Moderate deviation principle for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction

ABSTRACT

In this article, we obtain a central limit theorem and prove a moderate deviation principle for stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term.

KEYWORDS:

• Non-Lipschitz
• central limit theorems
• moderation deviations

1. Introduction

Consider the following stochastic reaction-diffusion system: (1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) with $ϵ\ge 0$, where $\overline{\mathcal{O}}$ is a bounded open set of ${\mathbb{R}}^d$, with $d\ge 1$, having a $C^{\mathrm{\infty }}$ boundary.

For each $i=1,\dots ,r$, ${\mathcal{A}}_i(\xi ,D)=\sum _{h,k=1}^d\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_h}(a_{hk}^i\times (\xi \left)\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_k}\right)-\sum _{h=1}^d{b}_h^i\left(\xi \right)\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_h},\xi \in \overline{\mathcal{O}},$ where the coefficients $a_{hk}^i$ are taken in $C^{\mathrm{\infty }}\left(\overline{\mathcal{O}}\right)$, the matrices $a^i\left(\xi \right):=\left[a_{hk}^i\right(\xi ){]}_{hk}$ are non-negative and symmetric for each $\xi \in \overline{\mathcal{O}}$ and fulfil a uniform ellipticity condition, i.e. $\underset{\xi \in \overline{\mathcal{O}}}{inf}\sum _{h,k=1}^d{a}^i\left(\xi \right)v_h{v}_k\ge {\lambda }_i|v{|}^2,v\in {\mathbb{R}}^d,$ for some positive constants ${\lambda }_i$, and coefficients $b_h^i$ are continuous. The operators ${\mathcal{B}}^i$ act on the boundary of $\mathcal{O}$ and are assumed to be either of Dirichlet or of co-normal type. The linear operators $Q_j$ are bounded on $L^2\left(\mathcal{O}\right)$ and may be equal to the identity operator if d = 1. Let $(\mathrm{\Omega },\mathcal{F},\mathbb{P})$ be a probability space with an increasing family $\{{\mathcal{F}}_t{\}}_{0\le t\le T}$ of the sub-σ-fields of $\mathcal{F}$ satisfying the usual conditions. The noises $\frac{\mathrm{\partial }{\omega }_j}{\mathrm{\partial }t}$ are independent cylindrical Wiener processes on $(\mathrm{\Omega },\mathcal{F},{\mathcal{F}}_t,\mathbb{P})$.

Moderate deviations trace back to Cramér (1938) established expansions of tail probabilities about sums of independent random variables defying the normal distribution. Comparing to large deviation, which offers a accurate estimate associated with the law of large number, moderate deviation is commonly applied to provide further estimates concerning the central limit theorem and the law of iterated logarithm.

Let $u^{ϵ}$ be the solution of a small perturbation of the equation of (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) ), that is (2) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i^{ϵ}}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i^{ϵ}(t,\xi )+f_i(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)}\\ {\displaystyle +\sqrt{ϵ}\sum _{j=1}^r{g}_{i,j}(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i^{ϵ}(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i^{ϵ}(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r.\end{array}$(2) The solution of Equation (2(2) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i^{ϵ}}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i^{ϵ}(t,\xi )+f_i(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)}\\ {\displaystyle +\sqrt{ϵ}\sum _{j=1}^r{g}_{i,j}(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i^{ϵ}(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i^{ϵ}(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r.\end{array}$(2) ) will tend to the solution, denoted by $u^0$, of the following deterministic equation, as $ϵ\to 0$, (3) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i^0}{\mathrm{\partial }t}}(t,\xi )={\mathcal{A}}_i{u}_i^0(t,\xi )+f_i(\xi ,u_1^0(t,\xi ),\dots ,\\ u_r^0(t,\xi )),t\ge 0,\xi \in \overline{\mathcal{O}},\\ u_i^0(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},\\ {\mathcal{B}}^i{u}_i^0(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r.\end{array}$(3) Assume that the mapping $f:=(f_1,\dots ,f_r):\overline{\mathcal{O}}\times {\mathbb{R}}^r\to {\mathbb{R}}^r$ is locally Lipschitz-continuous and has polynomial growth. The mapping $g:=\left[g_{i,j}\right]:\overline{\mathcal{O}}\times {\mathbb{R}}^r\to \mathcal{L}\left({\mathbb{R}}^r\right)$ is globally Lipschitz, without any assumption of boundedness and non-degeneracy. The existence and uniqueness of the solution for Equation (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) ) is proved in Cerrai (2003). The existence and uniqueness of the solutions for Equation (2(2) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i^{ϵ}}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i^{ϵ}(t,\xi )+f_i(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)}\\ {\displaystyle +\sqrt{ϵ}\sum _{j=1}^r{g}_{i,j}(\xi ,u_1^{ϵ}(t,\xi ),\dots ,u_r^{ϵ}(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i^{ϵ}(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i^{ϵ}(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r.\end{array}$(2) ) and Equation (3(3) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i^0}{\mathrm{\partial }t}}(t,\xi )={\mathcal{A}}_i{u}_i^0(t,\xi )+f_i(\xi ,u_1^0(t,\xi ),\dots ,\\ u_r^0(t,\xi )),t\ge 0,\xi \in \overline{\mathcal{O}},\\ u_i^0(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},\\ {\mathcal{B}}^i{u}_i^0(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r.\end{array}$(3) ) can be established as that of Equation (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) ). In Cerrai and Röckner (2004), Cerrai and Röckner had shown that the process $\{u^{ϵ}{\}}_{ϵ>0}$ obeys a large deviation principle (abbreviated LDP) in $C\left(\right[0,T];E)$, for any T>0. Also, they established the large deviations for invariant measures for $\{u^{ϵ}{\}}_{ϵ>0}$ in Cerrai and Röckner (2005). Based on these works, in this paper, we then consider the precise deviations of $u^{ϵ}$ from $u^0$ as $ϵ\to 0$ in the following way: the asymptotic behaviour of the trajectory $(u^{ϵ}-u^0)\left(t\right)/\left(\sqrt{ϵ}\lambda \right(ϵ\left)\right),t\in [0,T].$ We emphasize that

