightarrow\sigma_{0}$ and
$\tilde{\phi}_{k}\rightarrow\phi_{*}$ that

$\|\phi_{k}-u_{\alpha}(\sigma_{0},\sigma_{*},\theta_{-\tau}\omega,\phi_{*})\|\leqslant\|u_{\alpha}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k})-u_{\alpha}(\sigma_{0},\sigma_{*},\theta_{-\tau}\omega,\phi_{*})\|\rightarrow
0,$

as $n\rightarrow\infty$. The compactness is thus proved. ∎

Let $U\subset H$. We say that a mapping $U\ni u\mapsto\tilde{\Psi}(u)\in X$
for a Banach space $X$ is continuous in (norm of) $H$, if for a sequence
$\\{u^{(n)}\\}_{n\in\mathbb{N}^{+}}$ and $u$ in $U$,

$\|\tilde{\Psi}(u^{(n)})-\tilde{\Psi}(u)\|_{X}\rightarrow 0,\hskip
11.38092pt\mbox{as }\|u^{(n)}-u\|\rightarrow 0.$ (5.6)

######  Lemma 5.2.

Suppose that the assumptions (F1), (F2), (F3), (F4) and (F5) hold for $f$. Let
$\Psi\in{\mathcal{T}}$ and $u\in{\mathcal{A}}_{\alpha}(\tau,\omega)$, where
$\alpha\in\mathbb{R}$ and
${\mathcal{A}}_{\alpha}=\\{{\mathcal{A}}_{\alpha}(\tau,\omega)\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$
is the pullback random $(\ell^{2},\ell^{q})$-attractor given by Theorem 3.6.
Then the mappings

${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)\ni
u\mapsto\langle\tilde{F}(t,u),\Psi^{\prime}(u)\rangle$ (5.7)
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)\ni
u\mapsto\Psi^{\prime\prime}(u)(u,u)\hskip 11.38092pt\mbox{and}\hskip
11.38092pt{\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)\ni
u\mapsto(u,\Psi^{\prime}(u))$ (5.8)

are continuous in (norm of) $H$. Moreover, for the invariant sample measures
$\\{\mu_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ given in
Theorem 4.2, the mappings

$\varsigma\mapsto\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\hskip 11.38092pt\mbox{and}\hskip
11.38092pt\varsigma\mapsto\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (5.9)

belong to $L^{1}_{\rm loc}(\mathbb{R})$; and the mapping

$\varsigma\mapsto\int_{H}(u,\Psi^{\prime}(u))\mu_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (5.10)

belongs to $L^{2}_{\rm loc}(\mathbb{R})$.

###### Proof.

We first prove the results for
$\langle\tilde{F}(t,u),\Psi^{\prime}(u)\rangle$. To this end, we first
consider the continuity of the mapping

$V\ni u\mapsto\tilde{F}(t,u)\in V^{*}$ (5.11)

in norm of $H$ for each $t\in\mathbb{R}$. Take $u$, $v$, $w\in V$ with
$\|u\|$, $\|v\|\leqslant R$ for some $R>0$ and set $\tilde{w}=u-v$. For each
fixed $t\in\mathbb{R}$, we observe by (5.4) and the Differential Mean-Value
Theorem that

$\displaystyle\left|\langle\tilde{F}(t,u)-\tilde{F}(t,v),w\rangle\right|$
$\displaystyle\leqslant$
$\displaystyle\nu_{0}\left|\langle|Bu|^{p-2}\otimes(Bu)-|Bv|^{p-2}\otimes(Bv),Bw\rangle\right|+\lambda\|\tilde{w}\|\|w\|+\sum_{i\in\mathbb{Z}}|f_{i}(t,u_{i})-f_{i}(t,v_{i})||w_{i}|$
$\displaystyle\leqslant$ $\displaystyle
2(p-1)(\|Bu\|+\|Bv\|)^{p-2}\|B\tilde{w}\|\|w\|+\lambda\|\tilde{w}\|\|w\|+\left(\sum_{i\in\mathbb{Z}}\left|\frac{\partial
f_{i}}{\partial
u_{i}}(t,\xi_{i})\right|^{2}|\tilde{w}_{i}|^{2}\right)^{\frac{1}{2}}\|w\|$
$\displaystyle\leqslant$
$\displaystyle\left[(p-1)4^{p-1}R^{p-2}+\lambda+\|\psi_{5}(t)\|_{\infty}\right]\|\tilde{w}\|\|w\|,$

where we also used the trivial estimate (14) of [27]. Then we know that

$\displaystyle\|\tilde{F}(t,u)-\tilde{F}(t,v)\|_{V^{*}}=$
$\displaystyle\sup_{\|w\|_{V}\neq
0}\frac{\left|\langle\tilde{F}(t,u)-\tilde{F}(t,v),w\rangle\right|}{\|w\|_{V}}$
$\displaystyle\leqslant$ $\displaystyle\sup_{\|w\|_{V}\neq
0}\frac{\left|\langle\tilde{F}(t,u)-\tilde{F}(t,v),w\rangle\right|}{\|w\|}$
$\displaystyle\leqslant$
$\displaystyle\left[(p-1)4^{p-1}R^{p-2}+\lambda+\|\psi_{5}(t)\|_{\infty}\right]\|\tilde{w}\|,$

which indicates the continuity (5.11) in $H$ (also $V$ of course) immediately.

