#  Well-posedness and large deviations of fractional McKean-Vlasov stochastic
reaction-diffusion equations on unbounded domains

Zhang Chen∗, Bixiang Wang†
∗School of Mathematics, Shandong University, Jinan 250100, China
†Department of Mathematics, New Mexico Institute of Mining and Technology,
Socorro, NM 87801, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract. This paper is mainly concerned with the large deviation principle of
the fractional McKean-Vlasov stochastic reaction-diffusion equation defined on
$\displaystyle\mathbb{R}^{n}$ with polynomial drift of any degree. We first
prove the well-posedness of the underlying equation under a dissipative
condition, and then show the strong convergence of solutions of the
corresponding controlled equation with respect to the weak topology of
controls, by employing the idea of uniform tail-ends estimates of solutions in
order to circumvent the non-compactness of Sobolev embeddings on unbounded
domains. We finally establish the large deviation principle of the fractional
McKean-Vlasov equation by the weak convergence method without assuming the
time Hölder continuity of the non-autonomous diffusion coefficients.

Keywords. Large deviation principle, McKean-Vlasov equation, fractional
Laplacian, unbounded domain, compactness, tail-ends estimates.

AMS 2020 Mathematics Subject Classification. 60F10, 60H15, 37L55, 35R60.

##  1 Introduction

This paper deals with the well-posedness and the large deviation principle
(LDP) of the fractional McKean-Vlasov stochastic reaction-diffusion equation
driven by nonlinear noise defined on $\displaystyle\mathbb{R}^{n}$:

$\displaystyle\displaystyle
du^{\varepsilon}(t)+(-\triangle)^{\alpha}u^{\varepsilon}(t)dt$
$\displaystyle\displaystyle+f(t,x,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt=g(t,x,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt$
$\displaystyle\displaystyle+\sqrt{\varepsilon}\sum_{k=1}^{\infty}\left(\sigma_{1,k}(t,x)+\kappa(x)\sigma_{2,k}(t,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})\right)dW_{k}(t),$
(1.1)

with initial data

$\displaystyle\displaystyle u^{\varepsilon}(0)=u_{0},$ (1.2)

where $\displaystyle(-\triangle)^{\alpha}$ with $\displaystyle\alpha\in(0,1)$
is the fractional Laplace operator, $\displaystyle\varepsilon\in(0,1)$,
$\displaystyle\mathcal{L}_{u^{\varepsilon}(t)}$ is the distribution law of
$\displaystyle u^{\varepsilon}(t)$, $\displaystyle f$ is a nonlinear function
with arbitrary growth rate, $\displaystyle g$ is a Lipschitz function,
$\displaystyle\sigma_{1,k}:\mathbb{R}\to L^{2}(\mathbb{R}^{n})$ is given,
$\displaystyle\kappa\in L^{2}(\mathbb{R}^{n})\bigcap
L^{\infty}(\mathbb{R}^{n})$, and $\displaystyle\sigma_{2,k}$ is a nonlinear
diffusion term, and $\displaystyle\\{W_{k}\\}_{k\in\mathbb{N}}$ is a sequence
of independent standard real-valued Wiener processes on a complete filtered
probability space
$\displaystyle\left(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geqslant
0},\mathbb{P}\right)$.

The fractional differential equations have numerous applications in finance,
physics, biology and probability [1, 2, 3]. If the nonlinear drift and
diffusion terms do not depend on
$\displaystyle\mathcal{L}_{u^{\varepsilon}(t)}$, then (1) reduces to the
standard stochastic reaction-diffusion equation with fractional Laplacian. The
asymptotic behaviors of solutions of the standard fractional equations have
been extensively studied in the literature, see, e.g., [4, 5, 6] for
deterministic equations, and [7, 8, 9, 10, 11, 12, 13, 14] for stochastic
equations. We also refer the reader to [15] and [16] for global solutions of
stochastic reaction-diffusion equations with super-linear drift on bounded
domain and unbounded domain, respectively.

Recently, the McKean-Vlasov stochastic differential equations have received
considerable attention after the seminal works [17, 18]. Such stochastic
equations are also called mean field equations [19, 20] and describe the
limits of interacting particle systems [21, 22, 23, 24, 25], which depend not
only on the states of solutions, but also on the distribution law of
solutions. The existence of solutions of the McKean-Vlasov equations have been
intensively studied by many authors, see, e.g. [26, 27, 28, 29, 30, 31] and
the references therein. The reader is also referred to [32, 33] for the Bismut
formula and [34, 35] for the averaging principles of such systems. In this
paper, we investigate the LDP of the fractional McKean-Vlasov equation (1) on
$\displaystyle{\mathbb{R}}^{n}$.

The LDP is concerned with the exponential rate of convergence of solutions of
stochastic equations with small noise, which has applications in many fields
such as statistics, information theory and statistical physics [36, 37]. The
classical method for proving the LDP is based on the time discretization and
exponential tightness, see, e.g., [36, 37, 38, 39, 40, 41]. To avoid the
complicated time discretization, the weak convergence method for the LDP was
developed in [42, 43, 44] based on variational representations of positive
functionals of Brownian motion, which is effective for proving the LDP of a
large class of stochastic equations, see, e.g. [45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56] for equations driven by Brownian motion and [57, 58] for
equations driven by jump processes. In particular, in the case of unbounded
domains, the LDP of the fractional reaction-diffusion equation and the tamed
3D Navier-Stokes equations was examined in [59] and [60], respectively.

Currently, there are very few results on the LDP of the McKean-Vlasov
stochastic equations, see [58, 61, 62, 63, 64]. It is worth mentioning that in
[63, 64], the authors established the LDP for a class of McKean-Vlasov
stochastic equations with locally monotone nonlinearity by the weak
convergence method. The results of both papers apply to many stochastic
equations defined on bounded as well as unbounded domains, including the
reaction-diffusion equation and the Navier-Stokes equations.

However, the arguments of [63, 64] have restrictions on the growth order of
the nonlinear terms and require the non-autonomous diffusion terms to be
regular in time. Specifically, for the reaction-diffusion equation (1), the
assumptions in [63, 64] impose an upper bound on the growth order of
$\displaystyle f$ in its third argument and require
$\displaystyle\sigma_{1,k}$ and $\displaystyle\sigma_{2,k}$ to be Hölder
continuous in time. Consequently, the results of [63, 64] do not apply to (1)
when $\displaystyle f$ has a polynomial growth of arbitrary order in its third
argument or when $\displaystyle\sigma_{1,k}$ or $\displaystyle\sigma_{2,k}$ is
not Hölder continuous in time. In the present paper, we deal with this case
and prove the LDP of (1) when $\displaystyle f$ has an arbitrary growth order,
$\displaystyle\sigma_{1,k}$ and $\displaystyle\sigma_{2,k}$ are continuous in
time, but not Hölder continuous.

To achieve our goal, we need some kind of compactness related to the tightness
of probability distributions of a family of solutions. In the case of bounded
domains, such compactness can be obtained from the regularity of solutions and
the compactness of Sobolev embeddings. However, the McKean-Vlasov stochastic
equation (1) is defined on the unbounded domain $\displaystyle\mathbb{R}^{n}$
and the Sobolev embeddings on $\displaystyle\mathbb{R}^{n}$ are no longer
compact, which causes an essential difficulty for proving the LDP of (1) with
a polynomial drift $\displaystyle f$ of arbitrary degree. To solve the
problem, we will employ the idea of uniform tail-ends estimates of solutions
to establish the strong convergence of a family of solutions even though
$\displaystyle f$ has an arbitrary growth order and the diffusion terms are
not Hölder continuous in time. This approach was used in [59] for the standard
fractional reaction-diffusion equation on $\displaystyle\mathbb{R}^{n}$ where
the drift and diffusion terms do not depend on the probability distributions
of solutions.

More precisely, to prove the LDP of (1) with polynomial drift of any degree,
we first establish the existence and uniqueness of adapted solutions to
(1)-(1.2) by the decoupled method (i.e., the method of freezing
distributions), see Theorem 2.1. We then consider the controlled equation
associated with (1)-(1.2) for every control $\displaystyle v\in
L^{2}(0,T;l^{2})$, and prove the well-posedness by the approximation arguments
as in [63, 64] combined with more complicated analysis, see Theorem 3.2. To
overcome the non-compactness of Sobolev embeddings on
$\displaystyle\mathbb{R}^{n}$ and remove the assumptions of time Hölder
continuity on diffusion coefficients in [63, 64], we derive the uniform tail-
ends estimates of solutions in Lemma 3.1 and further prove the LDP of
solutions in the Polish space $\displaystyle
C([0,T],L^{2}({\mathbb{R}}^{n}))\bigcap
L^{2}(0,T;H^{\alpha}({\mathbb{R}}^{n}))$ by the weak convergence method, see
Theorem 4.1. Under a stronger dissipativeness condition on the drift term
$\displaystyle f$, we finally establish the LDP of solutions to (1)-(1.2) in
the space $\displaystyle C([0,T],L^{2}({\mathbb{R}}^{n}))\cap
L^{2}(0,T;H^{\alpha}({\mathbb{R}}^{n}))\cap
L^{p}(0,T;L^{p}({\mathbb{R}}^{n}))$ in Theorem 4.4.

The paper is organized as follows. We first prove the well-posedness of
(1)-(1.2) and the corresponding controlled equation in Section 2 and Section
3, respectively. We then establish the LDP of (1)-(1.2) in Section 4.

Hereafter, we denote by $\displaystyle H=L^{2}(\mathbb{R}^{n})$ with inner
product $\displaystyle(\cdot,\cdot)$ and norm $\displaystyle\|\cdot\|$. We
also write $\displaystyle V=H^{\alpha}(\mathbb{R}^{n})$ with inner product
$\displaystyle(\cdot,\cdot)_{V}$ and norm $\displaystyle\|\cdot\|_{V}$, where

$\displaystyle\|u\|_{V}^{2}=\|u\|^{2}+\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}{\frac{|u(x)-u(y)|^{2}}{|x-y|^{n+2\alpha}}}dxdy=\|u\|^{2}+{\frac{2}{C(n,\alpha)}}\|(-\triangle)^{\frac{\alpha}{2}}u\|^{2},$

where $\displaystyle C(n,\alpha)>0$ is a specific number depending on
$\displaystyle n$ and $\displaystyle\alpha$.

Denote by $\displaystyle\mathcal{P}(H)$ the space of all probability measures
on $\displaystyle H$ with weak topology, and let

$\displaystyle\mathcal{P}_{m}(H)=\left\\{\mu\in\mathcal{P}(H):\mu(\|\cdot\|^{m}):=\int_{H}\|\xi\|^{m}\mu(d\xi)<+\infty\right\\},$

Then $\displaystyle\mathcal{P}_{m}(H)$ is a Polish space with $\displaystyle
L^{m}$-Wasserstein distance given by:

$\displaystyle\mathbb{W}_{m}(\mu,\nu)=\inf_{\pi\in\mathcal{C}(\mu,\nu)}\left(\int_{H\times
H}\|\xi-\eta\|^{m}\pi(d\xi,d\eta)\right)^{\frac{1}{m}},\ \ \ \forall\
\mu,\nu\in\mathcal{P}_{m}(H),$

where $\displaystyle\mathcal{C}(\mu,\nu)$ is the set of all couplings of
$\displaystyle\mu$ and $\displaystyle\nu$.

##  2 Fractional McKean-Vlasov stochastic equations on
$\displaystyle\mathbb{R}^{n}$

In this section, we discuss the assumptions on the nonlinear drift and
diffusion terms in (1), and prove the existence and uniqueness of solutions.

###  2.1 Assumptions on nonlinear drift and diffusion terms

In the sequel, we assume that $\displaystyle
f:{\mathbb{R}}\times{\mathbb{R}}^{n}\times{\mathbb{R}}\times\mathcal{P}_{2}(H)\to{\mathbb{R}}$
is a continuous function which satisfies the following conditions:

$\displaystyle({\bf\Sigma 1})$ For any $\displaystyle t,u\in\mathbb{R}$,
$\displaystyle x\in\mathbb{R}^{n}$, $\displaystyle\mu$ and
$\displaystyle\mu_{i}\in\mathcal{P}_{2}(H)$ $\displaystyle(i=1,2)$,

$\displaystyle\displaystyle
f(t,x,u,\mu)u\geqslant\lambda_{1}|u|^{p}-\psi_{1}(t,x)(1+|u|^{2}+\mu(\|\cdot\|^{2})),$
(2.1)
$\displaystyle\displaystyle|f(t,x,u_{1},\mu_{1})-f(t,x,u_{2},\mu_{2})|\leqslant$
$\displaystyle\displaystyle\lambda_{2}\left(\psi_{2}(t,x)+|u_{1}|^{p-2}+|u_{2}|^{p-2}\right)|u_{1}-u_{2}|$
$\displaystyle\displaystyle+\psi_{3}(t,x)\mathbb{W}_{2}(\mu_{1},\mu_{2})),\ \
\ \forall\ u_{1},u_{2}\in\mathbb{R},$ (2.2)
$\displaystyle\displaystyle|f(t,x,u,\mu)|\leqslant\lambda_{3}|u|^{p-1}+\psi_{3}(t,x)\left(1+\sqrt{\mu(\|\cdot\|^{2})}\right),$
(2.3) $\displaystyle\displaystyle\frac{\partial f}{\partial
u}(t,x,u,\mu)\geqslant-\psi_{4}(t,x),$ (2.4)

where $\displaystyle p\geq 2$, $\displaystyle\lambda_{i}>0$ for $\displaystyle
i=1,2,3$, $\displaystyle\psi_{i}\in
L_{loc}^{\infty}(\mathbb{R},L^{\infty}(\mathbb{R}^{n}))\bigcap
L_{loc}^{1}(\mathbb{R},L^{1}(\mathbb{R}^{n}))$ for $\displaystyle i=1,2,3,4$.

######  Remark 2.1.

Given $\displaystyle t,u\in\mathbb{R}$, $\displaystyle x\in\mathbb{R}^{n}$ and
$\displaystyle\mu\in\mathcal{P}_{2}(H)$, let

$\displaystyle f(t,x,u,\mu)=|u|^{p-2}u+\varphi(t,x)\int_{H}h(\xi)\mu(d\xi),$

where $\displaystyle p\geq 2$,
$\displaystyle\varphi:\mathbb{R}\times\mathbb{R}^{n}\to\mathbb{R}$ is
continuous such that $\displaystyle\varphi\in
L_{loc}^{\infty}(\mathbb{R},L^{\infty}(\mathbb{R}^{n}))\bigcap
L_{loc}^{1}(\mathbb{R},L^{1}(\mathbb{R}^{n}))$, and $\displaystyle
h:H\rightarrow\mathbb{R}$ is Lipschitz continuous. Then one may verify that
$\displaystyle f$ satisfies conditions (2.1)-(2.4) based on the fact that
$\displaystyle\mathcal{P}_{2}(H)\subseteq\mathcal{P}_{1}(H)$ and
$\displaystyle\mathbb{W}_{1}(\cdot,\cdot)$ is the same as the Kantorovich-
Rubinstein distance (see Remark 6.5 in [65]).

We assume the nonlinear drift $\displaystyle
g:\mathbb{R}\times\mathbb{R}^{n}\times\mathbb{R}\times\mathcal{P}_{2}(H)\rightarrow\mathbb{R}$
is continuous which further satisfies: for all $\displaystyle
t,u_{1},u_{2}\in\mathbb{R},x\in\mathbb{R}^{n}$ and
$\displaystyle\mu_{1},\mu_{2}\in\mathcal{P}_{2}(H)$,

$\displaystyle({\bf\Sigma 2})$

$\displaystyle\displaystyle|g(t,x,0,\delta_{0})|\leqslant\psi_{g}(t,x),$ (2.5)
$\displaystyle\displaystyle|g(t,x,u_{1},\mu_{1})-g(t,x,u_{1},\mu_{2})|\leqslant\psi_{g}(t,x)\left(|u_{1}-u_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})\right),$
(2.6)

where $\displaystyle\psi_{g}(t,x)\in
L_{loc}^{\infty}(\mathbb{R},L^{\infty}(\mathbb{R}^{n}))\bigcap
L_{loc}^{1}(\mathbb{R},L^{1}(\mathbb{R}^{n}))$ and $\displaystyle\delta_{0}$
is the Dirac measure on $\displaystyle H$.

It follows from (2.5)-(2.6) that for all $\displaystyle
t,u\in\mathbb{R},x\in\mathbb{R}^{n}$ and
$\displaystyle\mu\in\mathcal{P}_{2}(H)$,

$\displaystyle\displaystyle|g(t,x,u,\mu)|\leqslant$
$\displaystyle\displaystyle|g(t,x,u,\mu)-g(t,x,0,\delta_{0})|+|g(t,x,0,\delta_{0})|$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\psi_{g}(t,x)\left(1+|u|+\sqrt{\mu(\|\cdot\|^{2})}\right).$
(2.7)

Finally, for the diffusion terms, we assume:

$\displaystyle({\bf\Sigma 3})$
$\displaystyle\sigma_{1}=\\{\sigma_{1,k}\\}_{k=1}^{\infty}:\mathbb{R}\rightarrow
L^{2}(\mathbb{R}^{n},l^{2})$ is continuous, and for every $\displaystyle
k\in\mathbb{N}$,
$\displaystyle\sigma_{2,k}:\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(H)\rightarrow\mathbb{R}$
is continuous and there exist nonnegative sequences
$\displaystyle\beta=\\{\beta_{k}\\}_{k=1}^{\infty}$ and
$\displaystyle\gamma=\\{\gamma_{k}\\}_{k=1}^{\infty}$ such that for any
$\displaystyle t,s\in\mathbb{R}$ and $\displaystyle\mu\in\mathcal{P}_{2}(H)$,

$\displaystyle\displaystyle|\sigma_{2,k}(t,s,\mu)|\leqslant\beta_{k}\left(1+\sqrt{\mu(\|\cdot\|^{2})}\right)+\gamma_{k}|s|.$
(2.8)

In addition, we assume that $\displaystyle\sigma_{2,k}(t,s,\mu)$ is Lipschitz
continuous in $\displaystyle s$ and $\displaystyle\mu$ uniformly for
$\displaystyle t\in\mathbb{R}$; that is, there exists a positive constant
$\displaystyle L_{\sigma,k}$ such that for any $\displaystyle
t,s_{1},s_{2}\in\mathbb{R}$ and
$\displaystyle\mu_{1},\mu_{2}\in\mathcal{P}_{2}(H)$,

$\displaystyle\displaystyle|\sigma_{2,k}(t,s_{1},\mu_{1})-\sigma_{2,k}(t,s_{2},\mu_{2})|\leqslant
L_{\sigma,k}\left(|s_{1}-s_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})\right),$ (2.9)

where
$\displaystyle\sum_{k=1}^{\infty}\left(\beta_{k}^{2}+\gamma_{k}^{2}+L_{\sigma,k}^{2}\right)<\infty$.