1. it is related to central limit theorem (abbreviated CLT) when $\lambda \left(ϵ\right)=1$, see Theorem 3.4 below;

2. it provides moderate deviation principle (MDP for short, cf. Dembo & Zeitouni, 2000) when $\lambda \left(ϵ\right)$ is the deviation scale verifying (4) $\lambda \left(ϵ\right)\to +\mathrm{\infty },\sqrt{ϵ}\lambda \left(ϵ\right)\to 0,\mathrm{a}\mathrm{s}ϵ\to 0,$(4) see Theorem 4.3 below. In the sequel, we assume that (4(4) $\lambda \left(ϵ\right)\to +\mathrm{\infty },\sqrt{ϵ}\lambda \left(ϵ\right)\to 0,\mathrm{a}\mathrm{s}ϵ\to 0,$(4) ) is in place.

Moderate deviation is an intermediate estimation between the large deviation with deviation scale $\lambda \left(ϵ\right)=1/\sqrt{ϵ}$ and the CLT with $\lambda \left(ϵ\right)=1$. The MDP provides rates of convergence and a way to construct asymptotic confidence interval, see Gao and Zhao (2011) and the references therein. In recent years, there is an increasing interest on the study of MDP. For independent and identically distributed random sequences, Chen (1990, 1991, 1993) and Ledoux (1992) found the necessary and sufficient conditions for MDP. Moderate deviation was discussed by Djellout and Guillin (2001), Gao (1996, 2003), and Wu (1995, 1999) for Markov processes as well as Guillin and Liptser (2006) for diffusion processes. For mean field interacting particle models, MDP was worked by Douc et al. (2001). Wang and Zhang (2015) and Wang et al. (2015) discussed moderate deviations for stochastic reaction-diffusion equation and 2D stochastic Navier-Stokes equations, respectively. Recently, there are several works on moderate deviations for SPDEs with jump, see Budhiraja et al. (2016) and Dong et al. (2017).

In this paper, we prove a CLT and establish a MDP for the class of reaction-diffusion systems of the form (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) ). Since we consider the solutions in spaces of continuous functions and there is no Itô formula, we will work directly on heat kernels. Also, because of the weak assumptions on drift and diffusion coefficients, we have to deal with the estimates on the solutions delicately.

The organization of this paper is as follows. The assumptions will be given in Section 2. Section 3 is devoted to prove the CLT. In Section 4, we establish the MDP.

Throughout this paper, c and $c_n$ are positive constants independent of ε and those values may be different from line to line.

2. Preliminaries

In this section, we formulate the equation and state the precise conditions on the coefficients.

Let H be the separable Hilbert space $L^2(\mathcal{O};{\mathbb{R}}^r)$ endowed with the scalar product $|x{|}_H:=\sum _{i=1}^r|x_i{|}_{L^2\left(\mathcal{O}\right)}$. We shall denote by A the realization in H of the differential operator $\mathcal{A}:=({\mathcal{A}}_1,\dots ,{\mathcal{A}}_r)$, endowed with the boundary conditions $\mathcal{B}:=({\mathcal{B}}_1,\dots ,{\mathcal{B}}_r)$, i.e. $D\left(A\right)=\{x\in W^{2,2}(\mathcal{O};{\mathbb{R}}^r):\mathcal{B}(\cdot ,D)x=0\mathrm{i}\mathrm{n}\mathrm{\partial }\mathcal{O}\}$, $Ax=\mathcal{A}(\cdot ,D)x$. Set $E:={\overline{D\left(A\right)}}^{C(\overline{\mathcal{O}};{\mathbb{R}}^r)}={\overline{D\left(A_1\right)}}^{C(\overline{\mathcal{O}};\mathbb{R})}\times \cdots \times {\overline{D\left(A_r\right)}}^{C(\overline{\mathcal{O}};\mathbb{R})}$ and ${\overline{D\left(A_i\right)}}^{C(\overline{\mathcal{O}};\mathbb{R})}$ coincides with $C(\overline{\mathcal{O}};\mathbb{R})$ or $C_0(\overline{\mathcal{O}};\mathbb{R})$.