Then we consider the continuity of the mapping $u\mapsto\Psi^{\prime}(u)$ from
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ to $V$ in norm of $H$. Let
$\\{u^{(n)}\\}_{n\in\mathbb{N}}$ be a sequence in
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ and
$\|u^{(n)}-u\|\rightarrow 0$ with $u\in H$ as $n\rightarrow\infty$. Noting
that ${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ is compact in the
spaces $\ell^{2}$ and $\ell^{q}$, we know that
$u\in{\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$. And each subsequence
$u^{(n_{k})}$ of $u^{(n)}$ has a new subsequence converging to $u^{*}$ in
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ in norm (or topology) of
$\ell^{q}$. Since $(\ell^{2},\ell^{q})$ is limit-identical, we know $u^{*}=u$.
It is easy to see that $u^{(n)}\rightarrow u$ in norm of $\ell^{q}$ and hence
of $V$. As a result, the condition (2) for the class ${\mathcal{T}}$ implies
that

$\|\Psi^{\prime}(u^{(n)})-\Psi^{\prime}(u)\|_{V}\rightarrow 0,$

and therefore the continuity of $u\mapsto\Psi^{\prime}(u)$ from
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ to $V$ (also $H$ obviously)
in norm of $H$. Finally the continuity of
$u\mapsto\langle\tilde{F}(t,u),\Psi^{\prime}(u)\rangle$ follows immediately.

Then we show the mapping

$\varsigma\mapsto\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$

belongs to $L^{1}_{\rm loc}(\mathbb{R})$. Due to again the fact that for each
$(\tau,\omega)\in\mathbb{R}\times\Omega$,
${\mathcal{A}}_{\alpha}(\tau,\omega)$ is compact in $H$ and that the invariant
sample measures $\mu_{\tau,\omega}$ is supported by
${\mathcal{A}}_{\alpha}(\tau,\omega)$, the mapping (5.9) is surely well
defined. Note also that

$\displaystyle\left|\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\right|\leqslant$
$\displaystyle\nu_{0}\left|\langle
Au,\Psi^{\prime}(u)\rangle\right|+\lambda\left|\left(u,\Psi^{\prime}(u)\right)\right|+\left|\langle
f(\varsigma,u),\Psi^{\prime}(u)\rangle\right|$ $\displaystyle\leqslant$
$\displaystyle
2^{p}\nu_{0}\|u\|^{p-1}\|\Psi^{\prime}(u)\|+\lambda\|u\|\|\Psi^{\prime}(u)\|$
$\displaystyle+\sum_{i\in\mathbb{Z}}\left(\psi_{3i}(\varsigma)|u_{i}|^{q-1}+\psi_{4i}(\varsigma)\right)|\Psi^{\prime}_{i}(u)|$
$\displaystyle\leqslant$
$\displaystyle\left(2^{p}\nu_{0}\|u\|^{p-1}+\lambda\|u\|+\|\psi_{4}(\varsigma)\|_{1}\right)\|\Psi^{\prime}(u)\|_{V}$
$\displaystyle+\sum_{i\in\mathbb{Z}}\psi_{3i}(\varsigma)\left(\frac{q-1}{q}|u_{i}|^{q}+\frac{1}{q}|\Psi^{\prime}(u)|^{q}\right)$
$\displaystyle\leqslant$
$\displaystyle\left(2^{p}\nu_{0}\|u\|^{p-1}+\lambda\|u\|+\|\psi_{4}(\varsigma)\|_{1}\right)\|\Psi^{\prime}(u)\|_{V}$
$\displaystyle+\frac{q-1}{q}\|\psi_{3}(\varsigma)\|_{1}\|u\|^{q}+\frac{1}{q}\|\psi_{3}(\varsigma)\|_{1}\|\Psi^{\prime}(u)\|_{V}^{q}.$
(5.12)

Now we consider an arbitrary interval $[s,t]\subset\mathbb{R}$. By (2) of the
definition of ${\mathcal{T}}$, we know $\|\Psi^{\prime}(u)\|_{V}$ is bounded;
by the compactness of ${\mathcal{U}}_{\alpha}[s,t]$, we also know that $\|u\|$
is bounded on ${\mathcal{U}}_{\alpha}[s,t]$. Hence we have a positive constant
${\mathscr{M}}_{\alpha}$ independent of $\varsigma$ such that

$\left|\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\right|\leqslant{\mathscr{M}}_{\alpha}(1+\|\psi_{3}(\varsigma)\|_{1}+\|\psi_{4}(\varsigma)\|_{1})$

and then

$\displaystyle\left|\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha}_{\varsigma,\theta_{\varsigma-t}\omega}({\rm
d}u)\right|=\left|\int_{{\mathcal{A}}_{\alpha_{n}}(\varsigma,\theta_{\varsigma-t}\omega)}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha}_{\varsigma,\theta_{\varsigma-t}\omega}({\rm
d}u)\right|$ $\displaystyle\leqslant$
$\displaystyle{\mathscr{M}}_{\alpha}(1+\|\psi_{3}(\varsigma)\|_{1}+\|\psi_{4}(\varsigma)\|_{1}),$
(5.13)

which ensures that (5.9) belongs to $L^{1}_{\rm loc}(\mathbb{R})$ by the
assumption (F3).