Let $\displaystyle l^{2}$ be the Hilbert space of square summable sequences of
real numbers, and $\displaystyle W$ be a cylindrical Wiener process in
$\displaystyle l^{2}$ with identity covariance operator which means that there
exists a separable Hilbert space $\displaystyle U$ such that $\displaystyle W$
takes values in $\displaystyle U$ and the embedding $\displaystyle
l^{2}\hookrightarrow U$ is Hilbert-Schmidt.

Given $\displaystyle t\in\mathbb{R}$, $\displaystyle u\in H$ and
$\displaystyle\mu\in\mathcal{P}_{2}(H)$, define
$\displaystyle\sigma(t,u,\mu):l^{2}\rightarrow H$ by

$\displaystyle\displaystyle\sigma(t,u,\mu)(\theta)(x)=\sum_{k=1}^{\infty}\left(\sigma_{1,k}(t,x)+\kappa(x)\sigma_{2,k}(t,u(x),\mu)\right)\theta_{k},\
\ \forall\ \theta=(\theta_{k})_{k=1}^{\infty}\in l^{2},x\in\mathbb{R}^{n}.$
(2.10)

It follows from (2.8) that the series in (2.10) is convergent in
$\displaystyle H$. Furthermore, the operator $\displaystyle\sigma(t,u,\mu)$ is
Hilbert-Schmidt from $\displaystyle l^{2}$ to $\displaystyle H$ and

$\displaystyle\displaystyle\|\sigma(t,u,\mu)\|_{L_{2}(l^{2},H)}^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{k=1}^{\infty}\|\sigma_{1,k}(t,x)+\kappa(x)\sigma_{2,k}(t,u(x),\mu)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\sum_{k=1}^{\infty}\|\sigma_{1,k}(t)\|^{2}+8\|\kappa\|^{2}\sum_{k=1}^{\infty}\beta_{k}^{2}\left(1+\mu(\|\cdot\|^{2})\right)+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\sum_{k=1}^{\infty}\gamma_{k}^{2}\|u\|^{2}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\|\sigma_{1}(t)\|_{L^{2}(\mathbb{R}^{n},l^{2})}^{2}+8\|\kappa\|^{2}\|\beta\|_{l^{2}}^{2}\left(1+\mu(\|\cdot\|^{2})\right)+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|\gamma\|_{l^{2}}^{2}\|u\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{\sigma,T}\left(1+\|u\|^{2}+\mu(\|\cdot\|^{2})\right),\ \ \ \forall\
t\in[0,T],$ (2.11)

where $\displaystyle L_{2}(l^{2},H)$ is the space of Hilbert-Schmidt operators
from $\displaystyle l^{2}$ to $\displaystyle H$ with norm
$\displaystyle\|\cdot\|_{L_{2}(l^{2},H)}$, $\displaystyle
M_{\sigma,T}=2\|\sigma_{1}\|_{C([0,T];L^{2}(\mathbb{R}^{n},l^{2}))}^{2}+8\|\kappa\|^{2}\|\beta\|_{l^{2}}^{2}+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|\gamma\|_{l^{2}}^{2}$.

On the other hand, by (2.9), we have for all $\displaystyle t\in\mathbb{R}$,
$\displaystyle u_{1},u_{2}\in H$ and
$\displaystyle\mu_{1},\mu_{2}\in\mathcal{P}_{2}(H)$,

$\displaystyle\displaystyle\|\sigma(t,u_{1},\mu_{1})-\sigma(t,u_{2},\mu_{2})\|^{2}_{L_{2}(l^{2},H)}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{k=1}^{\infty}\|\kappa(x)\left(\sigma_{2,k}(t,u_{1},\mu_{1})-\sigma_{2,k}(t,u_{2},\mu_{2})\right)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\sum_{k=1}^{\infty}L_{\sigma,k}^{2}\left[\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|u_{1}-u_{2}\|^{2}+\|\kappa\|^{2}\left(\mathbb{W}_{2}(\mu_{1},\mu_{2})\right)^{2}\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
L_{\sigma}\left[\|u_{1}-u_{2}\|^{2}+\left(\mathbb{W}_{2}(\mu_{1},\mu_{2})\right)^{2}\right],$
(2.12)

where $\displaystyle
L_{\sigma}=2\sum_{k=1}^{\infty}L_{\sigma,k}^{2}\left(\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}+\|\kappa\|^{2}\right)$.

With above notation, the stochastic equation (1) takes the form:

$\displaystyle\displaystyle
du^{\varepsilon}(t)+(-\triangle)^{\alpha}u^{\varepsilon}(t)dt+f(t,\cdot,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
g(t,\cdot,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt+\sqrt{\varepsilon}\sigma(t,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dW(t),$
(2.13)

with initial data

$\displaystyle\displaystyle u^{\varepsilon}(0)=u_{0}.$ (2.14)

###  2.2 Well-posedness of the stochastic equation

A solution of problem (2.1)-(2.14) is understood in the following sense.

######  Definition 2.1.

For $\displaystyle T>0$, $\displaystyle\varepsilon\in(0,1)$, $\displaystyle
u_{0}\in L^{2}(\Omega,\mathscr{F}_{0};H)$, a continuous $\displaystyle
H$-valued $\displaystyle\mathscr{F}_{t}$-adapted stochastic process
$\displaystyle u^{\varepsilon}$ is called a solution of (2.1)-(2.14) on
$\displaystyle[0,T]$ if

$\displaystyle u^{\varepsilon}\in L^{2}(\Omega,C([0,T],H))\bigcap
L^{2}(\Omega,L^{2}(0,T;V))\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))$

such that for all $\displaystyle t\in[0,T]$ and $\displaystyle\xi\in V\bigcap
L^{p}(\mathbb{R}^{n})$,

$\displaystyle\displaystyle(u^{\varepsilon}(t),\xi)+\int_{0}^{t}\left((-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s),(-\triangle)^{\frac{\alpha}{2}}\xi\right)ds+\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\xi(x)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle(u_{0},\xi)+\int_{0}^{t}\left(g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),\xi\right)ds+\sqrt{\varepsilon}\int_{0}^{t}\left(\xi,\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right),\
\ \ \mathbb{P}\text{-almost surely}.$

Note that if $\displaystyle u^{\varepsilon}$ is a solution of (2.1)-(2.14) in
the sense of Definition 2.1, then
$\displaystyle(-\triangle)^{\alpha}u^{\varepsilon}\in
L^{2}(\Omega,L^{2}(0,T;V^{\ast}))$ and $\displaystyle
f(\cdot,\cdot,u^{\varepsilon},\mathcal{L}_{u^{\varepsilon}})\in
L^{q}(\Omega,L^{q}(0,T;L^{q}(\mathbb{R}^{n})))$ with
$\displaystyle\frac{1}{p}+\frac{1}{q}=1$, and hence for all $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle
u^{\varepsilon}(t)+\int_{0}^{t}(-\triangle)^{\alpha}u^{\varepsilon}(s)ds+\int_{0}^{t}f(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})ds$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
u_{0}+\int_{0}^{t}g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})ds+\sqrt{\varepsilon}\int_{0}^{t}\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\quad\text{in}\
\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast},$ (2.15)

$\displaystyle\mathbb{P}$-almost surely.

The following theorem is concerned with the well-posedness of problem
(2.1)-(2.14).

######  Theorem 2.1.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$ hold. If
$\displaystyle T>0$, $\displaystyle\varepsilon\in(0,1)$, $\displaystyle
u_{0}\in L^{2}(\Omega,\mathscr{F}_{0};H)$, then problem (2.1)-(2.14) has a
unique solution $\displaystyle u^{\varepsilon}$ in the sense of Definition
2.1, which satisfies the energy equation: for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|u^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})u^{\varepsilon}(s)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\|u_{0}\|^{2}+2\int_{0}^{t}(g(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),u^{\varepsilon}(s))ds+\varepsilon\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle+2\sqrt{\varepsilon}\int_{0}^{t}\left(u^{\varepsilon}(s),\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right),\
\ \ \mathbb{P}\text{-almost surely}.$ (2.16)

Moreover, the following uniform estimates are valid:

$\displaystyle\displaystyle\|u^{\varepsilon}\|_{L^{2}(\Omega,C([0,T],H))}^{2}+\|u^{\varepsilon}\|_{L^{2}(\Omega,L^{2}(0,T;V))}^{2}+\|u^{\varepsilon}\|_{L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))}^{p}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{1}\left(1+\|u_{0}\|_{L^{2}(\Omega,\mathscr{F}_{0};H)}^{2}\right),$

where $\displaystyle M_{1}=M_{1}(T)>0$ is independent of $\displaystyle u_{0}$
and $\displaystyle\varepsilon$.

###### Proof.

We first prove the existence and uniqueness of solutions, and then derive the
uniform estimates of solutions.

Step 1: existence and uniqueness of solutions.

Given $\displaystyle u_{0}\in L^{2}(\Omega,\mathscr{F}_{0};H)$ and
$\displaystyle\mu\in C([0,T],\mathcal{P}_{2}(H))$, consider the stochastic
equation:

$\displaystyle\displaystyle
du_{\mu}^{\varepsilon}(t)+(-\triangle)^{\alpha}u_{\mu}^{\varepsilon}(t)dt+f_{\mu}(t,\cdot,u_{\mu}^{\varepsilon}(t))dt=g_{\mu}(t,\cdot,u_{\mu}^{\varepsilon}(t))dt+\sqrt{\varepsilon}\sigma_{\mu}(t,u_{\mu}^{\varepsilon}(t))dW(t),$
(2.17)

with initial condition:

$u_{\mu}^{\varepsilon}(0)=u_{0},$ (2.18)

where

$\displaystyle
f_{\mu}(t,\cdot,u_{\mu}^{\varepsilon}(t))=f(t,\cdot,u_{\mu}^{\varepsilon}(t),\mu(t)),\
g_{\mu}(t,\cdot,u_{\mu}^{\varepsilon}(t))=g(t,\cdot,u_{\mu}^{\varepsilon}(t),\mu(t)),\
\sigma_{\mu}(t,u_{\mu}^{\varepsilon}(t))=\sigma(t,u_{\mu}^{\varepsilon}(t),\mu(t)).$

By $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$, we find that
$\displaystyle f_{\mu},g_{\mu}$ and $\displaystyle\sigma_{\mu}$ satisfy all
the conditions in [11, Theorem 6.3], and hence problem (2.17)-(2.18) has a
unique solution

$\displaystyle u_{\mu}^{\varepsilon}\in L^{2}(\Omega,C([0,T],H))\bigcap
L^{2}(\Omega,L^{2}(0,T;V))\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n}))),$

which satisfies the energy equation (2.1) with
$\displaystyle\mathcal{L}_{u^{\varepsilon}(s)}$ replaced by
$\displaystyle\mu(s)$, and the uniform estimate:

$\displaystyle\displaystyle\|u_{\mu}^{\varepsilon}\|_{L^{2}(\Omega,C([0,T],H))}^{2}+\|u_{\mu}^{\varepsilon}\|_{L^{2}(\Omega,L^{2}(0,T;V))}^{2}+\|u_{\mu}^{\varepsilon}\|_{L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))}^{p}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{T,\mu}\left(1+\|u_{0}\|_{L^{2}(\Omega,\mathscr{F}_{0};H)}^{2}\right),$

where $\displaystyle M_{T,\mu}=M(T,\mu)>0$ is independent of $\displaystyle
u_{0}$ and $\displaystyle\varepsilon$.

By $\displaystyle u_{\mu}^{\varepsilon}\in L^{2}(\Omega,C([0,T],H))$ and the
Lebesgue dominated convergence theorem, we see that $\displaystyle
u_{\mu}^{\varepsilon}\in C([0,T],L^{2}(\Omega,H))$. Note that

$\displaystyle\displaystyle\mathbb{W}_{2}\left(\mathcal{L}_{u_{\mu}^{\varepsilon}(t)},\mathcal{L}_{u_{\mu}^{\varepsilon}(s)}\right)\leqslant\left(\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(t)-u_{\mu}^{\varepsilon}(s)\|^{2}\right]\right)^{\frac{1}{2}},\
\ \ \forall\ s,t\in[0,T],$

and hence $\displaystyle\mathcal{L}_{u_{\mu}^{\varepsilon}(\cdot)}\in
C([0,T],\mathcal{P}_{2}(H))$.

Define a map $\displaystyle\Phi^{u_{0}}:C([0,T],\mathcal{P}_{2}(H))\rightarrow
C([0,T],\mathcal{P}_{2}(H))$ by

$\displaystyle\displaystyle\Phi^{u_{0}}(\mu)(t)=\mathcal{L}_{u_{\mu}^{\varepsilon}(t)},\
\ \ t\in[0,T],\ \mu\in C([0,T],\mathcal{P}_{2}(H)),$ (2.19)

where $\displaystyle u_{\mu}^{\varepsilon}$ is the solution of (2.17)-(2.18).

Next, we will prove the existence of solutions of (2.1)-(2.14) by finding a
fixed point of $\displaystyle\Phi^{u_{0}}$. To that end, we need to show
$\displaystyle\Phi^{u_{0}}$ is a contractive map in the complete metric space
$\displaystyle(C([0,T],\mathcal{P}_{2}(H)),\ d_{C([0,T],\mathcal{P}_{2}(H))})$
where the metric $\displaystyle d_{C([0,T],\mathcal{P}_{2}(H))}$ is defined by

$\displaystyle
d_{C([0,T],\mathcal{P}_{2}(H))}(\mu,\nu)=\sup_{t\in[0,T]}e^{-\lambda
t}\mathbb{W}_{2}\left(\mu(t),\nu(t)\right),\quad\forall\ \mu,\nu\in
C([0,T],\mathcal{P}_{2}(H)).$

Let $\displaystyle\mu,\nu\in C([0,T],\mathcal{P}_{2}(H))$, and $\displaystyle
u_{\mu}^{\varepsilon}$ and $\displaystyle u_{\nu}^{\varepsilon}$ be the
solution of (2.17)-(2.18). By Itô’s formula, we have

$\displaystyle\displaystyle\|u_{\mu}^{\varepsilon}(t)-u_{\nu}^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)\|^{2}ds$
$\displaystyle\displaystyle+2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u_{\mu}^{\varepsilon}(s),\mu(s))-f(s,x,u_{\nu}^{\varepsilon}(s),\nu(s))\right)\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)dxds$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\int_{0}^{t}\left(g(s,\cdot,u_{\mu}^{\varepsilon}(s),\mu(s))-g(s,\cdot,u_{\nu}^{\varepsilon}(s),\nu(s)),u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)ds$
$\displaystyle\displaystyle+2\sqrt{\varepsilon}\int_{0}^{t}\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s),\left(\sigma(s,u_{\mu}^{\varepsilon}(s),\mu(s))-\sigma(s,u_{\nu}^{\varepsilon}(s),\nu(s))\right)dW(s)\right)$
$\displaystyle\displaystyle+\varepsilon\int_{0}^{t}\|\sigma(s,u_{\mu}^{\varepsilon}(s),\mu(s))-\sigma(s,u_{\nu}^{\varepsilon}(s),\nu(s))\|_{L_{2}(l^{2},H)}^{2}ds.$
(2.20)

Taking the expectation on both sides of (2.2), we obtain

$\displaystyle\displaystyle\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(t)-u_{\nu}^{\varepsilon}(t)\|^{2}\right]+2\int_{0}^{t}\mathbb{E}\left[\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)\|^{2}\right]ds$
$\displaystyle\displaystyle+2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u_{\mu}^{\varepsilon}(s),\mu(s))-f(s,x,u_{\nu}^{\varepsilon}(s),\nu(s))\right)\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)dxds\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\left(g(s,\cdot,u_{\mu}^{\varepsilon}(s),\mu(s))-g(s,\cdot,u_{\nu}^{\varepsilon}(s),\nu(s)),u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)ds\right]$
$\displaystyle\displaystyle+\varepsilon\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u_{\mu}^{\varepsilon}(s),\mu(s))-\sigma(s,u_{\nu}^{\varepsilon}(s),\nu(s))\|_{L_{2}(l^{2},H)}^{2}ds\right].$
(2.21)

For the third term on left-hand side of (2.2), by (2.1), (2.4) and the Hölder
inequality, we obtain

$\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u_{\mu}^{\varepsilon}(s),\mu(s))-f(s,x,u_{\nu}^{\varepsilon}(s),\nu(s))\right)\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)dxds\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u_{\mu}^{\varepsilon}(s),\mu(s))-f(s,x,u_{\nu}^{\varepsilon}(s),\mu(s))\right)\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)dxds\right]$
$\displaystyle\displaystyle+2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u_{\nu}^{\varepsilon}(s),\mu(s))-f(s,x,u_{\nu}^{\varepsilon}(s),\nu(s))\right)\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)dxds\right]$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\int_{0}^{t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds$
$\displaystyle\displaystyle-2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}|\psi_{3}(s,x)|\
\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\
\left|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right|dxds\right]$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\int_{0}^{t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds$
$\displaystyle\displaystyle-2\mathbb{E}\left[\int_{0}^{t}\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\
\|\psi_{3}(s)\|\
\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|ds\right]$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\int_{0}^{t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds$
$\displaystyle\displaystyle-\int_{0}^{t}\|\psi_{3}(s)\|^{2}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds-\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds.$
(2.22)