We set ${\mathcal{L}}_i(\xi ,D):=\sum _{h=1}^d\left(b_h^i\right(\xi )-\sum _{k=1}^d\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_k}\times a_{hk}^i(\xi \left)\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{\xi }_h},\xi \in \overline{\mathcal{O}},$ and define ${\mathcal{C}}_i:={\mathcal{A}}_i-{\mathcal{L}}_i$. Denoted by C the realization in H of the second-order elliptic operator $\mathcal{C}=({\mathcal{C}}_1,\dots ,{\mathcal{C}}_r)$, endowed with the boundary conditions $\mathcal{B}$. In Davies (1989), it is proved that C is a non-positive and self-adjoint operator which generates an analytic semigroup ${\mathrm{e}}^{tC}$ with dense domain given by ${\mathrm{e}}^{tC}=({\mathrm{e}}^{t{C}_1},\dots ,{\mathrm{e}}^{t{C}_r})$, where $C_i$ is the realization in $L^2\left(\mathcal{O}\right)$ of ${\mathcal{C}}_i$ with the boundary condition ${\mathcal{B}}_i$. Moreover, denoted by $L_p$ the realization of the operator $\mathcal{L}$ in $L^p(\mathcal{O};{\mathbb{R}}^r)$ for any $p\ge 1$. We see that $L_{p}x=L_{q}x,x\in D\left(L_p\right)\cap D\left(L_q\right)$ and denote all of them by L if there is no confusion.

The covariance operator Q of the Wiener process $W(\cdot )$ is a positive symmetric, trace class operator on H. Let $H_0=Q^{1/2}H$. Then $H_0$ is a Hilbert space with the inner product (5) $〈u,v{〉}_0=(Q^{-1/2}u,Q^{-1/2}v{)}_H\mathrm{\forall }u,v\in H_0.$(5) Let $|\cdot {|}_0$ denote the norm in $H_0$. Clearly, the embedding of $H_0$ in H is Hilbert-Schmidt, since Q is a trace class operator. Let $L_Q(H_0;H)$ denote the space of linear operators S such that $S{Q}^{1/2}$ is a Hilbert-Schmidt operator from H to H. Define the norm on the space $L_Q(H_0;H)$ by $|S{|}_{L_Q}=\sqrt{\mathrm{t}\mathrm{r}\left(SQ{S}^{\ast }\right)}$. Set $L_{(H_0,H)}$ the space of all bounded linear operators from $H_0$ into H and denote by $|\cdot {|}_{L_{(H_0,H)}}$ its norm.

The hypothesis concerns the eigenvalues of A:

Hypothesis 2.1

The complete orthonormal system of H which diagonalizes A is equi-bounded in the sup-norm.

Assume that $Q:=(Q_1,\dots ,Q_r):H\to H$ is a bounded linear operator which satisfies:

Hypothesis 2.2

Q is non-negative and diagonal with respect to the complete orthonormal basis which diagonalizes A, with eigenvalues $\left\{{\lambda }_n\right\}$. Moreover, if $d\ge 2$, there exists ϱ ($\varrho <\mathrm{\infty }$ if d = 2 and $\varrho <\frac{2d}{d-2}$ if d>2) such that $\Vert Q{\Vert }_{\varrho }:=(\sum _{k=1}^{\mathrm{\infty }}{\lambda }_n^{\varrho }{)}^{\frac{1}{\varrho }}<\mathrm{\infty }$.

The assumptions on the coefficients f and g are as follows.

Hypothesis 2.3

The mapping $g:[0,\mathrm{\infty })\times \overline{\mathcal{O}}\times {\mathbb{R}}^r\to \mathcal{L}\left({\mathbb{R}}^r\right)$ is continuous and $g(t,\xi ,\cdot ):{\mathbb{R}}^r\to \mathcal{L}\left({\mathbb{R}}^r\right)$ is Lipschitz-continuous, uniformly with respect to $\xi \in \overline{\mathcal{O}}$ and t in bounded sets of $[0,\mathrm{\infty })$, i.e., for some $\mathrm{\Psi }\in L_{loc}^{\mathrm{\infty }}[0,\mathrm{\infty }),$ $\underset{\xi \in \overline{\mathcal{O}}}{sup}\underset{\sigma ,\rho \in {\mathbb{R}}^r,\sigma \ne \rho }{sup}\frac{\Vert g(t,\xi ,\sigma )-g(t,\xi ,\rho ){\Vert }_{\mathcal{L}\left({\mathbb{R}}^r\right)}}{|\sigma -\rho |}\le \mathrm{\Psi }\left(t\right),t\ge 0$.