The continuity of the remaining two mappings in (5.8) is obvious, as long as
we notice that $\Phi^{\prime}(u)$ is continuous from
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ to $H$ and so is
$\Phi^{\prime\prime}(u)$ from
${\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)$ to $L(H\times H,\mathbb{R})$
by the definition of the class ${\mathcal{T}}$.

Next, we recall the definition of ${\mathcal{T}}$ that $\Psi^{\prime}(u)$ is
bounded from $V$ to $V$ and $\Psi^{\prime\prime}(u)$ is bounded from $V$ to
$L(H\times H,\mathbb{R})$. By the embedding $V\subset H$, we also know that
$\Psi^{\prime}(u)$ is bounded in $H$. Combining the boundedness of $\|u\|$ on
the compact set ${\mathcal{U}}_{\alpha}[s,t]$, one can directly see that
$|(u,\Psi^{\prime}(u))|$ and $|\Psi^{\prime\prime}(u)(u,u)|$ are uniformly
bounded on ${\mathcal{U}}_{\alpha}[s,t]$. Then similar to the analysis in
(5.13), we can obtain that the second mapping in (5.9) belongs to $L^{1}_{\rm
loc}(\mathbb{R})$ and the mapping (5.10) belongs to $L^{2}_{\rm
loc}(\mathbb{R})$. The proof is hence accomplished.∎

######  Remark 5.3.

Here the requirement that the integrand of the stochastic integral to belong
to $L^{2}_{\rm loc}(\mathbb{R})$ is to guarantee that the stochastic integral
makes sense; see [14, Section 2.3].

Now we are well prepared to show the stochastic Liouville type theorem.

######  Theorem 5.4.

Suppose that the assumptions (F1), (F2), (F3), (F4) and (F5) hold for $f$. Let
${\mathcal{A}}_{\alpha}$ be the pullback random
$(\ell^{2},\ell^{q})$-attractor of $\varphi_{\alpha}$ with respect to
${\mathcal{D}}_{\alpha}$ given in Theorem 3.6 and
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be
the invariant sample measures constructed in Theorem 4.2. Then the invariant
sample measures
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$
satisfy the stochastic Liouville type equation (1.4), for $s,t\in\mathbb{R}$
with $s\leqslant t$.

###### Proof.

Based on the definition of the class ${\mathcal{T}}$, and by (4.4), (4.6) and
(5.3), we know that for all $t\geqslant\tau$,

$\displaystyle\int_{H}\Psi(\phi)\mu^{\alpha}_{t,\theta_{t-\tau}\omega}({\rm
d}\phi)-\int_{H}\Psi(\phi)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}\phi)$
$\displaystyle=$
$\displaystyle\int_{{\mathcal{A}}_{\alpha}(t,\theta_{t-\tau}\omega)}\Psi(\phi)\mu^{\alpha}_{t,\theta_{t-\tau}\omega}({\rm
d}\phi)-\int_{{\mathcal{A}}_{\alpha}(s,\theta_{s-\tau}\omega)}\Psi(\phi)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}\phi)$ $\displaystyle=$
$\displaystyle\int_{{\mathcal{A}}_{\alpha}(s,\theta_{s-\tau}\omega)}\left(\Psi(u_{\alpha}(t,s,\theta_{-\tau}\omega,\phi))-\Psi(\phi)\right)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}\phi)$ $\displaystyle=$
$\displaystyle\int_{{\mathcal{A}}_{\alpha}(s,\theta_{s-\tau}\omega)}\int_{s}^{t}\langle\tilde{F}(\varsigma,u_{\alpha}(\varsigma)),\Psi^{\prime}(u_{\alpha}(\varsigma))\rangle{\rm
d}\varsigma\cdot\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}\phi)$
$\displaystyle+\alpha\int_{{\mathcal{A}}_{\alpha}(s,\theta_{s-\tau}\omega)}\int_{s}^{t}\left(u_{\alpha}(\varsigma),\Psi^{\prime}(u_{\alpha}(\varsigma))\right){\rm
d}\tilde{W}(\varsigma)\cdot\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}\phi)$
$\displaystyle+\frac{\alpha^{2}}{2}\int_{{\mathcal{A}}_{\alpha}(s,\theta_{s-\tau}\omega)}\int_{s}^{t}\Psi^{\prime\prime}(u_{\alpha}(\varsigma))\left(u_{\alpha}(\varsigma),u_{\alpha}(\varsigma)\right){\rm
d}\varsigma\cdot\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}\phi)$
$\displaystyle:=$ $\displaystyle M_{1}+M_{2}+M_{3},$ (5.14)

where $\tilde{W}(\varsigma)=W(-\tau+\varsigma)-W(-\tau)$ comes from the
stochastic datum $\theta_{-\tau}\omega$. Note by (3) of the definition of
NRDS’s and (2.10) that