For the first term on right-hand side of (2.2), by (2.6) and the Hölder
inequality, we obtain

$\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\left(g(s,\cdot,u_{\mu}^{\varepsilon}(s),\mu(s))-g(s,\cdot,u_{\nu}^{\varepsilon}(s),\nu(s)),u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)ds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\left|g(s,x,u_{\mu}^{\varepsilon}(s),\mu(s))-g(s,x,u_{\nu}^{\varepsilon}(s),\nu(s))\right|\left|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right|dxds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\mathbb{E}\left[\int_{0}^{t}\int_{\mathbb{R}^{n}}\psi_{g}(s,x)\left(|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)|+\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)|dxds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\mathbb{E}\left[\int_{0}^{t}\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}ds\right]$
$\displaystyle\displaystyle+2\mathbb{E}\left[\int_{0}^{t}\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\|\psi_{g}(s)\|\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|ds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\mathbb{E}\left[\int_{0}^{t}\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}ds\right]$
$\displaystyle\displaystyle+\int_{0}^{t}\|\psi_{g}(s)\|^{2}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds+\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds.$
(2.23)

For the second term on right-hand side of (2.2), by (2.1), we obtain that for
any $\displaystyle\varepsilon\in(0,1)$,

$\displaystyle\displaystyle\varepsilon\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u_{\mu}^{\varepsilon}(s),\mu(s))-\sigma(s,u_{\nu}^{\varepsilon}(s),\nu(s))\|_{L_{2}(l^{2},H)}^{2}ds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
L_{\sigma}\int_{0}^{t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds+L_{\sigma}\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds.$
(2.24)

It follows from (2.2)-(2.2) that

$\displaystyle\displaystyle\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(t)-u_{\nu}^{\varepsilon}(t)\|^{2}\right]+2\int_{0}^{t}\mathbb{E}\left[\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\right)\|^{2}\right]ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+L_{\sigma}\right)\int_{0}^{t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds$
$\displaystyle\displaystyle+\int_{0}^{t}\left(\|\psi_{3}(s)\|^{2}+\|\psi_{g}(s)\|^{2}\right)\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds+(2+L_{\sigma})\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{t}\left(c_{1}+\|\psi_{3}(s)\|^{2}+\|\psi_{g}(s)\|^{2}\right)\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(s)-u_{\nu}^{\varepsilon}(s)\|^{2}\right]ds$
$\displaystyle\displaystyle+(2+L_{\sigma})\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds,$
(2.25)

where $\displaystyle
c_{1}=2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+L_{\sigma}>0.$

Applying the Gronwall inequality to (2.2), we obtain that for any
$\displaystyle 0\leqslant t\leqslant T$,

$\displaystyle\displaystyle\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(t)-u_{\nu}^{\varepsilon}(t)\|^{2}\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle(2+L_{\sigma})\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds\cdot
e^{\int_{0}^{T}\left(c_{1}+\|\psi_{3}(s)\|^{2}+\|\psi_{g}(s)\|^{2}\right)ds}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
C_{T}\int_{0}^{t}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds,$
(2.26)

where $\displaystyle
C_{T}=(2+L_{\sigma})e^{\int_{0}^{T}\left(c_{1}+\|\psi_{3}(s)\|^{2}+\|\psi_{g}(s)\|^{2}\right)ds}>0$.
By (2.2), we obtain, for $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle e^{-2\lambda
t}\mathbb{E}\left[\|u_{\mu}^{\varepsilon}(t)-u_{\nu}^{\varepsilon}(t)\|^{2}\right]\leqslant$
$\displaystyle\displaystyle C_{T}\int_{0}^{t}e^{-2\lambda(t-s)}e^{-2\lambda
s}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{C_{T}}{2\lambda}\sup_{s\in[0,T]}e^{-2\lambda
s}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}.$ (2.27)

By (2.2) and the definition of $\displaystyle\mathbb{W}_{2}(\cdot,\cdot)$, we
have

$\displaystyle\displaystyle e^{-2\lambda
t}\mathbb{W}_{2}\left(\mathcal{L}_{u_{\mu}^{\varepsilon}(t)},\mathcal{L}_{u_{\nu}^{\varepsilon}(t)}\right)^{2}\leqslant\frac{C_{T}}{2\lambda}\sup_{s\in[0,T]}e^{-2\lambda
s}\left(\mathbb{W}_{2}\left(\mu(s),\nu(s)\right)\right)^{2}\ \ \text{for all}\
t\in[0,T].$ (2.28)

By (2.28) and the definition of $\displaystyle
d_{C([0,T],\mathcal{P}_{2}(H))}$, we obtain

$\displaystyle\displaystyle
d_{C([0,T],\mathcal{P}_{2}(H))}\left(\mathcal{L}_{u_{\mu}^{\varepsilon}},\mathcal{L}_{u_{\nu}^{\varepsilon}}\right)\leqslant\left(\frac{C_{T}}{2\lambda}\right)^{\frac{1}{2}}d_{C([0,T],\mathcal{P}_{2}(H))}\left(\mu,\nu\right).$
(2.29)

Choosing $\displaystyle\lambda>0$ such that
$\displaystyle\frac{C_{T}}{2\lambda}<\frac{1}{4}$, by (2.29) and the
definition of $\displaystyle\Phi^{u_{0}}$, we obtain

$\displaystyle\displaystyle{d_{C([0,T],\mathcal{P}_{2}(H))}}\left(\Phi^{u_{0}}(\mu),\Phi^{u_{0}}(\nu)\right)\leqslant\frac{1}{2}{d_{C([0,T],\mathcal{P}_{2}(H))}}\left(\mu,\nu\right),\quad\forall\
\mu,\nu\in{C([0,T],\mathcal{P}_{2}(H))},$

which shows $\displaystyle\Phi^{u_{0}}$ is a contractive map in
$\displaystyle{C([0,T],\mathcal{P}_{2}(H))}$ with metric $\displaystyle
d_{C([0,T],\mathcal{P}_{2}(H))})$. Therefore, $\displaystyle\Phi^{u_{0}}$ has
a unique fixed point $\displaystyle\bar{\mu}\in{C([0,T],\mathcal{P}_{2}(H))}$.
Then, $\displaystyle u_{\bar{\mu}}^{\varepsilon}$ is a solution of
(2.1)-(2.14) on $\displaystyle[0,T]$.

On the other hand, for every solution $\displaystyle u^{\varepsilon}$ of
(2.1)-(2.14), if we set $\displaystyle\mu(t)=\mathcal{L}_{u^{\varepsilon}(t)}$
for $\displaystyle t\in[0,T]$, then $\displaystyle\mu$ is a fixed point of
$\displaystyle\Phi^{u_{0}}$ in $\displaystyle{C([0,T],\mathcal{P}_{2}(H))}$,
which along with the uniqueness of fixed points of $\displaystyle\Phi^{u_{0}}$
and (2.2) implies the uniqueness of solutions of (2.1)- (2.14).

Step 2: uniform estimates of solutions of (2.1)- (2.14).

By Itô’s formula, we have

$\displaystyle\displaystyle\|u^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})u^{\varepsilon}(s)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\|u_{0}\|^{2}+2\int_{0}^{t}\left(g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),\
u^{\varepsilon}(s)\right)ds+\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle+2\int_{0}^{t}\left(u^{\varepsilon}(s),\
\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right),\ \
\ \text{for all}\ t\in[0,T].$ (2.30)

For the third term on the left-hand side of (2.2), by (2.1), we have

$\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})u^{\varepsilon}(s)dxds$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\int_{0}^{t}\|u^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds-2\int_{0}^{t}\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}\left(1+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds$
$\displaystyle\displaystyle-2\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\int_{0}^{t}\|u^{\varepsilon}(s)\|^{2}ds.$
(2.31)

For the second term on the right-hand side of (2.2), by (2.1) we obtain

$\displaystyle\displaystyle
2\int_{0}^{t}\left(g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),\
u^{\varepsilon}(s)\right)ds$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}\psi_{g}(t,x)\left(|u^{\varepsilon}(s)|+|u^{\varepsilon}(s)^{2}+|u^{\varepsilon}(s)|\sqrt{\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})}\right)dxds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
4\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\int_{0}^{t}\|u^{\varepsilon}(s)\|^{2}ds+\int_{0}^{t}\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})}(1+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2}))ds.$
(2.32)

Then by (2.2), (2.2) and (2.2), we get

$\displaystyle\displaystyle\|u^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{t}\|u^{\varepsilon}(s)\|^{p}_{L^{p}}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{0}\|^{2}+\int_{0}^{t}(2\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})})\left(1+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds$
$\displaystyle\displaystyle+2\left(\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\right)\int_{0}^{t}\|u^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle+\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds+2\int_{0}^{t}\left(u^{\varepsilon}(s),\
\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right).$
(2.33)

By (2.2), we obtain for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{0\leqslant r\leqslant
t}\left(\|u^{\varepsilon}(r)\|^{2}+2\int_{0}^{r}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{r}\|u^{\varepsilon}(s)\|^{p}_{L^{p}}ds\right)\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\mathbb{E}\left(\|u_{0}\|^{2}\right)+\int_{0}^{t}(2\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})})\left(1+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds$
$\displaystyle\displaystyle+2\left(\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\right)\int_{0}^{t}\mathbb{E}\left(\|u^{\varepsilon}(s)\|^{2}\right)ds$
$\displaystyle\displaystyle+\mathbb{E}\left(\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right)$
$\displaystyle\displaystyle+2\mathbb{E}\left(\sup_{0\leq r\leq
t}\int_{0}^{r}\left(u^{\varepsilon}(s),\
\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)\right).$
(2.34)

For the last two terms on the right-hand side of (2.2), by the Burkholder-
Davis-Gundy inequality and (2.1), we have for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right]$
$\displaystyle\displaystyle+2\mathbb{E}\left[\sup_{0\leq r\leq
t}\left|\int_{0}^{r}\left(u^{\varepsilon}(s),\
\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)\right|\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{0\leqslant
s\leqslant
t}\|u^{\varepsilon}(s)\|^{2}\right]+(1+2c^{2})\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{0\leqslant
s\leqslant
t}\|u^{\varepsilon}(s)\|^{2}\right]+(1+2c^{2})M_{\sigma,T}\mathbb{E}\left[\int_{0}^{t}\left(1+\|u^{\varepsilon}(s)\|^{2}+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds\right],$
(2.35)

where $\displaystyle c$ is a positive constant from the Burkholder-Davis-Gundy
inequality.

Then by (2.2)-(2.2), we obtain for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{0\leqslant r\leqslant
t}\left(\|u^{\varepsilon}(r)\|^{2}+2\int_{0}^{r}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{r}\|u^{\varepsilon}(s)\|^{p}_{L^{p}}ds\right)\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\mathbb{E}\left(\|u_{0}\|^{2}\right)+2\int_{0}^{t}(2\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})})\left(1+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds$
$\displaystyle\displaystyle+4\left(\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\right)\int_{0}^{t}\mathbb{E}\left(\|u^{\varepsilon}(s)\|^{2}\right)ds$
$\displaystyle\displaystyle+2(1+2c^{2})M_{\sigma,T}\mathbb{E}\left[\int_{0}^{t}\left(1+\|u^{\varepsilon}(s)\|^{2}+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds\right]$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\mathbb{E}\left(\|u_{0}\|^{2}\right)+2\int_{0}^{t}(2\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})})\left(1+\mathbb{E}\left(\|u^{\varepsilon}(s)\|^{2}\right)\right)ds$
$\displaystyle\displaystyle+4\left(\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\right)\int_{0}^{t}\mathbb{E}\left(\|u^{\varepsilon}(s)\|^{2}\right)ds$
$\displaystyle\displaystyle+2(1+2c^{2})M_{\sigma,T}\int_{0}^{t}\left(1+2\mathbb{E}(\|u^{\varepsilon}(s)\|^{2})\right)ds$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
2\mathbb{E}\left(\|u_{0}\|^{2}\right)+c_{1}+\int_{0}^{t}(c_{2}+4\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+2\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})})\mathbb{E}\left(\sup_{0\leq
r\leq s}\|u^{\varepsilon}(r)\|^{2}\right)ds,$ (2.36)

where

$\displaystyle\displaystyle
c_{1}=4\|\psi_{1}\|_{L^{1}(0,T;L^{1}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{1}(0,T;L^{1}(\mathbb{R}^{n}))}+2(1+2c^{2})M_{\sigma,T}T,$

and

$\displaystyle\displaystyle
c_{2}=4\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+8\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+4(1+2c^{2})M_{\sigma,T}.$

From (2.2) and Gronwall’s inequality, it follows that for all $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{0\leqslant r\leqslant
t}\left(\|u^{\varepsilon}(r)\|^{2}+2\int_{0}^{r}\|(-\triangle)^{\frac{\alpha}{2}}u^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{r}\|u^{\varepsilon}(s)\|^{p}_{L^{p}}ds\right)\right]$
$\displaystyle\displaystyle\leqslant\left(2\mathbb{E}\left[\|u_{0}\|^{2}\right]+c_{1}\right)e^{\int_{0}^{t}\left(c_{2}+4\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+2\|\psi_{g}(s)\|_{L^{1}(\mathbb{R}^{n})}\right)ds},$

which together with (2.2) completes the proof. ∎

##  3 Measure-dependent controlled equations on $\displaystyle\mathbb{R}^{n}$

In this section, we consider a measure-dependent controlled equation
corresponding to (2.1), which is useful for investigating the LDP of the
fractional McKean-Vlasov stochastic reaction-diffusion equations by the weak
convergence method. We first discuss the well-posedness and then derive the
uniform estimates of solutions for the controlled equation.

###  3.1 Well-posedness of controlled equations

If $\displaystyle\varepsilon=0$, then (2.1) reduces to the deterministic
reaction-diffusion equation on $\displaystyle\mathbb{R}^{n}$:

$\displaystyle\displaystyle\frac{\partial u^{0}(t)}{\partial
t}+(-\triangle)^{\alpha}u^{0}(t)+f(t,\cdot,u^{0}(t),\mathcal{L}_{u^{0}(t)})=g(t,\cdot,u^{0}(t),\mathcal{L}_{u^{0}(t)}),$
(3.1)

with initial data

$\displaystyle\displaystyle u^{0}(0)=u_{0},$ (3.2)

where the distribution $\displaystyle\mathcal{L}_{u^{0}(t)}$ of
$\displaystyle{u^{0}(t)}$ is the Dirac measure
$\displaystyle\delta_{u^{0}(t)}$.

Similar to Theorem 2.1, problem (3.1)-(3.2) is well-posed in $\displaystyle H$
as stated below.

######  Theorem 3.1.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 2})$ hold. If
$\displaystyle T>0$, $\displaystyle u_{0}\in H$, then problem (3.1)-(3.2) has
a unique solution $\displaystyle u^{0}\in C([0,T],H)\bigcap
L^{2}(0,T;V)\bigcap L^{p}(0,T;L^{p}(\mathbb{R}^{n}))$. Moreover, the following
uniform estimates are valid:

$\displaystyle\displaystyle\|u^{0}\|_{C([0,T],H)}^{2}+\|u^{0}\|_{L^{2}(0,T;V)}^{2}+\|u^{0}\|_{L^{p}(0,T;L^{p}(\mathbb{R}^{n}))}^{p}\leqslant
M_{0}\left(1+\|u_{0}\|^{2}\right),$

where $\displaystyle M_{0}=M_{0}(T)>0$ is independent of $\displaystyle
u_{0}$.

We now consider the controlled equation on $\displaystyle\mathbb{R}^{n}$ for a
control $\displaystyle v\in L^{2}(0,T;l^{2})$:

$\displaystyle\displaystyle\frac{\partial u_{v}(t)}{\partial
t}+(-\triangle)^{\alpha}u_{v}(t)+f(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})=g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})+\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),$
(3.3)

with initial data

$\displaystyle\displaystyle u_{v}(0)=u_{0},$ (3.4)

where $\displaystyle u^{0}(t)$ is the solution to (3.1)-(3.2).

######  Remark 3.1.

Note that the controlled equation (3.3) associated with the McKean-Vlasov
stochastic equation (2.1) depends on distributions of the solutions of the
deterministic equation (3.1). This is quite different from the usual
controlled equations associated with the stochastic equations which are
independent of distributions, see [63, Remark 2.2] for more details in this
regard.

The next theorem is concerned with the well-posedness of (3.3)-(3.4).

######  Theorem 3.2.

Suppose that $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$
hold. Then for every $\displaystyle v\in L^{2}(0,T;l^{2})$, problem
(3.3)-(3.4) has a unique solution $\displaystyle u_{v}\in C([0,T],H)\bigcap
L^{2}(0,T;V)\bigcap L^{p}(0,T;L^{p}(\mathbb{R}^{n})).$

Moreover, for any $\displaystyle\|u_{0}\|\vee\|u_{0,i}\|\leqslant R_{1}$, and
$\displaystyle\|v\|_{L^{2}(0,T;l^{2})}\vee\|v_{i}\|_{L^{2}(0,T;l^{2})}\leqslant
R_{2}$, $\displaystyle i=1,2$, the solutions $\displaystyle u_{v}(t)$ and
$\displaystyle u_{v_{i}}(t)$ of (3.3) with initial data $\displaystyle u_{0}$
and $\displaystyle u_{0,i}$ respectively, satisfy for any $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle\|u_{v}(t)\|^{2}+\int_{0}^{T}\|u_{v}(s)\|_{V}^{2}ds+\int_{0}^{T}\|u_{v}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds\leqslant
M_{2}$ (3.5)

and

$\displaystyle\displaystyle\|u_{v_{1}}(t)-u_{v_{2}}(t)\|^{2}+\int_{0}^{T}\|u_{v_{1}}(s)-u_{v_{2}}(s)\|_{V}^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{2}\left(\|u_{0,1}-u_{0,2}\|^{2}+\|v_{1}-v_{2}\|^{2}_{L^{2}(0,T;l^{2})}\right),$
(3.6)

where the constant $\displaystyle M_{2}>0$ depends on $\displaystyle
R_{1},R_{2}$ and $\displaystyle T$.