We set for any $x,y\in E$ and $t\ge 0$, $G(t,x)y\left(\xi \right):=g(t,\xi ,x(\xi \left)\right)y\left(\xi \right),$ and for $f=(f_1,\dots ,f_r)$, define for any $t\ge 0$ the composition operator $F(t,\cdot )$ by setting, for any $x:\overline{\mathcal{O}}\to {\mathbb{R}}^r$, $F(t,x)\left(\xi \right):=f(t,\xi ,x(\xi \left)\right),\xi \in \overline{\mathcal{O}}.$

Hypothesis 2.4

1. The mapping $F\left(t\right):E\to E$ is locally Lipschitz-continuous, locally uniformly for $t\ge 0$, and there exists $m\ge 1$ and $\mathrm{\Phi }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}[0,\mathrm{\infty })$ such that $|F(t,x){|}_E\le \mathrm{\Phi }\left(t\right)(1+|x{|}_E^m),x\in E,t\ge 0.$

2. There exists $\mathrm{\Lambda }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}[0,\mathrm{\infty })$ such that, for each $x,h\in E$, and for some ${\delta }_h\in \mathrm{\partial }|h{|}_E=\{h^{\ast }\in E\ast ;|h^{\ast }{|}_{E^{\ast }}=1,〈h^{\ast },h{〉}_E=|h{|}_E\},$ $t\ge 0,$ $〈F(t,x+h)-F(t,x),{\delta }_h{〉}_E\le \mathrm{\Lambda }\left(t\right)(1+|h{|}_E+|x{|}_E).$

3. One of the following two conditions holds:

   1. $\underset{\xi \in \overline{\mathcal{O}}}{sup}|g(t,\xi ,\sigma ){|}_{\mathcal{L}\left({\mathbb{R}}^r\right)}\le {\text{β}}_1\left(t\right)(1+|\sigma {|}^{\frac{1}{m}}),\sigma \in {\mathbb{R}}^r,t\ge 0$, where $m\ge 1$ is as in Hypothesis 2.4 (1) and $\text{β}\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}[0,\mathrm{\infty })$;

   2. there exists some a>0, $m\ge 1$ and ${\text{β}}_2\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}[0,\mathrm{\infty })$ such that, for each $x,h\in E$ and for some ${\delta }_h\in \mathrm{\partial }|h{|}_E$, $〈F(t,x+h)-F(t,x),{\delta }_h{〉}_E\le -a|h{|}_E^m+{\text{β}}_2\left(t\right)(1+|x{|}_E^m),t\ge 0.$

Then the system (1(1) $\{\begin{array}{l}{\displaystyle \frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }t}(t,\xi )={\mathcal{A}}_i{u}_i(t,\xi )+f_i(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)}\\ {\displaystyle +\sum _{j=1}^r{g}_{i,j}(\xi ,u_1(t,\xi ),\dots ,u_r(t,\xi \left)\right)Q_j\frac{\mathrm{\partial }{w}_j}{\mathrm{\partial }t}(t,\xi ),}\\ t\ge 0,\xi \in \overline{\mathcal{O}},\\ {\displaystyle u_i(0,\xi )=x_i\left(\xi \right),\xi \in \overline{\mathcal{O}},}\\ {\mathcal{B}}^i{u}_i(t,\xi )=0,t\ge 0,\xi \in \mathrm{\partial }\mathcal{O},1\le i\le r,\end{array}$(1) ) can be rewritten in the following type: (6) $\begin{array}{r}\{\begin{array}{l}\mathrm{d}u\left(t\right)=Au\left(t\right)\mathrm{d}t+F(t,u(t\left)\right)\mathrm{d}t\\ +G(t,u(t\left)\right)Q\mathrm{d}W\left(t\right),\\ u\left(s\right)=x,\end{array}\end{array}$(6) where the operator A can be written as A = C + L.

3. Central limit theorem

In this section, our task is to establish the CLT. We will need some results in Cerrai (2003), and we list them here for the convenience of readers.

Proposition 3.1

Under Hypotheses 2.1–2.3 on the coefficients, define the following mappings (7) $\begin{array}{rl}\psi \left(f\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}L\left(f\right)\left(r\right)\mathrm{d}r,t\in [s,T],\end{array}$(7) (8) $\begin{array}{rl}\gamma \left(u^{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\sqrt{ϵ}G(r,u^{ϵ}(r\left)\right)Q\mathrm{d}{W}_r.\end{array}$(8) Then, (1) (the inequality (3.7) in Cerrai, 2003) for any $p\ge 1$ and $f\in C\left(\right[s,t];D(L\left)\right)$ (9) $|\psi \left(f\right){|}_E\le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}|f(r){|}_E\mathrm{d}r$(9) where $ϵ(>0)$ is any constant;

(2) (the inequality (4.6) in Cerrai, 2003) there exists $p_{\ast }\ge 1$ such that γ maps $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$ into itself for any $p\ge p_{\ast }$ and for any $u,v\in L^p(\mathrm{\Omega };C([0,T];E\left)\right)$ (10) $|\gamma \left(u\right)-\gamma \left(v\right){|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}^{\gamma }\left(T\right)|u-v{|}_{L_{s,T,p}\left(E\right)},$(10) for some continuous increasing function $c_{s,p}^{\gamma }$ such that $c_{s,p}^{\gamma }\left(s\right)=0$.