$u_{\alpha}(\varsigma,s,\theta_{-\tau}\omega,u_{\alpha}(s,\sigma,\theta_{-\tau}\omega,\phi))=u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi).$
(5.15)

Then using (4.4), (5.15), Lemma 5.2, Fubini’s Theorem and the invariance of
$\mu^{\alpha}_{\tau,\omega}$ to $M_{1}$, we have

$\displaystyle M_{1}=$
$\displaystyle\operatorname*{LIM}_{\tau^{\prime}\rightarrow-\infty}\frac{1}{s-\tau^{\prime}}\int_{\tau^{\prime}}^{s}\int_{H}\int_{s}^{t}\langle\tilde{F}(\varsigma,u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi)),\Psi^{\prime}(u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi))\rangle{\rm
d}\varsigma\cdot\mu^{\alpha}_{\sigma,\theta_{\sigma-\tau}\omega}({\rm
d}\phi){\rm d}\sigma$ $\displaystyle=$
$\displaystyle\operatorname*{LIM}_{\tau^{\prime}\rightarrow-\infty}\frac{1}{s-\tau^{\prime}}\int_{\tau^{\prime}}^{s}\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi)),\Psi^{\prime}(u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi))\rangle\mu^{\alpha}_{\sigma,\theta_{\sigma-\tau}\omega}({\rm
d}\phi){\rm d}\varsigma{\rm d}\sigma$ $\displaystyle=$
$\displaystyle\operatorname*{LIM}_{\tau^{\prime}\rightarrow-\infty}\frac{1}{s-\tau^{\prime}}\int_{\tau^{\prime}}^{s}\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma{\rm d}\sigma$ $\displaystyle=$
$\displaystyle\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma,$ (5.16)

where we have replaced $t$, $\tau$ and $\omega$ in (4.6) by
$\varsigma-\sigma$, $\sigma$ and $\theta_{\sigma-\tau}\omega$, respectively,
to obtain the third equality. Similarly, for $M_{2}$ and $M_{3}$, by Lemma
5.2, we also have

$\displaystyle M_{2}=$
$\displaystyle\alpha\operatorname*{LIM}_{\tau^{\prime}\rightarrow-\infty}\frac{1}{s-\tau^{\prime}}\\!\\!\\!\int_{\tau^{\prime}}^{s}\\!\\!\\!\int_{H}\\!\\!\int_{s}^{t}\left(u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi),\Psi^{\prime}(u_{\alpha}(\varsigma,\sigma,\theta_{-\tau}\omega,\phi))\right){\rm
d}\tilde{W}(\varsigma)\cdot\mu^{\alpha}_{\sigma,\theta_{\sigma-\tau}\omega}({\rm
d}\phi){\rm d}\sigma$ $\displaystyle=$
$\displaystyle\alpha\int_{s}^{t}\int_{H}\left(u,\Psi^{\prime}(u)\right)\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\tilde{W}(\varsigma)\hskip 11.38092pt\mbox{and}$ (5.17)
$M_{3}=\frac{\alpha^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)\left(u,u\right)\mu^{\alpha}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma.$ (5.18)

Then the stochastic Liouville type equation (1.4) follows from the equalities
from (5.14) to (5.18). The proof is thus finished. ∎

###  5.2 Convergence of stochastic Liouville type equation

Since the stochastic Liouville type equation relies on the coefficient
$\alpha$ of the stochastic term, it is also an interesting problem to study
what will happen when a sequence
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}\subset\mathbb{R}$ tends to some number
$\alpha_{0}$ in $\mathbb{R}$. Actually, in Theorem 4.4, we have obtained the
weak convergence of a subsequence of
$\\{\mu^{\alpha_{n}}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\Omega}$ for
a sequence ${\alpha_{n}}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$ as $n\rightarrow\infty$.
Following this result, one can further deduce the following theorem.

######  Theorem 5.5.

Suppose that the assumptions (F1), (F2), (F3), (F4) and (F5) hold for $f$. Let
${\mathcal{A}}_{\alpha}$ be the pullback random
$(\ell^{2},\ell^{q})$-attractor of $\varphi_{\alpha}$ with respect to
${\mathcal{D}}_{\alpha}$ given in Theorem 3.6 and
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be
the invariant sample measures constructed in Theorem 4.2. Then for each
$\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$, each convergent
sequence $\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$, there exists a subsequence
$\\{\alpha_{n_{k}}\\}_{k\in\mathbb{N}^{+}}$ of
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ depending on $\tau$ and $\omega$, and
an invariant sample measure $\mu_{\tau,\omega}^{\alpha_{0}}$ of
$\varphi_{\alpha_{0}}$ such that for each $\Psi\in{\mathcal{T}}$ and
$s,t\in\mathbb{R}$ with $s\leqslant t$, the stochastic Liouville type equation