###### Proof.

Given $\displaystyle v\in L^{2}(0,T;l^{2})$, by $\displaystyle({\bf\Sigma
1})$-$\displaystyle({\bf\Sigma 3})$ and the argument of [11], one can verify
that for every $\displaystyle u_{0}\in H$, problem (3.3)-(3.4) has a unique
solution $\displaystyle u_{v}$ in the space

$\displaystyle C([0,T],H)\bigcap L^{2}(0,T;V)\bigcap
L^{p}(0,T;L^{p}(\mathbb{R}^{n})).$

Next, we derive the uniform estimates on the solutions of (3.3)-(3.4). Note
that

$\displaystyle\displaystyle\frac{d}{dt}\|u_{v}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(t)\|^{2}+2\int_{\mathbb{R}^{n}}f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})u_{v}(t)dx$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\left(g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),u_{v}(t)\right)+2\left(\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),u_{v}(t)\right).$
(3.7)

From (2.1) and Theorem 3.1, it follows that

$\displaystyle\displaystyle
2\int_{\mathbb{R}^{n}}f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})u_{v}(t)dx$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\|u_{v}(t)\|_{L^{p}(\mathbb{R}^{n})}^{p}-2\|\psi_{1}(t)\|_{L^{1}(\mathbb{R}^{n})}\left(1+\mathcal{L}_{u^{0}(t)}(\|\cdot\|^{2})\right)-2\|\psi_{1}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\|u_{v}(t)\|_{L^{p}(\mathbb{R}^{n})}^{p}-2\|\psi_{1}(t)\|_{L^{1}(\mathbb{R}^{n})}\left(1+\|u^{0}(t)\|^{2})\right)-2\|\psi_{1}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\|u_{v}(t)\|_{L^{p}(\mathbb{R}^{n})}^{p}-2\|\psi_{1}(t)\|_{L^{1}(\mathbb{R}^{n})}\left[1+M_{0}(1+R_{1}^{2})\right]-2\|\psi_{1}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v}(t)\|^{2}.$
(3.8)

By (2.1) and Theorem 3.1, we have

$\displaystyle\displaystyle
2\left(g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),u_{v}(t)\right)$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{v}(t)\|^{2}+\|g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|^{2}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{v}(t)\|^{2}+4\|\psi_{g}(t)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|u_{v}(t)\|^{2}+2\|\psi_{g}(t)\|^{2}\left(1+2M_{0}(1+R_{1}^{2})\right).$
(3.9)

By (2.1) and Theorem 3.1, we have

$\displaystyle\displaystyle
2\left(\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),u_{v}(t)\right)$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|^{2}_{L_{2}(l^{2},H)}+\|v(t)\|_{l^{2}}^{2}\|u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{\sigma,T}\left(1+\|u^{0}(t)\|^{2}\right)+\left(M_{\sigma,T}+\|v(t)\|_{l^{2}}^{2}\right)\|u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{\sigma,T}\left[1+M_{0}\left(1+R_{1}^{2}\right)\right]+\left(M_{\sigma,T}+\|v(t)\|_{l^{2}}^{2}\right)\|u_{v}(t)\|^{2}.$
(3.10)

Then by (3.1)-(3.1), we obtain

$\displaystyle\displaystyle\frac{d}{dt}\|u_{v}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(t)\|^{2}+2\lambda_{1}\|u_{v}(t)\|_{L^{p}(\mathbb{R}^{n})}^{p}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(1+2\|\psi_{1}(t)\|_{L^{\infty}(\mathbb{R}^{n})}+4\|\psi_{g}(t)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}+M_{\sigma,T}+\|v(t)\|_{l^{2}}^{2}\right)\|u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+2\|\psi_{1}(t)\|_{L^{1}(\mathbb{R}^{n})}\left[1+M_{0}(1+R_{1}^{2})\right]+2\|\psi_{g}(t)\|^{2}(1+2M_{0}(1+R_{1}^{2}))$
$\displaystyle\displaystyle+M_{\sigma,T}\left[1+M_{0}(1+R_{1}^{2})\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(c+\|v(t)\|_{l^{2}}^{2}\right)\|u_{v}(t)\|^{2}+c\left(1+\|\psi_{1}(t)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(t)\|^{2}\right),$
(3.11)

where $\displaystyle c=c(T,R_{1})>0$

By (3.1) and Gronwall’s inequality, we obtain for all $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle\|u_{v}(t)\|^{2}\leqslant$
$\displaystyle\displaystyle\|u_{0}\|^{2}e^{\int^{t}_{0}\left(c+\|v(s)\|_{l^{2}}^{2}\right)ds}+c\int_{0}^{t}e^{\int_{s}^{t}\left(c+\|v(r)\|_{l^{2}}^{2}\right)dr}\left(1+\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|^{2}\right)ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
R_{1}^{2}e^{cT+R_{2}^{2}}+ce^{cT+R_{2}^{2}}\left(T+\|\psi_{1}\|_{L^{1}(0,T;L^{1}(\mathbb{R}^{n}))}+\|\psi_{g}\|_{L^{2}(0,T;H)}^{2}\right).$
(3.12)

By (3.1)-(3.1) we see that there exists $\displaystyle
c_{1}=c_{1}(T,R_{1},R_{2})>0$ such that

$\displaystyle\displaystyle\|u_{v}(t)\|^{2}+\int_{0}^{t}2\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{t}\|u_{v}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds\leqslant
c_{1},\ \ \ \forall\ t\in[0,T].$ (3.13)

Then (3.5) follows from (3.13).

We now prove (3.2), which implies the uniqueness of solutions to (3.3)-(3.4).
Let $\displaystyle u_{1}(t)=u_{v_{1}}(t,u_{0,1})$ and $\displaystyle
u_{2}(t)=u_{v_{2}}(t,u_{0,2})$ be the solutions of (3.3)-(3.4) with initial
data $\displaystyle u_{0,1}$ and $\displaystyle u_{0,2}$ and controls
$\displaystyle v_{1}$ and $\displaystyle v_{2}$, respectively.

By (3.5), we find that if $\displaystyle\|u_{0,1}\|\vee\|u_{0,2}\|\leqslant
R_{1}$ and
$\displaystyle\|v_{1}\|_{L^{2}(0,T;l^{2})}\vee\|v_{2}\|_{L^{2}(0,T;l^{2})}\leqslant
R_{2}$, then for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|u_{1}(t)\|^{2}\leqslant M_{2}\ \text{ and }\
\|u_{2}(t)\|^{2}\leqslant M_{2}.$ (3.14)

By (3.3), we obtain

$\displaystyle\displaystyle\frac{d}{dt}\|u_{1}(t)-u_{2}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{1}(t)-u_{2}(t)\right)\|^{2}$
$\displaystyle\displaystyle+2\int_{\mathbb{R}^{n}}\left(f(t,x,u_{1}(t),\mathcal{L}_{u_{0}(t)})-f(t,x,u_{2}(t),\mathcal{L}_{u_{0}(t)})\right)\left(u_{1}(t)-u_{2}(t)\right)dx$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\left(g(t,\cdot,u_{1}(t),\mathcal{L}_{u_{0}(t)})-g(t,\cdot,u_{2}(t),\mathcal{L}_{u_{0}(t)}),\
u_{1}(t)-u_{2}(t)\right)$
$\displaystyle\displaystyle+2\left(\sigma(t,u_{1}(t),\mathcal{L}_{u_{0}(t)})v_{1}(t)-\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})v_{2}(t),\
u_{1}(t)-u_{2}(t)\right).$ (3.15)

For the third term on the left-and side of (3.1), by (2.4), we have

$\displaystyle\displaystyle-2\int_{\mathbb{R}^{n}}\left(f(t,x,u_{1}(t),\mathcal{L}_{u_{0}(t)})-f(t,x,u_{2}(t),\mathcal{L}_{u_{0}(t)})\right)\left(u_{1}(t)-u_{2}(t)\right)dx$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\|\psi_{4}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{1}(t)-u_{2}(t)\|^{2}.$
(3.16)

For the first term on the right-hand side of (3.1), by (2.6), we obtain

$\displaystyle\displaystyle
2\left(g(t,\cdot,u_{1}(t),\mathcal{L}_{u_{0}(t)})-g(t,\cdot,u_{2}(t),\mathcal{L}_{u_{0}(t)}),\
u_{1}(t)-u_{2}(t)\right)$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle
2\|g(t,\cdot,u_{1}(t),\mathcal{L}_{u_{0}(t)})-g(t,\cdot,u_{2}(t),\mathcal{L}_{u_{0}(t)})\|\|u_{1}(t)-u_{2}(t)\|$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\|u_{1}(t)-u_{2}(t)\|^{2}.$
(3.17)

For the second term on the right-hand side of (3.1), by (2.1), (3.14), and
Theorem 3.1, we get

$\displaystyle\displaystyle
2\left(\sigma(t,u_{1}(t),\mathcal{L}_{u_{0}(t)})v_{1}(t)-\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})v_{2}(t),\
u_{1}(t)-u_{2}(t)\right)$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle
2\|\left(\sigma(t,u_{1}(t),\mathcal{L}_{u_{0}(t)})-\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})\right)v_{1}(t)\|\|u_{1}(t)-u_{2}(t)\|$
$\displaystyle\displaystyle+2\|\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})\left(v_{1}(t)-v_{2}(t)\right)\|\|u_{1}(t)-u_{2}(t)\|$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\|\sigma(t,u_{1}(t),\mathcal{L}_{u_{0}(t)})-\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})\|_{L_{2}(l^{2},H)}\|v_{1}(t)\|_{l^{2}}\|u_{1}(t)-u_{2}(t)\|$
$\displaystyle\displaystyle+\|\sigma(t,u_{2}(t),\mathcal{L}_{u_{0}(t)})\|_{L_{2}(l^{2},H)}^{2}\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2L_{\sigma}^{\frac{1}{2}}\|v_{1}(t)\|_{l^{2}}\|u_{1}(t)-u_{2}(t)\|^{2}$
$\displaystyle\displaystyle+M_{\sigma,T}\left(1+\|u_{2}(t)\|^{2}+\|u^{0}(t)\|^{2}\right)\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2L_{\sigma}^{\frac{1}{2}}\|v_{1}(t)\|_{l^{2}}\|u_{1}(t)-u_{2}(t)\|^{2}$
$\displaystyle\displaystyle+M_{\sigma,T}\left[1+M_{2}+M_{0}\left(1+R_{1}^{2}\right)\right]\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{1}\left(1+\|v_{1}(t)\|_{l^{2}}\right)\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2},$
(3.18)

where $\displaystyle
c_{1}=2L_{\sigma}^{\frac{1}{2}}+M_{\sigma,T}\left[1+M_{2}+M_{0}\left(1+R_{1}^{2}\right)\right]$.

It follows from (3.1)-(3.1) that for $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\frac{d}{dt}\|u_{1}(t)-u_{2}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{1}(t)-u_{2}(t)\right)\|^{2}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(2\|\psi_{4}(t)\|_{L^{\infty}(\mathbb{R}^{n})}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+c_{1}+c_{1}\|v_{1}(t)\|_{l^{2}}\right)\|u_{1}(t)-u_{2}(t)\|^{2}$
$\displaystyle\displaystyle+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{2}\left(1+\|v_{1}(t)\|_{l^{2}}\right)\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}(t)-v_{2}(t)\|_{l^{2}}^{2},$
(3.19)

where $\displaystyle
c_{2}=2\|\psi_{4}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+2\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+c_{1}.$

By applying Gronwall’s inequality to (3.1), we get for all $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle\|u_{1}(t)-u_{2}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{0,1}-u_{0,2}\|^{2}e^{c_{2}\int_{0}^{t}\left(1+\|v_{1}(r)\|_{l^{2}}\right)dr}+\int_{0}^{t}e^{c_{2}\int_{s}^{t}\left(1+\|v_{1}(r)\|_{l^{2}}\right)dr}\|v_{1}(s)-v_{2}(s)\|_{l^{2}}^{2}ds,$

from which we obtain for any $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|u_{1}(t)-u_{2}(t)\|^{2}\leqslant$
$\displaystyle\displaystyle
e^{c_{2}T+c_{2}T^{\frac{1}{2}}R_{2}}\left(\|u_{0,1}-u_{0,2}\|^{2}+\|v_{1}-v_{2}\|_{L^{2}(0,T;l^{2})}^{2}\right).$
(3.20)

Integrating (3.1), we obtain by (3.20) that

$\displaystyle\displaystyle
2\int_{0}^{T}\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{1}(s)-u_{2}(s)\right)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{0,1}-u_{0,2}\|^{2}+c_{2}\left(T+T^{\frac{1}{2}}R_{2}\right)\sup_{t\in[0,T]}\|u_{1}(t)-u_{2}(t)\|^{2}+\|v_{1}-v_{2}\|_{L^{2}(0,T;l^{2})}^{2}.$
(3.21)

Then, (3.2) follows from (3.20) and (3.1). ∎

###  3.2 Uniform tail-estimates of solutions

In this subsection, we derive the uniform estimates on the tails of solutions
to (3.3)-(3.4) outside a bounded domain, which are crucial for establishing
the precompactness of solutions.

######  Lemma 3.1.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$ hold,
$\displaystyle T>0$ and $\displaystyle u_{0}\in H$. Then for any
$\displaystyle R>0$ and $\displaystyle\delta>0$, there exists $\displaystyle
m_{0}=m_{0}(u_{0},T,R,\delta)>0$ such that for all $\displaystyle v\in
L^{2}(0,T;l^{2})$ with $\displaystyle\|v\|_{L^{2}(0,T;l^{2})}\leqslant R$, the
solution $\displaystyle u_{v}$ to problem (3.3)-(3.4) satisfies

$\displaystyle\displaystyle\int_{|x|\geqslant m}|u_{v}(t,x)|^{2}dx<\delta,\ \
\forall\ m\geqslant m_{0},\ t\in[0,T].$

###### Proof.

Let $\displaystyle\zeta:\mathbb{R}^{n}\rightarrow[0,1]$ be a smooth function
such that $\displaystyle\zeta(x)=0$ for $\displaystyle|x|\leqslant\frac{1}{2}$
and $\displaystyle\zeta(x)=1$ for $\displaystyle|x|\geqslant 1$.

Given $\displaystyle m\in\mathbb{N}$, let
$\displaystyle\zeta_{m}(x)=\zeta(\frac{x}{m})$, then by (3.3), we have

$\displaystyle\displaystyle\frac{\partial}{\partial
t}\zeta_{m}u_{v}(t)+\zeta_{m}(-\triangle)^{\alpha}u_{v}(t)+\zeta_{m}f(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\zeta_{m}g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})+\zeta_{m}\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t).$

Then we get

$\displaystyle\displaystyle\frac{d}{dt}\|\zeta_{m}u_{v}(t)\|^{2}+2\left((-\triangle)^{\frac{\alpha}{2}}u_{v}(t),(-\triangle)^{\frac{\alpha}{2}}\left(\zeta_{m}^{2}u_{v}(t)\right)\right)$
$\displaystyle\displaystyle+2\int_{\mathbb{R}^{n}}f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})\zeta_{m}^{2}(x)u_{v}(t)dx$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\left(\zeta_{m}g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),\zeta_{m}u_{v}(t)\right)+2\left(\zeta_{m}\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),\zeta_{m}u_{v}(t)\right).$
(3.22)

For the second term on the left-hand side of (3.2), by (8.18) in
[wangbJDE2019], we find that there exists a positive constant $\displaystyle
c_{1}$ independent of $\displaystyle m$ such that for all $\displaystyle
t\geqslant 0$,

$\displaystyle\displaystyle-2\left((-\triangle)^{\frac{\alpha}{2}}u_{v}(t),(-\triangle)^{\frac{\alpha}{2}}\left(\zeta_{m}^{2}u_{v}(t)\right)\right)$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{1}C(n,\alpha)m^{-\alpha}\|u_{v}(t)\|^{2}+2c_{1}m^{-\alpha}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(t)\|^{2}.$
(3.23)

For the third term on the left-hand side of (3.2), by (2.1) and Theorem 3.1,
we have

$\displaystyle\displaystyle
2\int_{\mathbb{R}^{n}}f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})\zeta_{m}^{2}(x)u_{v}(t)dx$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\int_{\mathbb{R}^{n}}\zeta_{m}^{2}(x)\psi_{1}(t,x)(1+|u_{v}(t)|^{2}+\mathcal{L}_{u^{0}(t)}(\|\cdot\|^{2}))dx$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2(1+\|u^{0}(t)\|^{2}))\int_{|x|\geqslant\frac{m}{2}}|\psi_{1}(t,x)|dx-2\int_{\mathbb{R}^{n}}\zeta_{m}^{2}(x)\psi_{1}(t,x)|u_{v}(t)|^{2}dx$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\int_{|x|\geqslant\frac{m}{2}}|\psi_{1}(t,x)|dx-2\|\psi_{1}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|\zeta_{m}u_{v}(t)\|^{.}$
(3.24)

For the first term on the right-hand side of (3.2), by (2.1), Young’s
inequality and Theorem 3.1, we obtain

$\displaystyle\displaystyle
2\left(\zeta_{m}g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),\zeta_{m}u_{v}(t)\right)\leqslant\|\zeta_{m}g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|^{2}+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
4\int_{\mathbb{R}^{n}}\zeta^{2}_{m}(x)|\psi_{g}(t,x)|^{2}\left|u_{v}(t)\right|^{2}dx+4\int_{\mathbb{R}^{n}}\zeta^{2}_{m}(x)|\psi_{g}(t,x)|^{2}\|u^{0}(t)\|^{2}dx$
$\displaystyle\displaystyle+2\int_{\mathbb{R}^{n}}\zeta^{2}_{m}(x)|\psi_{g}(t,x)|^{2}dx+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(1+4\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}^{2}\right)\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+2\left[1+2M_{0}\left(1+\|u_{0}\|^{2}\right)\right]\int_{|x|\geqslant\frac{m}{2}}|\psi_{g}(t,x)|^{2}dx.$
(3.25)