Proposition 3.2

Theorem 5.3 in Cerrai, 2003

Under Hypotheses 2.1–2.4 on the coefficients, for any $x\in E$ and for any $p\ge 1$ and T>0 such a problem admits a unique mild solution in $L^p(\mathrm{\Omega };C([0,T];E)\cap L^{\mathrm{\infty }}(s,T;E\left)\right)$ which is the Banach space of all adapted processes u in $C\left(\right[0,T];E)\cap L^{\mathrm{\infty }}(s,T;E)$, such that $|u{|}_{L_{0,T,p}\left(E\right)}<\mathrm{\infty }$, where $|u{|}_{L_{s,T,p}\left(E\right)}:=\left[\mathbb{E}\underset{r\in [s,T]}{sup}|u\right(r){|}_E^p{]}^{\frac{1}{p}}$. Moreover, (11) $|u{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)(1+|x{|}_E).$(11)

Similarly as the proof of (11(11) $|u{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)(1+|x{|}_E).$(11) ), under Hypotheses 2.1–2.4, we have $|u^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)(1+|x{|}_E)$. The following result is concerned with the convergence of $u^{ϵ}$ as $ϵ\to 0$.

Proposition 3.3

Under Hypotheses 2.1–2.4, there exists a constant ${ϵ}_0>0$ such that, for all $0<ϵ\le {ϵ}_0$, (12) $|u^{ϵ}-u^0{|}_{L_{s,T,p}\left(E\right)}\le c\sqrt{ϵ}.$(12)

Proof.

Define (13) $F_n(t,x)\left(\xi \right)=\{\begin{array}{ll}F(t,x)\left(\xi \right),& \text{if }|x|\le n,\\ F(t,nx/|x|)\left(\xi \right),& \text{if }|x|>n.\end{array}$(13) It is immediate to check that $F_n$ is Lipschitz-continuous in E. Then if we consider (14) $\{\begin{array}{l}\mathrm{d}{u}_n^{ϵ}\left(t\right)=A{u}_n^{ϵ}\left(t\right)\mathrm{d}t+F_n(t,u_n^{ϵ}(t\left)\right)\mathrm{d}t\\ +\sqrt{ϵ}G(t,u_n^{ϵ}(t\left)\right)Q\mathrm{d}W\left(t\right),\\ u_n^{ϵ}\left(s\right)=x\end{array}$(14) and (15) $\{\begin{array}{l}\mathrm{d}{u}_n^0\left(t\right)=A{u}_n^0\left(t\right)\mathrm{d}t+F_n(t,u_n^0(t\left)\right)\mathrm{d}t,\\ u_n^0\left(s\right)=x,\end{array}$(15) there exist unique mild solutions $u_n^{ϵ}$ and $u_n^0$ in $L^p(\mathrm{\Omega };C((s,T];E)\cap L^{\mathrm{\infty }}(s,T;E\left)\right)$ for Equations (14(14) $\{\begin{array}{l}\mathrm{d}{u}_n^{ϵ}\left(t\right)=A{u}_n^{ϵ}\left(t\right)\mathrm{d}t+F_n(t,u_n^{ϵ}(t\left)\right)\mathrm{d}t\\ +\sqrt{ϵ}G(t,u_n^{ϵ}(t\left)\right)Q\mathrm{d}W\left(t\right),\\ u_n^{ϵ}\left(s\right)=x\end{array}$(14) ) and (15(15) $\{\begin{array}{l}\mathrm{d}{u}_n^0\left(t\right)=A{u}_n^0\left(t\right)\mathrm{d}t+F_n(t,u_n^0(t\left)\right)\mathrm{d}t,\\ u_n^0\left(s\right)=x,\end{array}$(15) ), respectively. Define $\begin{array}{rl}{\mathrm{\Lambda }}_n(u^{ϵ}-u^0)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\left[F_n\right(r,u^{ϵ}\left(r\right))\\ & -F_n(r,u^0(r\left)\right)]\mathrm{d}r.\end{array}$ So, $\begin{array}{rl}(u_n^{ϵ}-u_n^0)\left(t\right)& =\psi (u_n^{ϵ}-u_n^0)\left(t\right)+{\mathrm{\Lambda }}_n(u_n^{ϵ}-u_n^0)\left(t\right)\\ & +\gamma \left(u_n^{ϵ}\right)\left(t\right).\end{array}$ From (9(9) $|\psi \left(f\right){|}_E\le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}|f(r){|}_E\mathrm{d}r$(9) ), (16) $\begin{array}{rl}|\psi (u^{ϵ}-u^0){|}_E& \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\\ & \times |(u^{ϵ}-u^0)\left(r\right){|}_E\mathrm{d}r\\ & \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\mathrm{d}r\\ & \times \underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E,\end{array}$(16) and one obtains $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|(u_n^{ϵ}-u_n^0)\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|(u_n^{ϵ}-u_n^0)\left(r\right){|}_E+\underset{r\in [s,t]}{sup}|\gamma \left(u_n^{ϵ}\right)\left(r\right){|}_E\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r\end{array}$ in view of the Lipschitz continuity of $F_n$.