$\displaystyle\int_{H}\Psi(u)\mu^{\alpha_{n_{k}}}_{t,\theta_{t-\tau}\omega}({\rm
d}u)-\int_{H}\Psi(u)\mu^{\alpha_{n_{k}}}_{s,\theta_{s-\tau}\omega}({\rm d}u)$
$\displaystyle=$
$\displaystyle\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\varsigma+\alpha_{n_{k}}\int_{s}^{t}\int_{H}(u,\Psi^{\prime}(u))\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\tilde{W}(\varsigma)$
$\displaystyle+\frac{\alpha_{n_{k}}^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma$ (5.19)

converges termwise to the following stochastic Liouville type equation

$\displaystyle\int_{H}\Psi(u)\mu^{\alpha_{0}}_{t,\theta_{t-\tau}\omega}({\rm
d}u)-\int_{H}\Psi(u)\mu^{\alpha_{0}}_{s,\theta_{s-\tau}\omega}({\rm d}u)$
$\displaystyle=$
$\displaystyle\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\varsigma+\alpha_{0}\int_{s}^{t}\int_{H}(u,\Psi^{\prime}(u))\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\tilde{W}(\varsigma)$
$\displaystyle+\frac{\alpha_{0}^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma.$ (5.20)

To prove Theorem 5.5, we still need some auxiliary results.

######  Lemma 5.6.

Under the conditions of Theorem 5.5, for each $\tau\in\mathbb{R}$,
$\mathbb{P}$-a.s. $\omega\in\Omega$, $s,t\in\mathbb{R}$ with $s\leqslant t$
and a sequence $\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$ as $n\rightarrow\infty$, the
set

${\mathcal{U}}[s,t]:=\bigcup_{n=0}^{\infty}{\mathcal{U}}_{\alpha_{n}}[s,t]$

is precompact in $\ell^{2}$, where ${\mathcal{U}}_{\alpha}[s,t]$ is defined in
(5.5).

###### Proof.

We pick a sequence $\\{\phi_{k}\\}_{k\in\mathbb{N}^{+}}$ in
${\mathcal{U}}[s,t]$ arbitrarily with
$\phi_{k}\in{\mathcal{A}}_{\alpha_{n_{k}}}(\sigma_{k},\theta_{\sigma_{k}-\tau}\omega)$,
and it suffices to show that this sequence has a convergent subsequence with
its limit in $\ell^{2}$.

Similar to the proof of Lemma 5.1, we assume that
$\sigma_{k}\rightarrow\sigma_{0}\in[s,t]$ as $k\rightarrow\infty$ and take

$\sigma_{*}=\inf_{k\in\mathbb{N}}\sigma_{k}\in[s,t].$

Then we have
$\tilde{\phi}_{k}\in{\mathcal{A}}_{\alpha_{n_{k}}}(\sigma_{*},\theta_{\sigma_{*}-\tau}\omega)$
for each $k\in\mathbb{N}$ such that

$\phi_{k}=u_{\alpha_{n_{k}}}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k}).$

Since
$\tilde{\phi}_{k}\in\cup_{k\in\mathbb{N}^{+}}{\mathcal{A}}_{\alpha_{n_{k}}}(\sigma_{*},\theta_{\sigma_{*}-\tau}\omega)$,
which is precompact in $\ell^{2}$ (by Theorem 3.5 and Lemma 3.11), we can also
assume that

$\tilde{\phi}_{k}\rightarrow\phi_{*},\hskip 11.38092pt\mbox{as
}n\rightarrow\infty.$ (5.21)

Recalling the condition (4) of a continuous NRDS in Section 2 (or [27, Theorem
2.5]) and the uniform continuity in Lemma 3.8, we can infer from
$\alpha_{n_{k}}\rightarrow\alpha_{0}$, $\sigma_{k}\rightarrow\sigma_{0}$ and
$\tilde{\phi}_{k}\rightarrow\phi_{*}$ that

$\displaystyle\|\phi_{k}-u_{\alpha_{0}}(\sigma_{0},\sigma_{*},\theta_{-\tau}\omega,\phi_{*})\|\leqslant$
$\displaystyle\|u_{\alpha_{n_{k}}}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k})-u_{\alpha_{0}}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k})\|$
$\displaystyle+\|u_{\alpha_{0}}(\sigma_{k},\sigma_{*},\theta_{-\tau}\omega,\tilde{\phi}_{k})-u_{\alpha_{0}}(\sigma_{0},\sigma_{*},\theta_{-\tau}\omega,\phi_{*})\|\rightarrow
0,$

as $n\rightarrow\infty$. We have proved the conclusion. ∎

######  Lemma 5.7.

Under the conditions of Theorem 5.5, for each convergent sequence
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$, and $\Psi\in{\mathcal{T}}$,
the family ($n\in\mathbb{N}^{+}$) of mappings

$\varsigma\mapsto\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (5.22)

is dominated by a Lebesgue integrable function over $[s,t]$.