For the second term on the right-hand side of (3.2), by (2.10) and Theorem
3.1, we obtain

$\displaystyle\displaystyle
2\left(\zeta_{m}\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),\zeta_{m}u_{v}(t)\right)$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|\zeta_{m}\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|_{L_{2}(l^{2},H)}^{2}\|v(t)\|_{l^{2}}^{2}+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{k=1}^{\infty}\|\zeta_{m}\sigma_{1,k}(t)+\zeta_{m}\kappa\sigma_{2,k}(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|^{2}\|v(t)\|_{l^{2}}^{2}+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\sum_{k=1}^{\infty}\|\zeta_{m}\sigma_{1,k}(t)\|^{2}\|v(t)\|_{l^{2}}^{2}+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+4\sum_{k=1}^{\infty}\left[2\beta_{k}^{2}\left(1+\mathcal{L}_{u^{0}(t)}(\|\cdot\|^{2})\right)\|\zeta_{m}\kappa\|^{2}+\gamma_{k}^{2}\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|\zeta_{m}u_{v}(t)\|^{2}\right]\|v(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(t,x)|^{2}dx\|v(t)\|_{l^{2}}^{2}+\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+8\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\sum_{k=1}^{\infty}\beta_{k}^{2}\int_{|x|\geqslant\frac{m}{2}}|\kappa(x)|^{2}dx\|v(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\sum_{k=1}^{\infty}\gamma_{k}^{2}\|\zeta_{m}u_{v}(t)\|^{2}\|v(t)\|_{l^{2}}^{2}.$
(3.26)

Then from (3.2)-(3.2), it follows that for $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\frac{d}{dt}\|\zeta_{m}u_{v}(t)\|^{2}\leqslant$
$\displaystyle\displaystyle
c_{1}C(n,\alpha)m^{-\alpha}\|u_{v}(t)\|^{2}+2c_{1}m^{-\alpha}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+\left(2+4\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}^{2}+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\sum_{k=1}^{\infty}\gamma_{k}^{2}\|v(t)\|_{l^{2}}^{2}\right)\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+2\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\int_{|x|\geqslant\frac{m}{2}}\left(|\psi_{1}(t,x)|+|\psi_{g}(t,x)|^{2}\right)dx$
$\displaystyle\displaystyle+2\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(t,x)|^{2}dx\|v(t)\|_{l^{2}}^{2}$
$\displaystyle\displaystyle+8\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\sum_{k=1}^{\infty}\beta_{k}^{2}\int_{|x|\geqslant\frac{m}{2}}|\kappa(x)|^{2}dx\|v(t)\|_{l^{2}}^{2}.$
(3.27)

By Gronwall’s inequality and (3.2), we obtain for any $\displaystyle
t\in[0,T]$,

$\displaystyle\displaystyle\|\zeta_{m}u_{v}(t)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
e^{c_{2}\int_{0}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|\zeta_{m}u_{0}\|^{2}+c_{1}C(n,\alpha)m^{-\alpha}\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|u_{v}(s)\|^{2}ds$
$\displaystyle\displaystyle+2c_{1}m^{-\alpha}\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(s)\|^{2}ds$
$\displaystyle\displaystyle+2\left[1+2M_{0}(1+\|u_{0}\|^{2})\right]\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\int_{|x|\geqslant\frac{m}{2}}\left(|\psi_{1}(s,x)|+|\psi_{g}(s,x)|^{2}\right)dxds$
$\displaystyle\displaystyle+2\int_{0}^{T}\left(e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|v(s)\|_{l^{2}}^{2}\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(s,x)|^{2}dx\right)ds$
$\displaystyle\displaystyle+8\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\sum_{k=1}^{\infty}\beta_{k}^{2}\int_{|x|\geqslant\frac{m}{2}}|\kappa(x)|^{2}dx\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|\theta(r)\|_{l^{2}}^{2}\right)dr}\|v(s)\|_{l^{2}}^{2}ds,$
(3.28)

where $\displaystyle
c_{2}=\left(2+4\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}^{2}\right)+4\|\kappa\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\sum_{k=1}^{\infty}\gamma_{k}^{2}.$
For the first term on the right-hand side of (3.2), we have

$\displaystyle\displaystyle
e^{c_{2}\int_{0}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|\zeta_{m}u_{0}\|^{2}\leqslant
e^{c_{2}(T+R^{2})}\int_{|x|\geqslant\frac{m}{2}}|u_{0}(x)|^{2}\rightarrow\ 0,\
\ \ \text{as}\ m\to+\infty.$ (3.29)

For the second and third terms on the right-hand side of (3.2), by Theorem
3.2, we obtain

$\displaystyle\displaystyle
c_{1}C(n,\alpha)m^{-\alpha}\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|u_{v}(s)\|^{2}ds$
$\displaystyle\displaystyle+2c_{1}m^{-\alpha}\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(s)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{1}m^{-\alpha}e^{c_{2}(T+R^{2})}\left(C(n,\alpha)\vee
2\right)\int_{0}^{T}\left(\|u_{v}(s)\|^{2}+\|(-\triangle)^{\frac{\alpha}{2}}u_{v}(s)\|^{2}\right)ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{1}m^{-\alpha}e^{c_{2}(T+R^{2})}\left(C(n,\alpha)\vee 2\right)M_{2}$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle\ 0,\ \ \
\text{as}\ m\to+\infty.$ (3.30)

For the fourth term on the right-hand side of (3.2), by
$\displaystyle\psi_{1}\in L_{loc}^{1}(\mathbb{R},L^{1}(\mathbb{R}^{n}))$ and
$\displaystyle\psi_{g}\in L_{loc}^{2}(\mathbb{R},L^{2}(\mathbb{R}^{n}))$, we
obtain

$\displaystyle\displaystyle
2\left[1+2M_{0}(1+\|u_{0}\|^{2})\right]\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\int_{|x|\geqslant\frac{m}{2}}\left(|\psi_{1}(s,x)|+|\psi_{g}(s,x)|^{2}\right)dxds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\left[1+2M_{0}(1+\|u_{0}\|^{2})\right]e^{c_{2}\left(T+R^{2}\right)}\int_{0}^{T}\int_{|x|\geqslant\frac{m}{2}}\left(|\psi_{1}(s,x)|+|\psi_{g}(s,x)|^{2}\right)dxds$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle\ 0,\ \ \
\text{as}\ m\to+\infty.$ (3.31)

For the fifth term on the right-hand side of (3.2), we have

$\displaystyle\displaystyle
2\int_{0}^{T}\left(e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|v(s)\|_{l^{2}}^{2}\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(s,x)|^{2}dx\right)ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2e^{c_{2}\left(T+R^{2}\right)}\int_{0}^{T}\left(\|v(s)\|_{l^{2}}^{2}\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(s,x)|^{2}dx\right)ds.$
(3.32)

Since $\displaystyle\sigma_{1}:[0,T]\rightarrow L^{2}(\mathbb{R}^{n},l^{2})$
is continuous due to $\displaystyle({\bf\Sigma 3})$, the set
$\displaystyle\\{\sigma_{1}(s):s\in[0,T]\\}$ is a compact subset of
$\displaystyle L^{2}(\mathbb{R}^{n},l^{2})$, which implies that for any
$\displaystyle\delta>0$, there exists $\displaystyle k\in\mathbb{N}$ and
$\displaystyle s_{1},\cdots,s_{k}\in[0,T]$ such that the open balls centered
at $\displaystyle\sigma_{1}(s_{i})\in L^{2}(\mathbb{R}^{n},l^{2})$
$\displaystyle(i=1,\cdots,k)$ with radius $\displaystyle\frac{\delta}{2}$ form
a finite open cover of the set $\displaystyle\\{\sigma_{1}(s):s\in[0,T]\\}$ in
$\displaystyle L^{2}(\mathbb{R}^{n},l^{2})$. Therefore, we infer that there
exists $\displaystyle m_{0}>0$ such that for every $\displaystyle m\geqslant
m_{0}$ and $\displaystyle s\in[0,T]$

$\displaystyle\displaystyle\left(\int_{|x|\geqslant\frac{m}{2}}\sum_{k=1}^{\infty}|\sigma_{1,k}(s,x)|^{2}dx\right)^{\frac{1}{2}}<\delta.$
(3.33)

By (3.2) and (3.33), we obtain

$\displaystyle\displaystyle
2\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|v(r)\|_{l^{2}}^{2}\right)dr}\|v(s)\|_{l^{2}}^{2}\sum_{k=1}^{\infty}\int_{|x|\geqslant\frac{m}{2}}|\sigma_{1,k}(s,x)|^{2}dxds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle 2\delta
e^{c_{2}\left(T+R^{2}\right)}\int_{0}^{T}\|v(s)\|_{l^{2}}^{2}ds\leqslant
2\delta R^{2}e^{c_{2}\left(T+R^{2}\right)}.$ (3.34)

For the sixth term on the right-hand side of (3.2), by $\displaystyle\kappa\in
L^{2}(\mathbb{R}^{n})$ we get

$\displaystyle\displaystyle
8\left[1+M_{0}(1+\|u_{0}\|^{2})\right]\sum_{k=1}^{\infty}\beta_{k}^{2}\int_{|x|\geqslant\frac{m}{2}}|\kappa(x)|^{2}dx\int_{0}^{T}e^{c_{2}\int_{s}^{T}\left(1+\|\theta(r)\|_{l^{2}}^{2}\right)dr}\|v(s)\|_{l^{2}}^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
8\left[1+M_{0}(1+\|u_{0}\|^{2})\right]R^{2}e^{c_{2}\left(T+R^{2}\right)}\sum_{k=1}^{\infty}\beta_{k}^{2}\int_{|x|\geqslant\frac{m}{2}}|\kappa(x)|^{2}dx$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle\ 0,\ \ \
\text{as}\ m\to+\infty.$ (3.35)

It follows from (3.2)-(3.2) and (3.2)-(3.2) that there exists $\displaystyle
m_{1}=m_{1}(u_{0},T,R,\delta)>0$ such that for all $\displaystyle m\geqslant
m_{1}$ and $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\int_{|x|\geqslant
m}|u_{v}(t,x)|^{2}dx<c_{2}\delta,$

where $\displaystyle c_{2}=c_{2}(u_{0},T,R)>0$ is independent of
$\displaystyle\delta$, as desired. ∎

##  4 Large deviations of McKean-Vlasov stochastic equations

In this section, we prove the LDP of the solutions of the fractional McKean-
Vlasov stochastic reaction-diffusion equation (1)-(1.2) on
$\displaystyle\mathbb{R}^{n}$ by the weak convergence method.

###  4.1 Weak convergence theory of large deviations

We now review the basic concepts and results of the weak converge theory for
the LDP.

Let $\displaystyle\mathbb{S}$ be a polish space. For any
$\displaystyle\varepsilon>0$, let
$\displaystyle\mathcal{G}^{\varepsilon}:C([0,T],U)\rightarrow\mathbb{S}$ be a
measurable map. Denote by $\displaystyle
X^{\varepsilon}=\mathcal{G}^{\varepsilon}(W)$ for all
$\displaystyle\varepsilon>0$.

######  Definition 4.1.

A function $\displaystyle I:\mathbb{S}\rightarrow[0,\infty]$ is called a rate
function on $\displaystyle\mathbb{S}$ if it is lower semicontinuous in
$\displaystyle\mathbb{S}$. A rate function $\displaystyle I$ on
$\displaystyle\mathbb{S}$ is said to be a good rate function on
$\displaystyle\mathbb{S}$ if for every $\displaystyle 0\leqslant C<\infty$,
the level set $\displaystyle\\{x\in\mathbb{S}:I(x)\leqslant C\\}$ is a compact
subset of $\displaystyle\mathbb{S}$.

######  Definition 4.2.

The family $\displaystyle\\{X^{\varepsilon}\\}$ is said to satisfy the LDP in
$\displaystyle\mathbb{S}$ with a rate function $\displaystyle
I:\mathbb{S}\rightarrow[0,\infty]$, if for every Borel subset $\displaystyle
A$ of $\displaystyle\mathbb{S}$,

$\displaystyle\displaystyle-\inf_{x\in
A^{\circ}}I(x)\leqslant\liminf_{\varepsilon\rightarrow
0}\varepsilon\log\mathbb{P}(X^{\varepsilon}\in
A)\leqslant\limsup_{\varepsilon\rightarrow
0}\varepsilon\log\mathbb{P}(X^{\varepsilon}\in
A)\leqslant-\inf_{x\in\overline{A}}I(x),$

where $\displaystyle A^{\circ}$ and $\displaystyle\overline{A}$ are the
interior and the closure of $\displaystyle A$ in $\displaystyle\mathbb{S}$,
respectively.

Given $\displaystyle N>0$, denote by

$S_{N}=\left\\{v\in
L^{2}(0,T;l^{2}):\int_{0}^{T}\|v(t)\|^{2}_{l^{2}}dt\leqslant N\right\\}.$

Then $\displaystyle S_{N}$ is a polish space endowed with the weak topology.
Let $\displaystyle\mathcal{A}$ be the space of all $\displaystyle
l^{2}$-valued stochastic processes $\displaystyle v$ which are progressively
measurable with respect to $\displaystyle\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$
and $\displaystyle\int_{0}^{T}\|v(t)\|_{l^{2}}^{2}dt<\infty$
$\displaystyle\mathbb{P}$-almost surely. Denote by
$\displaystyle\mathcal{A}_{N}=\\{v\in\mathcal{A}:v(\omega)\in S_{N}\ \text{for
almost all }\ \omega\in\Omega\\}$.

In order to prove the LDP of $\displaystyle X^{\varepsilon}$, we will assume
that the family $\displaystyle\left\\{\mathcal{G}^{\varepsilon}\right\\}$
fulfills the following conditions: there exists a measurable map
$\displaystyle\mathcal{G}^{0}:C([0,T],U)\rightarrow\mathbb{S}$ such that:

$\displaystyle({\bf H1})$ For every $\displaystyle N<\infty$ and sequence
$\displaystyle\\{v_{i}:i\in\mathbb{N}\\}\subseteq S_{N}$, if $\displaystyle
v_{i}$ converges to $\displaystyle v$ in $\displaystyle S_{N}$ as
$\displaystyle\varepsilon\rightarrow 0$, then
$\displaystyle\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v_{i}(s)ds\big{)}$
converges to
$\displaystyle\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v(s)ds\big{)}$ in
$\displaystyle\mathbb{S}$.

$\displaystyle({\bf H2})$ For every $\displaystyle N<\infty$,
$\displaystyle\\{v^{\varepsilon}:\varepsilon>0\\}\subseteq\mathcal{A}^{N}$ and
$\displaystyle\delta>0$,

$\displaystyle\lim_{\varepsilon\to
0}\mathbb{P}\left(\rho\Big{(}\mathcal{G}^{\varepsilon}\big{(}W+\varepsilon^{-\frac{1}{2}}\int_{0}^{\cdot}v^{\varepsilon}(t)dt\big{)},\
\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v^{\varepsilon}(t)dt\big{)}\Big{)}>\delta\right)=0,$

where $\displaystyle\rho(\cdot,\cdot)$ stands for the metric in
$\displaystyle\mathbb{S}$.

We recall the following theorem from [54, Theorem 3.2] and [58, Theorem 4.4]
which is essentially established by Budhiraja and Dupuis [43, Theorem 4.4].

######  Proposition 4.1.

[54, 58] Under $\displaystyle({\bf H1})$-$\displaystyle({\bf H2})$, the family
$\displaystyle\\{X^{\varepsilon}\\}$ satisfies the LDP in
$\displaystyle\mathbb{S}$ with rate function $\displaystyle I$ given by,

$\displaystyle\displaystyle I(x)=\inf_{v\in
L^{2}(0,T;l^{2}):x=\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v(t)dt\big{)}}\left\\{\frac{1}{2}\int_{0}^{T}\|v(t)\|_{l^{2}}^{2}dt\right\\},\
\ \ x\in\mathbb{S},$

with convention $\displaystyle\inf\\{\emptyset\\}=\infty$.

###  4.2 Main results

In this subsection, we present the result on the LDP of (2.1)-(2.14) in
$\displaystyle C([0,T],H)\bigcap L^{2}(0,T;V)$ as follows.

######  Theorem 4.1.

Suppose that $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$
hold, and $\displaystyle u^{\varepsilon}$ is the solution of (2.1)-(2.14).
Then as $\displaystyle\varepsilon\rightarrow 0$, the family
$\displaystyle\\{u^{\varepsilon}\\}$ satisfies the LDP in $\displaystyle
C([0,T],H)\bigcap L^{2}(0,T;V)$ with good rate function given by

$\displaystyle\displaystyle
I(\phi)=\inf\left\\{\frac{1}{2}\int_{0}^{T}\|v(t)\|_{l^{2}}^{2}dt:v\in
L^{2}(0,T;l^{2}),\ u_{v}=\phi\right\\},$ (4.1)

where $\displaystyle\phi\in C([0,T],H)\bigcap L^{2}(0,T;V)$, and
$\displaystyle u_{v}$ is the solution to (3.3)-(3.4). As usual, the infimum of
the empty set is taken to be $\displaystyle\infty$.

To prove Theorem 4.1, we will verify the conditions of Proposition 4.1, for
which we need the following lemma.

######  Lemma 4.1.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$ hold.
Then for any $\displaystyle\varepsilon\in(0,1)$ and $\displaystyle u_{0}\in
H$, there exists a Borel-measurable map

$\displaystyle\mathcal{G}^{\varepsilon}{\large}:\ C([0,T],U)\rightarrow
L^{2}(\Omega,C([0,T],H))\bigcap L^{2}(\Omega,L^{2}(0,T;V))\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))$

such that $\displaystyle\mathcal{G}^{\varepsilon}(W)=u^{\varepsilon}$ almost
surely, where $\displaystyle u^{\varepsilon}$ is the solution to (2.1)-(2.14).