If we take $t_0>0$ such that $c_s^{\psi }(s+t_0)<\frac{1}{2}$, this yields $\begin{array}{rl}& \underset{r\in [s,s+t_0]}{sup}|(u_n^{ϵ}-u_n^0)\left(r\right){|}_E\\ & \le 2\underset{r\in [s,s+t_0]}{sup}|\gamma \left(u_n^{ϵ}\right)\left(r\right){|}_E\\ & +2c_n(s+t_0){\int }_s^{s+t_0}\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r.\end{array}$ Gronwall's inequality implies $\begin{array}{rl}& \underset{r\in [s,s+t_0]}{sup}|(u_n^{ϵ}-u_n^0)\left(r\right){|}_E\\ & \le 2\underset{r\in [s,s+t_0]}{sup}|\gamma \left(u_n^{ϵ}\right)\left(r\right){|}_E{\mathrm{e}}^{2c_n(s+t_0)t_0}\\ & =:c(s+t_0,n)\underset{r\in [s,s+t_0]}{sup}|\gamma \left(u_n^{ϵ}\right)\left(r\right){|}_E.\end{array}$ Therefore, due to (10(10) $|\gamma \left(u\right)-\gamma \left(v\right){|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}^{\gamma }\left(T\right)|u-v{|}_{L_{s,T,p}\left(E\right)},$(10) ), $\begin{array}{rl}|u_n^{ϵ}-u_n^0{|}_{L_{s,s+t_0,p}\left(E\right)}& =\left[\mathbb{E}\underset{r\in [s,s+t_0]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & \le c(s+t_0,n)|\gamma \left(u_n^{ϵ}\right)\left(r\right){|}_{L_{s,s+t_0,p}\left(E\right)}\\ & \le c(s+t_0,n)c_{s,p}^{\gamma }|u_n^{ϵ}{|}_{L_{s,s+t_0,p}\left(E\right)}\cdot \sqrt{ϵ}.\end{array}$ Next we can repeat the same arguments in the intervals $[s+t_0,s+2t_0]$, $[s+2t_0,s+3t_0]$ and so on and for any T>s, (17) $|u_n^{ϵ}-u_n^0{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)\sqrt{ϵ}$(17) with the help of (11(11) $|u{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)(1+|x{|}_E).$(11) ) for $u_n^{ϵ}$ and $u_n^0$.

Define ${\tau }_n=inf\{t\ge s:|(u_n^{ϵ}-u_n^0\left)\right(t){|}_E\ge n\}$ and $inf\mathrm{\varnothing }=+\mathrm{\infty }$. We can see that ${\tau }_n$ is non-decreasing and denote $\tau :=\underset{n\in \mathbb{N}}{sup}{\tau }_n$. So (17(17) $|u_n^{ϵ}-u_n^0{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)\sqrt{ϵ}$(17) ) with p = 1 yields $\begin{array}{rl}P(\tau <\mathrm{\infty })& =\underset{T\to \mathrm{\infty }}{lim}P(\tau \le T)\\ & =\underset{T\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim}P({\tau }_n\le T)\\ & =\underset{T,n\to \mathrm{\infty }}{lim}P\left(\underset{t\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(t){|}_E\ge n)\\ & \le \underset{T,n\to \mathrm{\infty }}{lim}\frac{1}{n}\mathbb{E}\left[\underset{t\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(t){|}_E\ge n]\\ & =0,\end{array}$ i.e. $P(\tau =\mathrm{\infty })=1$. Thus, Fatou's lemma implies (18) $\begin{array}{rl}& \mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le \underset{n\to \mathrm{\infty }}{lim}\mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le (c_{s,p}\sqrt{ϵ}{)}^p,\end{array}$(18) which completes the proof.

Let $V^{ϵ}\left(t\right):=(u^{ϵ}-u^0)/\sqrt{ϵ}$ be the solution of the following SPDE: $\begin{array}{rl}\mathrm{d}{V}^{ϵ}\left(t\right)& =A{V}^{ϵ}\left(t\right)\mathrm{d}t\\ & +\left[F\right(t,u^{ϵ}\left(t\right))-F(t,u^0\left(t\right)\left)\right]/\sqrt{ϵ}\mathrm{d}t\\ & +G(t,u^{ϵ}(t\left)\right)Q\mathrm{d}W\left(t\right),\end{array}$ with initial value $V^{ϵ}\left(0\right)=0$. Let $V^0$ be the solution of the following SPDE: $\begin{array}{rl}\mathrm{d}{V}^0\left(t\right)& =A{V}^0\left(t\right)\mathrm{d}t+F^{\prime }\left(u^0\right(t\left)\right)V^0\left(t\right)\mathrm{d}t\\ & +G(t,u^0(t\left)\right)Q\mathrm{d}W\left(t\right),\\ V^0\left(0\right)& =0,\end{array}$ where we need an additional assumption about $F^{\prime }$.