###### Proof.

By the compactness of $\overline{{\mathcal{U}}[s,t]}$ by Lemma 5.6, we know
that $\|u\|$ is bounded on $\overline{{\mathcal{U}}[s,t]}$. Hence by (5.12),
we have a positive constant ${\mathscr{M}}$ independent of $s$ and $n$ such
that

$\left|\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\right|\leqslant{\mathscr{M}}(1+\|\psi_{3}(\varsigma)\|_{1}+\|\psi_{4}(\varsigma)\|_{1})$

and then

$\displaystyle\left|\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\right|=\left|\int_{{\mathcal{A}}_{\alpha_{n}}(\varsigma,\theta_{\varsigma-\tau}\omega)}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\right|$ $\displaystyle\leqslant$
$\displaystyle{\mathscr{M}}(1+\|\psi_{3}(\varsigma)\|_{1}+\|\psi_{4}(\varsigma)\|_{1}),$
(5.23)

which is Lebesgue integrable over $[s,t]$ by the assumption
$\psi_{3},\psi_{4}\in L^{1}_{\rm loc}(\mathbb{R},\ell^{1})$. We can
immediately obtain the final result from (5.2). ∎

######  Lemma 5.8.

Under the conditions of Theorem 5.5, for each convergent sequence
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$, and $\Psi\in{\mathcal{T}}$,
the family ($n\in\mathbb{N}^{+}$) of mappings

$\varsigma\mapsto\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (5.24)

is uniformly bounded over $[s,t]$.

###### Proof.

Indeed by the condition (3) of the definition of ${\mathcal{T}}$, we can
easily see that for all $u\in\overline{{\mathcal{U}}[s,t]}\subset V$,
$\Psi^{\prime\prime}(u)$ is bounded in $L(H\times H,\mathbb{R})$. Thus we
obtain

$\left|\Psi^{\prime\prime}(u)(u,u)\right|\leqslant\max_{u\in\overline{{\mathcal{U}}[s,t]}}\|\Psi^{\prime\prime}(u)\|_{L(H\times
H,\mathbb{R})}\|u\|^{2},$

which is uniformly bounded by a positive constant ${\mathscr{N}}$ independent
of $s$ and $n$ on the compact set $\overline{{\mathcal{U}}[s,t]}$. Therefore

$\left|\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\right|=\left|\int_{{\mathcal{A}}_{\alpha_{n}}(\varsigma,\theta_{\varsigma-\tau}\omega)}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{n}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\right|\leqslant{\mathscr{N}}.$

The proof is complete. ∎

Now we are ready to show Theorem 5.5 as follows.

Proof of Theorem 5.5. By Theorem 4.4, we have known that for each
$\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$ and each convergent
sequence $\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}\in\mathbb{R}$, there exists a subsequence
$\\{n_{k}\\}_{k\in\mathbb{N}^{+}}$ of $\\{n\\}_{n\in\mathbb{N}^{+}}$ such that
the invariant sample measures $\mu^{\alpha_{n_{k}}}_{\tau,\omega}$ weakly
converges to an invariant sample measures $\mu^{\alpha_{0}}_{\tau,\omega}$ of
$\varphi_{\alpha_{0}}$. Then by definition it is natural that as
$k\rightarrow\infty$,

$\int_{H}\Psi(u)\mu^{\alpha_{n_{k}}}_{t,\theta_{t-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\Psi(u)\mu^{\alpha_{0}}_{t,\theta_{t-\tau}\omega}({\rm
d}u)\hskip 11.38092pt\mbox{and}$
$\int_{H}\Psi(u)\mu^{\alpha_{n_{k}}}_{s,\theta_{s-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\Psi(u)\mu^{\alpha_{0}}_{s,\theta_{s-\tau}\omega}({\rm
d}u),$

for the left sides of (5.19) and (5.20).

For the right sides, the weak convergence of
$\mu^{\alpha_{n_{k}}}_{\tau,\omega}$ to $\mu^{\alpha_{0}}_{\tau,\omega}$
ensures the convergence

$\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u).$

Combining the consequence in Lemma 5.7, we can deduce from the (Lebesgue)
Dominated Convergence Theorem the convergence

$\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\varsigma\rightarrow\int_{s}^{t}\int_{H}\langle\tilde{F}(\varsigma,u),\Psi^{\prime}(u)\rangle\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma.$

The convergence

$\frac{\alpha_{n_{k}}^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\varsigma\rightarrow\frac{\alpha_{0}^{2}}{2}\int_{s}^{t}\int_{H}\Psi^{\prime\prime}(u)(u,u)\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\varsigma$

is a similar consequence by Lemma 5.8.

Now each term in (5.19) except the term including stochastic integral
converges to the corresponding term in (5.20). Moreover, the equations (5.19)
and (5.20) are both obtained by Theorem 5.4. One can trivially infers that the
remaining term including stochastic integral in (5.19) also converges as
follows,

$\alpha_{n_{k}}\int_{s}^{t}\int_{H}(u,\Psi^{\prime}(u))\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm
d}\tilde{W}(\varsigma)\rightarrow\alpha_{0}\int_{s}^{t}\int_{H}(u,\Psi^{\prime}(u))\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u){\rm d}\tilde{W}(\varsigma),$

as $k\rightarrow\infty$. This proves the termwise convergence now.∎

######  Remark 5.9.

As a special case of Theorem 5.5, if we let $\alpha_{0}=0$, then the equation
(2.4) is a deterministic nonautonomous lattice equation. Thus the convergence
presented in Theorem 5.5 is actually from a stochastic Liouville type equation
to a deterministic Liouville type equation.