Furthermore, for every control $\displaystyle v\in L^{2}(0,T;l^{2})$,
$\displaystyle
u_{v}^{\varepsilon}=\mathcal{G}^{\varepsilon}\left(W+\varepsilon^{-\frac{1}{2}}\int_{0}^{\cdot}v(t)dt\right)$
is the unique solution of

$\left\\{\begin{aligned}
&du_{v}^{\varepsilon}(t)+(-\triangle)^{\alpha}u_{v}^{\varepsilon}(t)dt+f(t,\cdot,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt\\\
&=g(t,\cdot,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt+\sigma(t,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})v(t)dt+\sqrt{\varepsilon}\sigma(t,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dW(t),\\\
&u_{v}^{\varepsilon}(0)=u_{0}\in H.\end{aligned}\right.$ (4.2)

###### Proof.

For a fixed $\displaystyle\mu\in C([0,T],\mathcal{P}_{2}(H))$,
$\displaystyle\varepsilon\in(0,1)$ and $\displaystyle u_{0}\in H$, consider
the decoupled stochastic equation:

$\displaystyle\displaystyle
dy_{\mu}^{\varepsilon}(t)+(-\triangle)^{\alpha}y_{\mu}^{\varepsilon}(t)dt+f(t,\cdot,y_{\mu}^{\varepsilon}(t),\mu(t))dt$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
g(t,\cdot,y_{\mu}^{\varepsilon}(t),\mu(t))dt+\sqrt{\varepsilon}\sigma(t,y_{\mu}^{\varepsilon}(t),\mu(t))dW(t),$
(4.3)

with initial data

$\displaystyle\displaystyle y_{\mu}^{\varepsilon}(0)=u_{0}.$ (4.4)

Since $\displaystyle\mu\in C([0,T],\mathcal{P}_{2}(H))$ is fixed, problem
(4.2)-(4.4) is similar to the classical stochastic equation independent of
distributions as discussed in [11], which means that under conditions
$\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$, for every
$\displaystyle\varepsilon\in(0,1)$ and $\displaystyle u_{0}\in H$, problem
problem (4.2)-(4.4) has a unique solution

$\displaystyle y_{\mu}^{\varepsilon}\in L^{2}(\Omega,C([0,T],H))\bigcap
L^{2}(\Omega,L^{2}(0,T;V))\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n}))).$

Consequently, there exists a Borel-measurable map (see, e.g., [58, Theorem
3.6]):

$\displaystyle\mathcal{G}_{\mu}^{\varepsilon}:\ C([0,T],U)\rightarrow
L^{2}(\Omega,C([0,T],H))\bigcap L^{2}(\Omega,L^{2}(0,T;V))\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))$

such that $\displaystyle
y^{\varepsilon}_{\mu}=\mathcal{G}_{\mu}^{\varepsilon}(W)$.

Let $\displaystyle u^{\varepsilon}$ be the solution of (2.1)-(2.14) and
$\displaystyle\mu^{\varepsilon}(t)=\mathcal{L}_{u^{\varepsilon}(t)}$ for
$\displaystyle t\in[0,T]$. Then by Theorem 2.1, we find that $\displaystyle
u^{\varepsilon}=y^{\varepsilon}_{\mu^{\varepsilon}}$ and hence $\displaystyle
u^{\varepsilon}=\mathcal{G}^{\varepsilon}_{\mu^{\varepsilon}}(W)$. Denote by
$\displaystyle\mathcal{G}^{\varepsilon}=\mathcal{G}^{\varepsilon}_{\mu^{\varepsilon}}$.
Then $\displaystyle u^{\varepsilon}=\mathcal{G}^{\varepsilon}(W)$, and for
every control $\displaystyle v\in L^{2}(0,T;l^{2})$, $\displaystyle
u_{v}^{\varepsilon}=\mathcal{G}^{\varepsilon}\left(W+\varepsilon^{-\frac{1}{2}}\int_{0}^{\cdot}v(t)dt\right)$
is the unique solution of

$\left\\{\begin{aligned}
&du_{v}^{\varepsilon}(t)+(-\triangle)^{\alpha}u_{v}^{\varepsilon}(t)dt+f(t,\cdot,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt\\\
&=g(t,\cdot,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dt+\sigma(t,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})v(t)dt+\sqrt{\varepsilon}\sigma(t,u_{v}^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dW(t),\\\
&u_{v}^{\varepsilon}(0)=u_{0}\in H,\end{aligned}\right.$

which concludes the proof. ∎

Now, define $\displaystyle\mathcal{G}^{0}:C([0,T],U)\rightarrow
C([0,T],H)\bigcap L^{2}(0,T;V)\bigcap
L^{p}(\Omega,L^{p}(0,T;L^{p}(\mathbb{R}^{n})))$ by, for every
$\displaystyle\zeta\in C([0,T],U)$,

$\mathcal{G}^{0}(\zeta)=\left\\{\begin{array}[]{ll}&u_{v},\ \ \ \text{if}\
\zeta=\int_{0}^{\cdot}v(t)dt\ \text{for\ some}\ v\in L^{2}(0,T;l^{2}),\\\ &0,\
\ \ \text{otherwise},\end{array}\right.$ (4.5)

where $\displaystyle u_{v}$ is the solution to (3.3)-(3.4).

In the next subsection, we will show $\displaystyle\mathcal{G}^{0}$ satisfies
$\displaystyle({\bf H1})$ in Proposition 4.1.

###  4.3 Convergence of solutions of controlled equations

We now prove that $\displaystyle\mathcal{G}^{0}$ satisfies the condition
$\displaystyle({\bf H1})$; more precisely, we will show the convergence of
solutions of the controlled equation (3.3) in the space $\displaystyle
C([0,T],H)\bigcap L^{2}(0,T;V)$ with respect to controls in the weak topology
of $\displaystyle L^{2}(0,T;l^{2})$. The argument is similar to the case of
stochastic partial differential equations which are independent of
distributions. For the reader’s convenience, we sketch the proof with
necessary modifications.

######  Theorem 4.2.

Suppose that $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$
hold, $\displaystyle u_{0}\in H$, and $\displaystyle v,v_{i}\in
L^{2}(0,T;l^{2})$ for $\displaystyle i\in\mathbb{N}$. If $\displaystyle
v_{i}\rightarrow v$ weakly in $\displaystyle L^{2}(0,T;l^{2})$, then
$\displaystyle\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v_{i}(s)ds\big{)}$
converges to
$\displaystyle\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v(s)ds\big{)}$ strongly in
$\displaystyle C([0,T],H)\bigcap L^{2}(0,T;V)$.

###### Proof.

Let $\displaystyle u_{v}$ and $\displaystyle u_{v_{i}}$ be the solutions of
(3.3)-(3.4) corresponding to controls $\displaystyle v$ and $\displaystyle
v_{i}$, respectively. By (4.5), we have $\displaystyle
u_{v}=\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v(s)ds\big{)}$ and $\displaystyle
u_{v_{i}}=\mathcal{G}^{0}\big{(}\int_{0}^{\cdot}v_{i}(s)ds\big{)}$. Therefore,
we only need to prove $\displaystyle u_{v_{i}}$ converges to $\displaystyle
u_{v}$ strongly in $\displaystyle C([0,T],H)\bigcap L^{2}(0,T;V)$.

Step 1: equicontinuity of $\displaystyle\\{u_{v_{i}}\\}_{i=1}^{\infty}$ in
$\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$. Since
$\displaystyle v_{i}\rightarrow v$ weakly in $\displaystyle L^{2}(0,T;l^{2})$,
we know that $\displaystyle\\{v_{i}\\}_{i=1}^{\infty}$ is bounded in
$\displaystyle L^{2}(0,T;l^{2})$. By Theorem 3.2, there exists $\displaystyle
C_{1}=C_{1}(T)>0$ such that for all $\displaystyle i\in\mathbb{N}$,

$\displaystyle\displaystyle\|u_{v_{i}}\|_{C([0,T],H)}+\|u_{v_{i}}\|_{L^{2}(0,T;V)}+\|u_{v_{i}}\|_{L^{p}(0,T;L^{p}(\mathbb{R}^{n}))}+\|v_{i}\|_{L^{2}(0,T;l^{2})}\leqslant
C_{1}.$ (4.6)

By (2.4), (4.6) and Theorem 3.1, one find that there exists $\displaystyle
C_{2}=C_{2}(T)>0$ such that

$\displaystyle\displaystyle\|f(\cdot,\cdot,u_{v_{i}},\mathcal{L}_{u^{0}(\cdot)})\|_{L^{q}(0,T;L^{q}(\mathbb{R}^{n}))}^{q}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}\left|f(t,x,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})\right|^{q}dxdt$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}\left[\lambda_{3}|u_{v_{i}}(t)|^{p-1}+\psi_{3}(t,x)\left(1+\|u^{0}(t)\|\right)\right]^{q}dxdt$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{T}\int_{\mathbb{R}^{n}}\left\\{\lambda_{3}|u_{v_{i}}(t)|^{p-1}+\psi_{3}(t,x)\left[1+M_{0}^{\frac{1}{2}}(1+\|u_{0}\|)\right]\right\\}^{q}dxdt$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2^{q-1}\lambda_{3}^{q}\int_{0}^{T}\int_{\mathbb{R}^{n}}|u_{v_{i}}(t)|^{q(p-1)}dxdt+2^{q-1}\left[1+M_{0}^{\frac{1}{2}}(1+\|u_{0}\|)\right]^{q}\int_{0}^{T}\int_{\mathbb{R}^{n}}|\psi_{3}(t,x)|^{q}dxdt$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2^{q-1}\lambda_{3}^{q}\|u_{v_{i}}\|_{L^{p}(0,T;L^{p}(\mathbb{R}^{n}))}^{p}+2^{q-1}\left[1+M_{0}^{\frac{1}{2}}(1+\|u_{0}\|)\right]^{q}\int_{0}^{T}\int_{\mathbb{R}^{n}}|\psi_{3}(t,x)|^{q}dxdt\leq
C_{2},$ (4.7)

where $\displaystyle C_{2}=C_{2}(T)>0$. Similar to (2.2), by (3.3)-(3.4) we
have

$\displaystyle\displaystyle
u_{v_{i}}(t)+\int_{0}^{t}(-\triangle)^{\alpha}u_{v_{i}}(s)ds+\int_{0}^{t}f(s,\cdot,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})ds$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
u_{0}+\int_{0}^{t}g(s,\cdot,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})ds+\int_{0}^{t}\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})v_{i}(s)ds$
(4.8)

in $\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$. Then for
all $\displaystyle 0\leqslant s\leqslant t\leqslant T$,

$\displaystyle\displaystyle\|u_{v_{i}}(t)-u_{v_{i}}(s)\|_{\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}}$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{s}^{t}\|(-\triangle)^{\alpha}u_{\theta_{i}}(r)\|_{V^{\ast}}dr+\int_{s}^{t}\|f(r,\cdot,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|_{L^{q}(\mathbb{R}^{n})}dr$
$\displaystyle\displaystyle+\int_{s}^{t}\|g(r,\cdot,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|dr+\int_{s}^{t}\|\sigma(r,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})v_{i}(r)\|dr.$
(4.9)

For the first term on the right-hand side of (4.3), by (4.6) and the Hölder
inequality, we obtain

$\displaystyle\displaystyle\int_{s}^{t}\|(-\triangle)^{\alpha}u_{v_{i}}(r)\|_{V^{\ast}}dr\leqslant\int_{s}^{t}\|u_{v_{i}}(r)\|_{V}dr\leqslant
C_{1}(t-s)^{\frac{1}{2}}.$ (4.10)

For the second term on the right-hand side of (4.3), by the Hölder inequality
and (4.3), we have

$\displaystyle\displaystyle\int_{s}^{t}\|f(r,\cdot,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|_{L^{q}(\mathbb{R}^{n})}dr\leqslant(t-s)^{\frac{1}{p}}\|f(\cdot,\cdot,u_{v_{i}},\mathcal{L}_{u^{0}(\cdot)})\|_{L^{q}(0,T;L^{q}(\mathbb{R}^{n}))}\leqslant
C_{2}^{\frac{1}{q}}(t-s)^{\frac{1}{p}}.$ (4.11)

For the third term on the right-hand side of (4.3), by (2.1) and Theorem 3.1,
we obtain

$\displaystyle\displaystyle\int_{s}^{t}\|g(r,\cdot,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|dr$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle(t-s)^{\frac{1}{2}}\|g(\cdot,\cdot,u_{v_{i}},\mathcal{L}_{u^{0}(\cdot)})\|_{L^{2}(0,T;H)}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle(t-s)^{\frac{1}{2}}\left\\{\int_{0}^{T}\left[4\|\psi_{g}(r)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|u_{v_{i}}(r)\|^{2}+4M_{0}(1+\|u_{0}\|^{2})\|\psi_{g}(r)\|^{2}+2\|\psi_{g}(r)\|^{2}\right]dr\right\\}^{\frac{1}{2}}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle(t-s)^{\frac{1}{2}}\left\\{2\sqrt{T}C_{1}\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+(2\sqrt{M_{0}(1+\|u_{0}\|^{2})}+\sqrt{2})\|\psi_{g}\|_{L^{2}(0,T;l^{2}(\mathbb{R}^{n}))}\right\\}.$
(4.12)

For the fourth term on the right-hand side of (4.3), by (2.1), (4.6) and
Theorem 3.1, we obtain

$\displaystyle\displaystyle\int_{s}^{t}\|\sigma(r,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})v_{i}(r)\|dr$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{s}^{t}\|\sigma(r,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|_{L_{2}(l^{2},H)}\|v_{i}(r)\|_{l^{2}}dr$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(\int_{s}^{t}\|\sigma(r,u_{v_{i}}(r),\mathcal{L}_{u^{0}(r)})\|_{L_{2}(l^{2},H)}^{2}dr\right)^{\frac{1}{2}}\|v_{i}\|_{L^{2}(0,T;l^{2})}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left[\int_{s}^{t}M_{\sigma,T}\left(1+\|u_{v_{i}}(r)\|^{2}+\mathcal{L}_{u^{0}(r)}(\|\cdot\|^{2})\right)dr\right]^{\frac{1}{2}}\|v_{i}\|_{L^{2}(0,T;l^{2})}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left\\{M_{\sigma,T}\left[1+C_{1}^{2}+M_{0}\left(1+\|u_{0}\|^{2}\right)\right]\right\\}^{\frac{1}{2}}C_{1}|t-s|^{\frac{1}{2}}.$
(4.13)

Then from (4.3)-(4.3), it follows that

$\displaystyle\displaystyle\|u_{v_{i}}(t)-u_{v_{i}}(s)\|_{\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}}\leqslant C_{3}(t-s)^{\frac{1}{p}},$
(4.14)

where $\displaystyle C_{3}=C_{3}(T)>0$.

From (4.6) and (4.3), we know that there exist

$\displaystyle z\in L^{\infty}(0,T;H)\bigcap L^{2}(0,T;V)\bigcap
L^{p}(0,T;L^{p}(\mathbb{R}^{n})),\ \varphi\in
L^{q}(0,T;L^{q}(\mathbb{R}^{n})),$

and a subsequence of $\displaystyle\\{u_{v_{i}}\\}_{i=1}^{\infty}$ (not
relabeled) such that

$\displaystyle\displaystyle u_{v_{i}}\rightarrow z\ \text{weak-star in}\
L^{\infty}(0,T;H),$ (4.15) $\displaystyle\displaystyle u_{v_{i}}\rightarrow z\
\text{weakly in}\ L^{2}(0,T;V),$ (4.16) $\displaystyle\displaystyle
u_{v_{i}}\rightarrow z\ \text{weakly in}\ L^{p}(0,T;L^{p}(\mathbb{R}^{n})),$
(4.17) $\displaystyle\displaystyle
f(\cdot,\cdot,u_{v_{i}},\mathcal{L}_{u^{0}(\cdot)})\rightarrow\varphi\
\text{weakly in}\ L^{q}(0,T;L^{q}(\mathbb{R}^{n})).$ (4.18)

Next, we prove the strong convergence of
$\displaystyle\\{u_{v_{i}}\\}_{i=1}^{\infty}$ in $\displaystyle L^{2}(0,T;H)$.

Step 2: proving

$\displaystyle\displaystyle u_{v_{i}}\rightarrow z\ \ \ \text{strongly in}\ \
L^{2}(0,T;H).$ (4.19)

We firstly prove $\displaystyle u_{v_{i}}\rightarrow z\ \text{strongly in}\
C\left([0,T],\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}\right).$ By
Step 1, we know $\displaystyle\\{u_{v_{i}}\\}_{i=1}^{\infty}$ is
equicontinuous in $\displaystyle\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}$ on $\displaystyle[0,T]$. So by the
Arzelà-Ascoli Theorem, we only need to show that for each $\displaystyle
t\in[0,T]$, $\displaystyle\\{u_{v_{i}}(t)\\}_{i=1}^{\infty}$ is precompact in
$\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$.

Let $\displaystyle\zeta_{m}$ be the cut-off function as introduced in the
proof of Lemma 3.1. Given $\displaystyle t\in[0,T]$, let
$\displaystyle\widetilde{u}_{v_{i}}^{m}(t,x)=\zeta_{m}(x)u_{v_{i}}(t,x)$ and
$\displaystyle\widehat{u}_{v_{i}}^{m}(t,x)=(1-\zeta_{m}(x))u_{v_{i}}(t,x)$ for
$\displaystyle x\in\mathbb{R}^{n}$. Then $\displaystyle
u_{v_{i}}(t,x)=\widetilde{u}_{v_{i}}^{m}(t,x)+\widehat{u}_{v_{i}}^{m}(t,x).$
for every $\displaystyle m\in\mathbb{N}$. Moreover, by (4.6), we have

$\displaystyle\displaystyle\|\widehat{u}_{v_{i}}^{m}(t)\|\leqslant\|u_{v_{i}}\|_{C([0,T],H)}\leqslant
C_{1},\ \ \ \forall\ i,m\in\mathbb{N},\ t\in[0,T].$ (4.20)

Notice that $\displaystyle\widehat{u}_{v_{i}}^{m}(t,x)=0$ for
$\displaystyle|x|\geqslant m$ and $\displaystyle t\in[0,T]$. By (4.20) and the
compactness of the embedding $\displaystyle H^{\alpha}(|x|<m)\hookrightarrow
L^{2}(|x|<m)$, we infer that
$\displaystyle\\{\widehat{u}_{v_{i}}^{m}(t)\\}_{i=1}^{\infty}$ is precompact
in $\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$.