Hypothesis 3.1

$F^{\prime }:[0,T]\times E\to L\left(E\right)$ is Fréchet derivative of F w.r.t. the second variable satisfying $|F^{\prime }(t,u){|}_{L\left(E\right)}\le F_0^{\prime }|u|+F_1^{\prime }$ for some positive $F_0^{\prime }$, $F_1^{\prime }$ and there exists ${\mathrm{\Lambda }}^{\prime }\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}[0,\mathrm{\infty })$ such that, for each $x,h\in E$ and $t\ge 0,$ $〈F^{\prime }(t,x+h)-F^{\prime }(t,x),{\delta }_h{〉}_E\le {\mathrm{\Lambda }}^{\prime }\left(t\right)(1+|h{|}_E+|x{|}_E)$.

We now come to the main result of this section.

Theorem 3.4

Central Limit Theorem

Under Hypotheses 2.1–3.1, $V^{ϵ}\left(t\right)$ converges to $V^0$ in the space $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$ in probability, that is, there exists a constant ${ϵ}_0>0$ such that, for all $0<ϵ\le {ϵ}_0$, (19) $|V^{ϵ}-V^0{|}_{L_{s,T,p}\left(E\right)}\le c\sqrt{ϵ}.$(19)

Proof.

Define $F_n^{\prime }(t,x)\left(\xi \right)=\{\begin{array}{ll}{F}^{\prime }(t,x)\left(\xi \right),& \text{if }|x|\le n,\\ F^{\prime }(t,nx/|x|)\left(\xi \right),& \text{if }|x|>n,\end{array}$ and we see that $F_n^{\prime }$ is Lipschitz-continuous in E. Then if we consider $\begin{array}{rl}\mathrm{d}{V}_n^{ϵ}\left(t\right)& =A{V}_n^{ϵ}\left(t\right)\mathrm{d}t\\ & +\left[F_n\right(t,u^{ϵ}\left(t\right))-F_n(t,u^0\left(t\right)\left)\right]/\sqrt{ϵ}\mathrm{d}t\\ & +G(t,u^{ϵ}(t\left)\right)Q\mathrm{d}W\left(t\right),V_n^{ϵ}\left(s\right)=0\end{array}$ and $\begin{array}{rl}\mathrm{d}{V}_n^0\left(t\right)& =A{V}_n^0\left(t\right)\mathrm{d}t+F_n^{\prime }(t,u^0(t\left)\right)V_n^0\left(t\right)\mathrm{d}t\\ & +G(t,u^0(t\left)\right)Q\mathrm{d}W\left(t\right),V_n^0\left(s\right)=0.\end{array}$ There exists unique mild solutions $V_n^{ϵ}$ and $V_n^0$ in $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$, respectively. Define the following mappings $\begin{array}{rl}\psi (V^{ϵ}-V^0)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}L(V^{ϵ}-V^0)\left(r\right)\mathrm{d}r;\\ {\mathrm{\Lambda }}_n(V^{ϵ}-V^0)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\left\{\right[F_n(r,u^{ϵ}(r\left)\right)\\ & -F_n(r,u^0(r\left)\right)]/\sqrt{ϵ}\\ & -F_n^{\prime }(r,u^0(r\left)\right)\}\mathrm{d}r;\\ \gamma (V^{ϵ}-V^0)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\left[G\right(r,u^{ϵ}\left(r\right))\\ & -G(r,u^0(r\left)\right)]Q\mathrm{d}{W}_r.\end{array}$ Then $\begin{array}{rl}(V_n^{ϵ}-V_n^0)\left(t\right)& =\psi (V_n^{ϵ}-V_n^0)\left(t\right)+{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(t\right)\\ & +\gamma (V_n^{ϵ}-V_n^0)\left(t\right).\end{array}$ Due to inequality (16(16) $\begin{array}{rl}|\psi (u^{ϵ}-u^0){|}_E& \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\\ & \times |(u^{ϵ}-u^0)\left(r\right){|}_E\mathrm{d}r\\ & \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\mathrm{d}r\\ & \times \underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E,\end{array}$(16) ), one obtains $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & +\underset{r\in [s,t]}{sup}|{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & +\underset{r\in [s,t]}{sup}|\gamma (V_n^{ϵ}-V_n^0)\left(r\right){|}_E.\end{array}$ If we take $t_1>0$ such that $c_s^{\psi }(s+t_1)<\frac{1}{2}$, this yields (20) $\begin{array}{rl}& \underset{r\in [s,s+t_1]}{sup}|(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le 2\underset{r\in [s,s+t_1]}{sup}|{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & +2\underset{r\in [s,s+t_1]}{sup}|\gamma (V_n^{ϵ}-V_n^0)\left(r\right){|}_E.