##  6 Convergence of sample statistical solutions

In [33] the authors developed statistical solutions for deterministic
equations into sample statistical solutions for stochastic differential
equations, which is defined as follows in our framework.

######  Definition 6.1.

A family of Borel probability measures
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ in
$H$ is called a sample statistical solution (in $H$) of (5.2) if the following
conditions are satisfied: for each $\tau\in\mathbb{R}$ and $\mathbb{P}$-a.s.
$\omega\in\Omega$,

  1. (1)

The function
$s\mapsto\int_{H}\Psi(u)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}u)$ is
continuous for every $\Psi\in{\mathcal{C}}_{\rm b}(H)$;

  2. (2)

For almost all $s\in\mathbb{R}$, the functions

$u\mapsto\langle\tilde{F}(s,u),\phi\rangle,\hskip
11.38092ptu\mapsto(u,\phi)\hskip 11.38092pt\mbox{and}\hskip
11.38092ptu\mapsto\Phi(u,u)$

are $\mu^{\alpha}_{s,\theta_{s-\tau}\omega}$-integrable for every $\phi\in V$
and bilinear mapping $\Phi\in L(H\times H,\mathbb{R})$; moreover, the mappings

$s\mapsto\int_{H}\langle\tilde{F}(s,u),\phi\rangle\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}u)\hskip 11.38092pt\mbox{and}\hskip
11.38092pts\mapsto\int_{H}\Phi(u,u)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}u)$

belongs to $L_{\rm loc}^{1}(\mathbb{R})$, and the mapping

$s\mapsto\int_{H}\left(u,\phi\right)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm
d}u)$

belongs to $L^{2}_{\rm loc}(\mathbb{R})$;

  3. (3)

For each $\Psi\in{\mathcal{T}}$, the stochastic Liouville type equation (1.4)
holds.

Then we can prove that the invariant sample measures obtained above are
actually a sample statistical solution for (5.2).

######  Theorem 6.2.

Suppose that the assumptions (F1), (F2), (F3), (F4) and (F5) hold for $f$.
Then the family of Borel probability measures
$\\{\mu_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ obtained in
Theorem 4.2 is a sample statistical solution of the equation (5.2).

###### Proof.

We check the conditions of Definition 6.1 one by one. For the condition (1),
we let $\Psi\in{\mathcal{C}}_{b}(H)$. Then by (4.4) and (4.6), for
$t,\,s,\,s_{*}\in\mathbb{R}$ with $t,\,s>s_{*}$ and $s,\,s_{*}$ fixed, we have

$\displaystyle\left|\int_{H}\Psi(u)\mu^{\alpha}_{t,\theta_{t-\tau}\omega}({\rm
d}u)-\int_{H}\Psi(u)\mu^{\alpha}_{s,\theta_{s-\tau}\omega}({\rm d}u)\right|$
$\displaystyle\leqslant$
$\displaystyle\int_{{\mathcal{A}}(s_{*},\theta_{s_{*}-\tau}\omega)}\left|\Psi(u_{\alpha}(t,s_{*},\theta_{-\tau}\omega,u))-\Psi(u_{\alpha}(s,s_{*},\theta_{-\tau}\omega,u))\right|\mu^{\alpha}_{s_{*},\theta_{s_{*}-\tau}\omega}({\rm
d}u).$ (6.1)

Since ${\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)$ is compact, by
the continuity stated in (4) of the definition of NRDS’s, we know the
continuity of $s\mapsto u_{\alpha}(s,s_{*},\theta_{-\tau}\omega,\phi)$ is
uniform for all
$\phi\in{\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)$. Hence for
every $\delta>0$, there exists $\overline{\delta}>0$ independent of
$\phi\in{\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)$, such that
when $|t-s|<\overline{\delta}$,

$\|u_{\alpha}(t,s_{*},\theta_{-\tau}\omega,\phi)-u_{\alpha}(s,s_{*},\theta_{-\tau}\omega,\phi)\|<\delta,$
(6.2)

for all $\phi\in{\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)$.
Furthermore, for all
$\phi\in{\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)$ and
$\varsigma\in[s-1,s+1]$, we know that

$u_{\alpha}(\varsigma,s_{*},\theta_{-\tau}\omega,\phi)\in{\mathcal{U}}_{\alpha}[s-1,s+1],$

which is compact by Lemma 5.1. Also, $\Psi\in{\mathcal{C}}_{\rm b}(H)$ implies
that $\Psi$ is uniformly continuous on ${\mathcal{U}}_{\alpha}[s-1,s+1]$. This
means that, for each $\varepsilon>0$, there is $\delta>0$ such that when
$u,\,v\in{\mathcal{U}}_{\alpha}[s,t]$ and $\|u-v\|<\delta$,

$|\Psi(u)-\Psi(v)|<\varepsilon.$ (6.3)

As a result, by (6.1), (6.2) and (6.3), it can be seen that for each
$\varepsilon>0$, there exists $\delta>0$ and $\overline{\delta}\in(0,1)$, such
that when $|t-s|<\overline{\delta}$, (6.2) holds true and then

$\eqref{3.70z}<\int_{{\mathcal{A}}_{\alpha}(s_{*},\theta_{s_{*}-\tau}\omega)}\varepsilon\mu^{\alpha}_{s_{*},\theta_{s_{*}-\tau}\omega}({\rm
d}\phi)=\varepsilon,$ (6.4)

which indicates the condition (1) of Definition 6.1.