On the other hand, by Lemma 3.1 and (4.6), we find that for every
$\displaystyle\delta>0$, there exists $\displaystyle
m_{0}=m_{0}(u_{0},T,\delta)>0$ such that for all $\displaystyle m\geqslant
m_{0}$, $\displaystyle i\in\mathbb{N}$ and $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|\widetilde{u}_{v_{i}}^{m}(t)\|_{\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}}^{2}\leqslant\|\widetilde{u}_{v_{i}}^{m}(t)\|_{H}^{2}\leqslant\int_{|x|\geqslant{\frac{1}{2}}m}|u_{v_{i}}(t,x)|^{2}dx<\frac{1}{4}\delta^{2}.$

Then we infer that
$\displaystyle\\{u_{v_{i}}(t)\\}_{i=1}^{\infty}=\\{\widetilde{u}_{v_{i}}^{m}(t)+\widehat{u}_{v_{i}}^{m}(t)\\}_{i=1}^{\infty}$
has a finite open cover with radius $\displaystyle\delta$ in
$\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$, and hence
$\displaystyle\\{u_{v_{i}}(t)\\}_{i=1}^{\infty}$ is precompact in
$\displaystyle\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}$ for
$\displaystyle t\in[0,T]$, which along with (4.15) implies that

$\displaystyle\displaystyle u_{v_{i}}\rightarrow z\ \ \text{strongly in}\
C\left([0,T],\left(V\bigcap L^{p}(\mathbb{R}^{n})\right)^{\ast}\right).$
(4.21)

Since

$\displaystyle\displaystyle\int_{0}^{T}\|u_{v_{i}}(t)-z(t)\|^{2}dt$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{T}\left\langle
u_{v_{i}}(t)-z(t),u_{v_{i}}(t)-z(t)\right\rangle_{\left(V\bigcap
L^{p}(\mathbb{R}^{n}),\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}\right)}dt$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(\int_{0}^{T}\|u_{v_{i}}(t)-z(t)\|_{V\bigcap
L^{p}(\mathbb{R}^{n})}^{2}dt\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|u_{v_{i}}(t)-z(t)\|_{\left(V\bigcap
L^{p}(\mathbb{R}^{n})\right)^{\ast}}^{2}dt\right)^{\frac{1}{2}},$

it follows from (4.6) and (4.21) that

$\displaystyle\displaystyle u_{v_{i}}\rightarrow z\ \ \text{strongly in}\ \
L^{2}(0,T;H).$

Step 3: proving $\displaystyle u_{v}=z$.

By (3.3), we obtain that for any $\displaystyle t\in[0,T]$ and
$\displaystyle\xi\in V\bigcap L^{p}(\mathbb{R}^{n})$,

$\displaystyle\displaystyle\left(u_{v_{i}}(t),\xi\right)+\int_{0}^{t}\left((-\triangle)^{\frac{\alpha}{2}}u_{v_{i}}(s),(-\triangle)^{\frac{\alpha}{2}}\xi\right)ds$
$\displaystyle\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})\xi(x)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(u_{0},\xi\right)+\int_{0}^{t}\left(g(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)}),\xi\right)ds+\int_{0}^{t}\left(\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})v_{i}(s),\xi\right)ds.$
(4.22)

It follows from (4.19) in Step 2 that (up to a subsequence)

$\displaystyle u_{v_{i}}\rightarrow z\ \ \ \text{a.e. on }\
(0,T)\times\mathbb{R}^{n}.$

Then by the continuity of $\displaystyle f$, we get

$\displaystyle\displaystyle
f(t,x,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})\rightarrow
f(t,x,z,\mathcal{L}_{u^{0}(t)})\ \ \ \text{a.e. on }\
(0,T)\times\mathbb{R}^{n}.$ (4.23)

By (4.18), (4.23) and Mazur’s theorem, we have

$\displaystyle\displaystyle\varphi=f(t,x,z,\mathcal{L}_{u^{0}(t)})\ \ \
\text{a.e. on }\ (0,T)\times\mathbb{R}^{n},$ (4.24)

and hence

$\displaystyle\displaystyle
f(\cdot,\cdot,u_{v_{i}},\mathcal{L}_{u^{0}(\cdot)})\rightarrow
f(\cdot,\cdot,z,\mathcal{L}_{u^{0}(\cdot)})\ \ \text{weakly in}\
L^{q}(0,T;L^{q}(\mathbb{R}^{n})).$ (4.25)

Taking the limit of the third term on the left-hand side of (4.21), we obtain

$\displaystyle\displaystyle\lim_{i\to\infty}\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})\xi(x)dxds=\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,z(s),\mathcal{L}_{u^{0}(s)})\xi(x)dxds$
(4.26)

For the second term on the right-hand side of (4.3), by (2.6) and (4.19), we
have

$\displaystyle\displaystyle\left|\int_{0}^{t}\left(g(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})-g(s,z(s),\mathcal{L}_{u^{0}(s)}),\xi\right)ds\right|$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\|g(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})-g(s,z(s),\mathcal{L}_{u^{0}(s)})\|\|\xi\|ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\|\xi\|t^{\frac{1}{2}}\left(\int_{0}^{t}\|u_{v_{i}}(s)-z(s)\|ds\right)^{\frac{1}{2}}\rightarrow\
0,\ \ \ \text{as}\ i\to\infty.$ (4.27)

For the third term on the right-hand side of (4.3), we have

$\displaystyle\displaystyle\int_{0}^{t}\left(\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})v_{i}(s),\xi\right)ds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{t}\left(\left(\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})-\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\right)v_{i}(s),\xi\right)ds$
$\displaystyle\displaystyle+\int_{0}^{t}\left(\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})v_{i}(s),\xi\right)ds.$
(4.28)

For the first term on the right-hand side of (4.3), by (2.1), (4.6) and
(4.19), we have

$\displaystyle\displaystyle\|\int_{0}^{t}\left(\left(\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})-\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\right)v_{i}(s),\xi\right)ds\|$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\|\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})-\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\|_{L_{2}(l^{2},H)}\|v_{i}(s)\|_{l^{2}}\|\xi\|ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
L_{\sigma}^{\frac{1}{2}}\|\xi\|\int_{0}^{t}\|u_{v_{i}}(s)-z(s)\|\|v_{i}(s)\|_{l^{2}}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
L_{\sigma}^{\frac{1}{2}}\|\xi\|\left(\int_{0}^{T}\|u_{v_{i}}(s)-z(s)\|^{2}ds\right)^{\frac{1}{2}}\left(\int_{0}^{T}\|v_{i}(s)\|_{l^{2}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle\ 0,\ \ \
\text{as}\ i\to+\infty.$ (4.29)

For the second term on the right-hand side of (4.3), we define an operator
$\displaystyle\mathcal{G}:L^{2}(0,T;l^{2})\rightarrow H$ by

$\displaystyle\mathcal{G}(\tilde{v})=\int_{0}^{t}\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\tilde{v}(s)ds,\
\ \ \forall\ \tilde{v}\in L^{2}(0,T;l^{2}).$

By (2.1), (4.15) and Theorem 3.1, we get for any $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|\int_{0}^{t}\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\tilde{v}(s)ds\|$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\|\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\|_{L_{2}(l^{2},H)}\|\tilde{v}(s)\|_{l^{2}}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(\int_{0}^{t}\|\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right)^{\frac{1}{2}}\left(\int_{0}^{t}\|\tilde{v}(s)\|_{l^{2}}^{2}ds\right)^{\frac{1}{2}}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left[\int_{0}^{t}M_{\sigma,T}\left(1+\|z(s)\|^{2}+\mathcal{L}_{u^{0}(s)}(\|\cdot\|^{2})\right)ds\right]^{\frac{1}{2}}\|\tilde{v}\|_{L^{2}(0,T;l^{2})}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left\\{TM_{\sigma,T}\left[1+\|z\|_{L^{\infty}(0,T;H)}^{2}+M_{0}\left(1+\|u_{0}\|^{2}\right)\right]\right\\}^{\frac{1}{2}}\|\tilde{v}\|_{L^{2}(0,T;l^{2})},$
(4.30)

which shows that $\displaystyle\mathcal{G}$ is a bounded linear operator, and
thus is weakly continuous. Since $\displaystyle v_{i}\rightarrow v$ weakly in
$\displaystyle L^{2}(0,T;l^{2})$, it follows that
$\displaystyle\mathcal{G}(v_{i})$ converges to $\displaystyle\mathcal{G}(v)$
weakly in $\displaystyle H$. So we have

$\displaystyle\displaystyle\lim_{i\to\infty}\int_{0}^{t}\left(\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})v_{i}(s),\xi\right)ds=\int_{0}^{t}\left(\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})v(s),\xi\right)ds.$
(4.31)

By (4.3), (4.3) and (4.31), we obtain

$\displaystyle\displaystyle\lim_{i\to\infty}\int_{0}^{t}\left(\sigma(s,u_{v_{i}}(s),\mathcal{L}_{u^{0}(s)})v_{i}(s),\xi\right)ds=\int_{0}^{t}\left(\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})v(s),\xi\right)ds.$
(4.32)

Taking the limit of (4.3), by (4.26), (4.3) and (4.32), we obtain

$\displaystyle\displaystyle\left(z(t),\xi\right)+\int_{0}^{t}\left((-\triangle)^{\frac{\alpha}{2}}z(s),(-\triangle)^{\frac{\alpha}{2}}\xi\right)ds$
$\displaystyle\displaystyle+\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,z(s),\mathcal{L}_{u^{0}(s)})\xi(x)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(u_{0},\xi\right)+\int_{0}^{t}\left(g(s,z(s),\mathcal{L}_{u^{0}(s)}),\xi\right)ds+\int_{0}^{t}\left(\sigma(s,z(s),\mathcal{L}_{u^{0}(s)})v(s),\xi\right)ds,$

which shows that $\displaystyle z$ is a solution to (3.3)-(3.4). Then by the
uniqueness of solutions to (3.3)-(3.4) in Lemma 3.2, we obtain $\displaystyle
u_{v}=z$.

Step 4: proving $\displaystyle u_{v_{i}}\rightarrow u_{v}$ strongly in
$\displaystyle C([0,T],H)\bigcap L^{2}(0,T;V)$.

By (3.3), we obtain

$\displaystyle\displaystyle\frac{d}{dt}\|u_{v_{i}}(t)-u_{v}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{v_{i}}(t)-u_{v}(t)\right)\|^{2}$
$\displaystyle\displaystyle+2\int_{\mathbb{R}^{n}}\left(f(t,x,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})-f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})\right)\left(u_{v_{i}}(t)-u_{v}(t)\right)dx$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\left(g(t,\cdot,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})-g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),\
u_{v_{i}}(t)-u_{v}(t)\right)$
$\displaystyle\displaystyle+2\left(\sigma(t,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})v_{i}(t)-\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),\
u_{v_{i}}(t)-u_{v}(t)\right)$ (4.33)

with $\displaystyle u_{v_{i}}(0)-u_{v}(0)=0$. For the third term on the left-
hand side of (4.3), by (2.4), we have

$\displaystyle\displaystyle
2\int_{\mathbb{R}^{n}}\left(f(t,x,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})-f(t,x,u_{v}(t),\mathcal{L}_{u^{0}(t)})\right)\left(u_{v_{i}}(t)-u_{v}(t)\right)dx$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\|\psi_{4}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v_{i}}(t)-u_{v}(t)\|^{2}.$
(4.34)

For the first term on the right-hand side of (4.3), by (2.6), we obtain

$\displaystyle\displaystyle
2\left(g(t,\cdot,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})-g(t,\cdot,u_{v}(t),\mathcal{L}_{u^{0}(t)}),\
u_{v_{i}}(t)-u_{v}(t)\right)$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle
2\|\psi_{g}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v_{i}}(t)-u_{v}(t)\|^{2}.$
(4.35)

For the second term on the right-hand side of (4.3), by (2.1) and (4.6), we
obtain

$\displaystyle\displaystyle
2\left(\sigma(t,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})v_{i}(t)-\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})v(t),\
u_{v_{i}}(t)-u_{v}(t)\right)$ $\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle
2\left(\|\sigma(t,u_{v_{i}}(t),\mathcal{L}_{u^{0}(t)})\|_{L_{2}(l^{2},H)}\|v_{i}(t)\|_{l^{2}}+\|\sigma(t,u_{v}(t),\mathcal{L}_{u^{0}(t)})\|_{L_{2}(l^{2},H)}\|v(t)\|_{l^{2}}\right)\|u_{v_{i}}(t)-u_{v}(t)\|$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\hat{C}_{1}\left(\|v_{i}(t)\|_{l^{2}}+\|v(t)\|_{l^{2}}\right)\|u_{v_{i}}(t)-u_{v}(t)\|,$
(4.36)

where $\displaystyle\hat{C}_{1}=\hat{C}_{1}(T)>0$.

By (4.3)-(4.3), we obtain

$\displaystyle\displaystyle\frac{d}{dt}\|u_{v_{i}}(t)-u_{v}(t)\|^{2}+2\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{v_{i}}(t)-u_{v}(t)\right)\|^{2}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\left(\|\psi_{4}(t)\|_{L^{\infty}(\mathbb{R}^{n})}+\|\psi_{g}(t)\|_{L^{\infty}(\mathbb{R}^{n})}\right)\|u_{v_{i}}(t)-u_{v}(t)\|^{2}$
$\displaystyle\displaystyle+2\hat{C}_{1}\left(\|v_{i}(t)\|_{l^{2}}+\|v(t)\|_{l^{2}}\right)\|u_{v_{i}}(t)-u_{v}(t)\|.$
(4.37)

Integrating (4.3) on $\displaystyle(0,t)$, by Hölder’s inequality, (4.6), Step
2 and Step 3, we get for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\|u_{v_{i}}(t)-u_{v}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}\left(u_{v_{i}}(s)-u_{v}(s)\right)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\left(\|\psi_{4}(t)\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}+\|\psi_{g}(t)\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}\right)\|u_{v_{i}}-u_{v}\|_{L^{2}(0,T;H)}^{2}$
$\displaystyle\displaystyle+2\sqrt{2}\hat{C}_{1}\left(\|v_{i}\|_{L^{2}(0,T;l^{2}}+\|v\|_{L^{2}(0,T;l^{2}}\right)\|u_{v_{i}}-u_{v}\|_{L^{2}(0,T;H)}$
$\displaystyle\displaystyle\rightarrow$ $\displaystyle\displaystyle\ 0,\ \ \
i\to\infty,$ (4.38)

which concludes the proof. ∎

###  4.4 Convergence of solutions of stochastic equations

In this subsection, we prove $\displaystyle\mathcal{G}^{\varepsilon}$ and
$\displaystyle\mathcal{G}^{0}$ satisfy $\displaystyle({\bf H2})$ in
Proposition 4.1; more precisely, we will show the convergence of solutions of
the stochastic equation (4.2) in $\displaystyle C([0,T],H)\bigcap
L^{2}(0,T;V)$, for which we need the following uniform estimates.