For (2) of Definition 6.1, we can recall Lemma 5.2. The conclusions for
$\langle\tilde{F}(s,u),\phi\rangle$ and $(u,\phi)$ can be obtained by defining
$\Psi(u)\equiv\phi$. The conclusions for $\Phi(u,u)$ can be obtained by
defining $\Psi(u)=\Phi(u,u)$.

The condition (3) of Definition 6.1 follows from Theorem 5.4. The theorem is
proved here. ∎

According to the convergence consequences of invariant sample measures and
stochastic Liouville type equations, we can incidentally obtain the
convergence of sample statistical solutions

######  Theorem 6.3.

Suppose that the assumptions (F1), (F2), (F3), (F4) and (F5) hold for $f$. Let
$\varphi_{\alpha}$ be the NRDS defined as 2.10 generated by (2.4) and
$\\{\mu^{\alpha}_{\tau,\omega}\\}_{(\tau,\omega)\in\mathbb{R}\times\Omega}$ be
the sample statistical solution of $\varphi_{\alpha}$ given in Theorem 6.2.
Then for each $\tau\in\mathbb{R}$, $\mathbb{P}$-a.s. $\omega\in\Omega$ and
each sequence $\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ in $\mathbb{R}$ with
$\alpha_{n}\rightarrow\alpha_{0}$ as $n\rightarrow\infty$, there exists a
subsequence $\\{\alpha_{n_{k}}\\}_{k\in\mathbb{N}^{+}}$ of
$\\{\alpha_{n}\\}_{n\in\mathbb{N}^{+}}$ depending on $\tau$ and $\omega$, and
a sample statistical solution $\mu_{\tau,\omega}^{\alpha_{0}}$ of
$\varphi_{\alpha_{0}}$ such that $\mu_{\tau,\omega}^{\alpha_{n_{k}}}$
converges to $\mu_{\tau,\omega}^{\alpha_{0}}$ as $k\rightarrow+\infty$ in the
following sense:

  1. (1)

As a family of invariant sample measures,
$\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{n_{k}}}$ converges weakly to
$\mu_{t,\theta_{t-\tau}\omega}^{\alpha_{0}}$ for every $t\in\mathbb{R}$;

  2. (2)

As $k\rightarrow\infty$,

$\int_{H}\Psi(u)\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\Psi(u)\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (6.5)

in ${\mathcal{C}}([s,t])$ with maximum norm for each interval
$[s,t]\in\mathbb{R}$ and $\Psi\in{\mathcal{C}}_{\rm b}(H)$;

  3. (3)

For all $\phi\in V$ and $\Phi\in L(H\times H,\mathbb{R})$, as
$k\rightarrow\infty$,

$\int_{H}\langle\tilde{F}(\varsigma,u),\phi\rangle\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\langle\tilde{F}(\varsigma,u),\phi\rangle\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ $\mbox{and}\hskip
11.38092pt\int_{H}\Phi(u,u)\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\Phi(u,u)\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$

in $L^{1}([s,t])$ and

$\int_{H}\left(u,\phi\right)\mu^{\alpha_{n_{k}}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)\rightarrow\int_{H}\left(u,\phi\right)\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}({\rm
d}u)$ (6.6)

in $L^{2}([s,t])$ for each interval $[s,t]\in\mathbb{R}$;

  4. (4)

As $k\rightarrow\infty$, the stochastic Liouville type equation (5.19)
converges to (5.20) termwise.

###### Proof.

The conclusion (1) follows from Theorem 4.4. We next apply Arzelà-Ascoli
theorem to show the conclusion (2). Observe that in the procedure of proving
(6.4), as long as the compact set ${\mathcal{U}}_{\alpha}[s,t]$ is replaced by
$\overline{{\mathcal{U}}[s,t]}$ (defined in Lemma 5.6), which is compact,
(6.4) is valid uniformly for all $\alpha=\alpha_{n}$, $n\in\mathbb{N}$. This
is indeed the equicontinity of the sequence in (6.5). The uniform boundedness
is also obvious, since $\Psi$ has a uniform bound on ${\mathcal{U}}[s,t]$. The
reason why the limit integral is with respect to
$\mu^{\alpha_{0}}_{\varsigma,\theta_{\varsigma-\tau}\omega}$ is the weak
convergence in (1). Hence the convergence (6.5) holds in ${\mathcal{C}}[s,t]$
(perhaps for a subsequence).

The conclusion (3) follows from Lemmas 5.6 and 5.7, where the convergence
(6.6) in $L^{2}([s,t])$ is a similar result by the proof procedure of Lemma
5.8 with a slight modification. The conclusion (4) has been proved in Theorem
5.5. ∎

## Data Availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

## Acknowledgements

Our work was supported by grants from the National Natural Science Foundation
of China (NNSFC Nos. 11801190 and 12101462).

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