######  Lemma 4.2.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$ hold.
Then for every $\displaystyle R>0$, $\displaystyle T>0$ and $\displaystyle
N>0$, there exists $\displaystyle M_{3}=M_{3}(R,T,N)>0$ such that for any
$\displaystyle u_{0}\in H$ with $\displaystyle\|u_{0}\|\leqslant R$ and any
$\displaystyle v\in\mathcal{A}_{N}$, the solution $\displaystyle
u_{v}^{\varepsilon}$ of (4.2) satisfies for all
$\displaystyle\varepsilon\in(0,1)$,

$\displaystyle\displaystyle\mathbb{E}\left[\|u_{v}^{\varepsilon}\|^{2}_{C([0,T],H)}\right]+\mathbb{E}\left[\|u_{v}^{\varepsilon}\|^{2}_{L^{2}(0,T;V)}\right]+\mathbb{E}\left[\|u_{v}^{\varepsilon}\|^{2}_{L^{p}(0,T;L^{p}(\mathbb{R}^{n}))}\right]\leqslant
M_{3}.$ (4.39)

###### Proof.

It follows from (4.2) and Itô’s formula that for all $\displaystyle
t\in[0,T]$, $\displaystyle P$-almost surely,

$\displaystyle\displaystyle\|u_{v}^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds+2\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})u_{v}^{\varepsilon}(s)dxds$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\|u_{0}\|^{2}+2\int_{0}^{t}\left(g(s,\cdot,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),u_{v}^{\varepsilon}(s)\right)ds+2\int_{0}^{t}\left(\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})v(s),u_{v}^{\varepsilon}(s)\right)ds$
$\displaystyle\displaystyle+\varepsilon\int_{0}^{t}\left\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\right\|^{2}_{L_{2}(l^{2},H)}ds+2\sqrt{\varepsilon}\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right).$
(4.40)

For the third term on the left-hand side of (4.4), by (2.1) and Theorem 2.1,
we have

$\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}f(s,x,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})u_{v}^{\varepsilon}(s)dxds$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds-2\int_{0}^{t}\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}\left(1+\mathbb{E}\left[\|u^{\varepsilon}(s)\|^{2}\right]\right)dsds$
$\displaystyle\displaystyle-2\int_{0}^{t}\|\psi_{1}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle\geqslant$ $\displaystyle\displaystyle
2\lambda_{1}\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds-2\left[1+M_{1}(1+\|u_{0}\|^{2})\right]\int_{0}^{t}\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}dsds$
$\displaystyle\displaystyle-2\int_{0}^{t}\|\psi_{1}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\|u_{v}^{\varepsilon}(s)\|^{2}ds.$
(4.41)

For the second term on the right-hand side of (4.4), by (2.1) and Theorem 2.1,
we have

$\displaystyle\displaystyle
2\int_{0}^{t}\left(g(s,\cdot,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)}),u_{v}^{\varepsilon}(s)\right)ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\|g(s,\cdot,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|^{2}ds+\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
4\int_{0}^{t}\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\|u_{v}^{\varepsilon}(s)\|^{2}ds+2(1+2\mathbb{E}\left[\|u^{\varepsilon}(s)\|^{2}\right])\int_{0}^{t}\|\psi_{g}(s)\|^{2}ds+\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\left(1+4\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds+2\left[1+2M_{1}(1+\|u_{0}\|^{2})\right]\int_{0}^{t}\|\psi_{g}(s)\|^{2}ds.$
(4.42)

For the third term on the right-hand side of (4.4), by (2.1) and Theorem 2.1,
we have

$\displaystyle\displaystyle
2\int_{0}^{t}\left(\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})v(s),u_{v}^{\varepsilon}(s)\right)ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}\|v(s)\|_{l^{2}}^{2}ds+\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}M_{\sigma,T}\left(1+\mathbb{E}\left[\|u^{\varepsilon}(s)\|^{2}\right]\right)\|v(s)\|_{l^{2}}^{2}ds+\int_{0}^{t}\left(1+M_{\sigma,T}\|v(s)\|_{l^{2}}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
M_{\sigma,T}\left[1+M_{1}(1+\|u_{0}\|^{2})\right]\int_{0}^{t}\|v(s)\|_{l^{2}}^{2}ds+\int_{0}^{t}\left(1+M_{\sigma,T}\|v(s)\|_{l^{2}}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds.$
(4.43)

For the fourth term on the right-hand side of (4.4), by (2.1) and Theorem 2.1,
we have

$\displaystyle\displaystyle\varepsilon\int_{0}^{t}\left\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\right\|^{2}_{L_{2}(l^{2},H)}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\varepsilon\int_{0}^{t}M_{\sigma,T}\left(1+\|u_{v}^{\varepsilon}(s)\|^{2}+\mathbb{E}\left[\|u^{\varepsilon}(s)\|^{2}\right]\right)ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle\varepsilon
TM_{\sigma,T}\left[1+M_{1}(1+\|u_{0}\|^{2})\right]+\varepsilon
M_{\sigma,T}\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^{2}ds.$ (4.44)

From (4.4)-(4.4), it follows that for all $\displaystyle t\in[0,T]$,
$\displaystyle P$-almost surely,

$\displaystyle\displaystyle\|u_{v}^{\varepsilon}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\|u_{0}\|^{2}+\int_{0}^{t}\left(2+4\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}^{2}+2\|\psi_{1}(s)\|_{L^{\infty}(\mathbb{R}^{n})}+\varepsilon
M_{\sigma,T}+M_{\sigma,T}\|v(s)\|_{l^{2}}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle+\left[1+2M_{1}(1+\|u_{0}\|^{2})\right]\left(2\int_{0}^{t}\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}ds+2\int_{0}^{t}\|\psi_{g}(s)\|^{2}ds+M_{\sigma,T}\int_{0}^{t}\|v(s)\|_{l^{2}}^{2}ds\right)$
$\displaystyle\displaystyle+\varepsilon
TM_{\sigma,T}\left[1+M_{1}(1+\|u_{0}\|^{2})\right]+2\sqrt{\varepsilon}\int_{0}^{t}\left(u_{v}^{\varepsilon}(s),\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right).$
(4.45)

If $\displaystyle u_{0}\in H$ with $\displaystyle\|u_{0}\|\leqslant R$, and
$\displaystyle v\in\mathcal{A}_{N}$, then we obtain by (4.4) that for all
$\displaystyle 0\leqslant r\leqslant t\leqslant T$, $\displaystyle P$-almost
surely,

$\displaystyle\displaystyle\|u_{v}^{\varepsilon}(r)\|^{2}+2\int_{0}^{r}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{r}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
R^{2}+\int_{0}^{r}\left(2+4\|\psi_{g}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}^{2}+2\|\psi_{1}\|_{L^{\infty}(0,T;L^{\infty}(\mathbb{R}^{n}))}^{2}+\varepsilon
M_{\sigma,T}+M_{\sigma,T}\|v(s)\|_{l^{2}}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds$
$\displaystyle\displaystyle+\left[1+2M_{1}(1+R^{2})\right]\left(2\int_{0}^{T}\|\psi_{1}(s)\|_{L^{1}(\mathbb{R}^{n})}ds+2\int_{0}^{T}\|\psi_{g}(s)\|^{2}ds+M_{\sigma,T}N\right)$
$\displaystyle\displaystyle+\varepsilon
TM_{\sigma,T}\left[1+M_{1}(1+R^{2})\right]+2\sqrt{\varepsilon}\int_{0}^{r}\left(u_{v}^{\varepsilon}(s),\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{1}+c_{1}\int_{0}^{r}\left(1+\|v(s)\|_{l^{2}}^{2}\right)\|u_{v}^{\varepsilon}(s)\|^{2}ds+K(t),$
(4.46)

where $\displaystyle c_{1}=c_{1}(R,N,T)>0$, $\displaystyle
K(t)=2\sqrt{\varepsilon}\sup_{r\in[0,t]}\left|\int_{0}^{r}\left(u_{v}^{\varepsilon}(s),\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)\right|.$

By Gronwall’s inequality and (4.4), we obtain for all $\displaystyle
r\in[0,t]$,

$\displaystyle\displaystyle\|u_{v}^{\varepsilon}(r)\|^{2}+2\int_{0}^{r}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds+2\lambda_{1}\int_{0}^{r}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(c_{1}+K(t)\right)e^{c_{1}\int_{0}^{r}\left(1+\|v(s)\|_{l^{2}}^{2}\right)ds}$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\left(c_{1}+K(t)\right)e^{c_{1}(T+N)},\ \ \
\mathbb{P}\text{-almost surely}.$ (4.47)

Then by (4.4), we obtain for all $\displaystyle t\in[0,T]$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{r\in[0,t]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]+2\mathbb{E}\left[\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds\right]+2\lambda_{1}\mathbb{E}\left[\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\left(c_{1}+\mathbb{E}\left[K(t)\right]\right)e^{c_{1}(T+N)}$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{2}+c_{2}\mathbb{E}\left[K(t)\right],\ \ \ \mathbb{P}\text{-almost surely},$
(4.48)

where $\displaystyle c_{2}=2\left(c_{1}+1\right)e^{c_{1}(T+N)}$. By the
Burkholder inequality and (2.1), we have

$\displaystyle\displaystyle c_{2}\mathbb{E}\left[K(t)\right]=$
$\displaystyle\displaystyle
2c_{2}\sqrt{\varepsilon}\mathbb{E}\left[\sup_{r\in[0,t]}\left|\int_{0}^{r}\left(u_{v}^{\varepsilon}(s),\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)\right|\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
6c_{2}\sqrt{\varepsilon}\mathbb{E}\left[\left(\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|^{2}\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right)^{\frac{1}{2}}\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{s\in[0,t]}\|u_{v}^{\varepsilon}(s)\|^{2}\right]+18c_{2}^{2}\varepsilon\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|^{2}_{L_{2}(l^{2},H)}ds\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{s\in[0,t]}\|u_{v}^{\varepsilon}(s)\|^{2}\right]+18c_{2}^{2}\varepsilon
M_{\sigma,T}\mathbb{E}\left[\int_{0}^{t}\left(1+\|u_{v}^{\varepsilon}(s)\|^{2}+\mathcal{L}_{u^{\varepsilon}(s)}(\|\cdot\|^{2})\right)ds\right]$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{s\in[0,t]}\|u_{v}^{\varepsilon}(s)\|^{2}\right]+18c_{2}^{2}\varepsilon
M_{\sigma,T}\left[\int_{0}^{t}\left(1+\mathbb{E}\left[\|u_{v}^{\varepsilon}(s)\|^{2}\right]+\mathbb{E}\left[\|u^{\varepsilon}(s)\|^{2}\right]\right)ds\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{r\in[0,t]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]+18c_{2}^{2}\varepsilon
TM_{\sigma,T}\left[1+M_{1}(1+\|u_{0}\|^{2})\right]$
$\displaystyle\displaystyle+18c_{2}^{2}\varepsilon
M_{\sigma,T}\int_{0}^{t}\mathbb{E}\left[\sup_{r\in[0,s]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]ds.$
(4.49)

By (4.4)-(4.4), we obtain that for all $\displaystyle t\in[0,T]$ and
$\displaystyle\varepsilon\in(0,1)$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{r\in[0,t]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]+4\mathbb{E}\left[\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds\right]+4\lambda_{1}\mathbb{E}\left[\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
c_{3}+c_{3}\int_{0}^{t}\mathbb{E}\left[\sup_{r\in[0,s]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]ds,$
(4.50)

where $\displaystyle c_{3}=c_{3}(R,N,T)>0$.

By (4.4) and Gronwall’s inequality, we have for all $\displaystyle t\in[0,T]$
and $\displaystyle\varepsilon\in(0,1)$,

$\displaystyle\displaystyle\mathbb{E}\left[\sup_{r\in[0,t]}\|u_{v}^{\varepsilon}(r)\|^{2}\right]+4\mathbb{E}\left[\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}u_{v}^{\varepsilon}(s)\|^{2}ds\right]+4\lambda_{1}\mathbb{E}\left[\int_{0}^{t}\|u_{v}^{\varepsilon}(s)\|_{L^{p}(\mathbb{R}^{n})}^{p}ds\right]\leqslant
c_{3}e^{c_{3}t},$

which implies (4.39). ∎

The next lemma is concerned with the convergence of solutions of (2.1)-(2.14)
as $\displaystyle\varepsilon\to 0$.

######  Lemma 4.3.

Suppose $\displaystyle({\bf\Sigma 1})$-$\displaystyle({\bf\Sigma 3})$ hold and
$\displaystyle u_{0}\in H$. Let $\displaystyle u^{\varepsilon}$ and
$\displaystyle u^{0}$ be the solutions of (2.1)-(2.14) and (3.1)-(3.2),
respectively. Then there exists $\displaystyle C=C(T)>0$ such that for all
$\displaystyle\varepsilon\in(0,1)$,

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\|u^{\varepsilon}(t)-u^{0}(t)\|^{2}\right]\leqslant
C\varepsilon\left(1+\|u_{0}\|^{2}\right).$

###### Proof.

By (2.1)-(2.14) and (3.1)-(3.2), we find that $\displaystyle
u^{\varepsilon}-u^{0}$ satisfies the equation:

$\displaystyle\displaystyle
d\left(u^{\varepsilon}(t)-u^{0}(t)\right)+(-\triangle)^{\alpha}\left(u^{\varepsilon}(t)-u^{0}(t)\right)dt$
$\displaystyle\displaystyle+\left(f(t,\cdot,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})-f(t,\cdot,u^{0}(t),\mathcal{L}_{u^{0}(t)})\right)dt$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\left(g(t,\cdot,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})-g(t,\cdot,u^{0}(t),\mathcal{L}_{u^{0}(t)})\right)dt+\sqrt{\varepsilon}\sigma(t,u^{\varepsilon}(t),\mathcal{L}_{u^{\varepsilon}(t)})dW(t),$
(4.51)

with initial condition $\displaystyle u^{\varepsilon}(0)-u^{0}(0)=0$.

Applying Itô’s formula to (4.4), we have

$\displaystyle\displaystyle\|u^{\varepsilon}(t)-u^{0}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}\left(u^{\varepsilon}(s)-u^{0}(s)\right)\|^{2}ds$
$\displaystyle\displaystyle+2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-f(s,x,u^{0}(s),\mathcal{L}_{u^{0}(s)})\right)\left(u^{\varepsilon}(s)-u^{0}(s)\right)dxds$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\int_{0}^{t}\left(g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-g(s,\cdot,u^{0}(s),\mathcal{L}_{u^{0}(s)}),u^{\varepsilon}(s)-u^{0}(s)\right)ds$
$\displaystyle\displaystyle+\varepsilon\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle+2\sqrt{\varepsilon}\int_{0}^{t}\left(u^{\varepsilon}(s)-u^{0}(s),\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right).$
(4.52)

For the third term on the left-hand side of (4.4), by (2.1), (2.4) and
Hölder’s inequality, we have

$\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-f(s,x,u^{0}(s),\mathcal{L}_{u^{0}(s)})\right)\left(u^{\varepsilon}(s)-u^{0}(s)\right)dxds$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-f(s,x,u^{0}(s),\mathcal{L}_{u^{\varepsilon}(s)})\right)\left(u^{\varepsilon}(s)-u^{0}(s)\right)dxds$
$\displaystyle\displaystyle+2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left(f(s,x,u^{0}(s),\mathcal{L}_{u^{\varepsilon}(s)})-f(s,x,u^{0}(s),\mathcal{L}_{u^{0}(s)})\right)\left(u^{\varepsilon}(s)-u^{0}(s)\right)dxds$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\int_{0}^{t}\int_{\mathbb{R}^{n}}\psi_{4}(s,x)|u^{\varepsilon}(s)-u^{0}(s)|^{2}dxds$
$\displaystyle\displaystyle-2\int_{0}^{t}\int_{\mathbb{R}^{n}}\psi_{3}(s,x)\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)|u^{\varepsilon}(s)-u^{0}(s)|dxds$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-2\int_{0}^{t}\|\psi_{4}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}ds$
$\displaystyle\displaystyle-2\int_{0}^{t}\|\psi_{3}(s)\|\|u^{\varepsilon}(s)-u^{0}(s)\|\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)ds$
$\displaystyle\displaystyle\geqslant$
$\displaystyle\displaystyle-\int_{0}^{t}\left(2\|\psi_{4}(s)\|_{L^{\infty}(\mathbb{R}^{n})}+\|\psi_{3}(s)\|^{2}\right)\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}ds-\int_{0}^{t}\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)^{2}ds.$
(4.53)

For the first term on the right-hand side of (4.4), by (2.6) and Hölder’s
inequality, we obtain

$\displaystyle\displaystyle
2\int_{0}^{t}\left(g(s,\cdot,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-g(s,\cdot,u^{0}(s),\mathcal{L}_{u^{0}(s)}),u^{\varepsilon}(s)-u^{0}(s)\right)ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}\left|g(s,x,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-g(s,x,u^{0}(s),\mathcal{L}_{u^{0}(s)})\right||u^{\varepsilon}(s)-u^{0}(s)|dxds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\int_{0}^{t}\int_{\mathbb{R}^{n}}\psi_{g}(s,x)\left(|u^{\varepsilon}(s)-u^{0}(s)|+\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)\right)|u^{\varepsilon}(s)-u^{0}(s)|dxds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\int_{0}^{t}\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}ds+2\int_{0}^{t}\|\psi_{g}(s)\|\|u^{\varepsilon}(s)-u^{0}(s)\|\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\left(2\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|^{2}\right)\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}ds+\int_{0}^{t}\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)^{2}ds.$
(4.54)

Then from (4.4)-(4.4), it follows that

$\displaystyle\displaystyle\|u^{\varepsilon}(t)-u^{0}(t)\|^{2}+2\int_{0}^{t}\|(-\triangle)^{\frac{\alpha}{2}}\left(u^{\varepsilon}(s)-u^{0}(s)\right)\|^{2}ds$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\int_{0}^{t}\left(2\|\psi_{4}(s)\|_{L^{\infty}(\mathbb{R}^{n})}+\|\psi_{3}(s)\|^{2}+2\|\psi_{g}(s)\|_{L^{\infty}(\mathbb{R}^{n})}+\|\psi_{g}(s)\|^{2}\right)\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}ds$
$\displaystyle\displaystyle+2\int_{0}^{t}\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)^{2}ds+\varepsilon\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle+2\sqrt{\varepsilon}\int_{0}^{t}\left(u^{\varepsilon}(s)-u^{0}(s),\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right).$
(4.55)

For the third term on the right-hand side of (4.4), by (2.1) and (2.1), we
obtain

$\displaystyle\displaystyle\varepsilon\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
2\varepsilon\int_{0}^{t}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})-\sigma(s,u^{0}(s),\mathcal{L}_{u^{0}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle+2\varepsilon\int_{0}^{t}\|\sigma(s,u^{0}(s),\mathcal{L}_{u^{0}(s)})\|_{L_{2}(l^{2},H)}^{2}ds$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle 2\varepsilon
L_{\sigma}\int_{0}^{t}\left(\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}+\mathbb{W}_{2}\left(\mathcal{L}_{u^{\varepsilon}(s)},\mathcal{L}_{u^{0}(s)}\right)^{2}\right)ds$
$\displaystyle\displaystyle+2\varepsilon
TM_{\sigma,T}\left(1+2\sup_{s\in[0,T]}\|u^{0}(s)\|^{2}\right).$ (4.56)

For the fourth term on the right-hand side of (4.4), by the Burkholder
inequality and (4.4), we obtain

$\displaystyle\displaystyle
2\mathbb{E}\left[\sqrt{\varepsilon}\sup_{r\in[0,t]}\left|\int_{0}^{r}\left(u^{\varepsilon}(s)-u^{0}(s),\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})dW(s)\right)\right|\right]$
$\displaystyle\displaystyle\leqslant$ $\displaystyle\displaystyle
6\sqrt{\varepsilon}\mathbb{E}\left[\left(\int_{0}^{t}\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}\|\sigma(s,u^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|_{L_{2}(l^{2},H)}^{2}ds\right)^{\frac{1}{2}}\right]$
$\displaystyle\displaystyle\leqslant$
$\displaystyle\displaystyle\frac{1}{2}\mathbb{E}\left[\sup_{s\in[0,t]}\|u^{\varepsilon}(s)-u^{0}(s)\|^{2}\right]+18\varepsilon\mathbb{E}\left[\int_{0}^{t}\|\sigma(s,u_{v}^{\varepsilon}(s),\mathcal{L}_{u^{\varepsilon}(s)})\|^{2}_{L_{2}(l^{2},H)}ds\right]$
$\displaystyle\displaystyle\leqslant$