\end{array}$(20) Consider $\begin{array}{rl}& \left[F_n\right(r,u^{ϵ}\left(r\right))-F_n(r,u^0\left(r\right)\left)\right]/\sqrt{ϵ}-F_n^{\prime }(r,u^0(r\left)\right)V_n^0\\ & =\left[F_n\right(r,u^{ϵ}\left(r\right))-F_n(r,u^0\left(r\right)\left)\right]/\sqrt{ϵ}+F_n^{\prime }(r,u^0(r\left)\right)\\ & \times [V_n^{ϵ}-V_n^0-V_n^{ϵ}]\\ & =F_n^{\prime }(r,u^0(r\left)\right)[V_n^{ϵ}-V_n^0]\\ & +\left[F_n\right(r,u^{ϵ}\left(r\right))-F_n(r,u^0\left(r\right))\\ & -F_n^{\prime }(r,u^0(r\left)\right)(u_n^{ϵ}-u_n^0)]/\sqrt{ϵ}.\end{array}$ There exists a random field ${\eta }^{ϵ}(r,x)$ taking values in $(0,1)$ such that $\begin{array}{rl}& F_n(r,u^{ϵ}(r\left)\right)-F_n(r,u^0(r\left)\right)\\ & =F_n^{\prime }(r,u_n^0+{\eta }^{ϵ}(u_n^{ϵ}-u_n^0\left)\right)\cdot (u_n^{ϵ}-u_n^0).\end{array}$ Then, $\begin{array}{rl}& \left[F_n\right(r,u^{ϵ}\left(r\right))-F_n(r,u^0\left(r\right)\left)\right]/\sqrt{ϵ}-F_n^{\prime }(r,u^0(r\left)\right)V_n^0\\ & =F_n^{\prime }(r,u^0(r\left)\right)[V_n^{ϵ}-V_n^0]\\ & +\left[F_n^{\prime }\right(r,u_n^0+{\eta }^{ϵ}(u_n^{ϵ}-u_n^0))-F_n^{\prime }(r,u^0\left(r\right)\left)\right]\\ & \times (u_n^{ϵ}-u_n^0)/\sqrt{ϵ}.\end{array}$ In view of last inequality, the Lipschitz continuity of $F_n^{\prime }$ and Hypothesis 3.1, it yields (21) $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|F_n^{\prime }\left(u^0\right(r^{\prime }\left)\right)(V_n^{ϵ}-V_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E^2/\sqrt{ϵ}\mathrm{d}r\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}\left(F_0^{\prime }|u^0\right(r^{\prime }){|}_E+F_1^{\prime })\\ & \times |(V_n^{ϵ}-V_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E^2/\sqrt{ϵ}\mathrm{d}r.\end{array}$(21) Inequalities (20(20) $\begin{array}{rl}& \underset{r\in [s,s+t_1]}{sup}|(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le 2\underset{r\in [s,s+t_1]}{sup}|{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & +2\underset{r\in [s,s+t_1]}{sup}|\gamma (V_n^{ϵ}-V_n^0)\left(r\right){|}_E.\end{array}$(20) ), (21(21) $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|{\mathrm{\Lambda }}_n(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|F_n^{\prime }\left(u^0\right(r^{\prime }\left)\right)(V_n^{ϵ}-V_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E^2/\sqrt{ϵ}\mathrm{d}r\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}\left(F_0^{\prime }|u^0\right(r^{\prime }){|}_E+F_1^{\prime })\\ & \times |(V_n^{ϵ}-V_n^0)\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|(u_n^{ϵ}-u_n^0)\left(r^{\prime }\right){|}_E^2/\sqrt{ϵ}\mathrm{d}r.\end{array}$(21) ) and Gronwall's inequality implies $\begin{array}{rl}& \underset{r\in [s,s+t_1]}{sup}|(V_n^{ϵ}-V_n^0)\left(r\right){|}_E\\ & \le \left[2c_n\right(s+t_1\left){\int }_s^{s+t_1}\underset{r^{\prime }\in [s,r]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r^{\prime }){|}_E^2/\sqrt{ϵ}\mathrm{d}r\\ & +2\underset{r\in [s,s+t_1]}{sup}|\gamma (V_n^{ϵ}-V_n^0)\left(r\right){|}_E\phantom{{\int }_s^{s+t_1}}]\\ & \times \exp \left\{\phantom{{\int }_s^{s+t_1}}{\int }_s^{s+t_1}2c_n\right(s+t_1)\\ & \times \underset{r^{\prime }\in [s,r]}{sup}\left(F_0^{\prime }|u^0\right(r^{\prime }){|}_E+F_1^{\prime })\mathrm{d}r\}.\end{array}$

Furthermore, $\begin{array}{rl}& \{\mathbb{E}\left[2c_n\right(s+t_1\left){\int }_s^{s+t_1}\underset{r^{\prime }\in [s,r]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r^{\prime }){|}_E^2/\sqrt{ϵ}\mathrm{d}r\\ & {+{2\underset{r\in [s,s+t_1]}{sup}|\gamma (V_n^{ϵ}-V_n^0)\left(r\right){|}_E]}^p\}}^{\frac{1}{p}}\\ & \le 2c_n(s+t_1)\{\mathbb{E}\left[{\int }_s^{s+t_1}\underset{r^{\prime }\in [s,r]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r^{\prime }){|}_E^2\\ & {-{\sqrt{ϵ}\mathrm{d}r]}^p\phantom{{\int }_s^{s+t_1}\underset{r^{\prime }\in [s,r]}{sup}}\}}^{\frac{1}{p}}\\ & +2{\left\{\mathbb{E}{\left[\underset{r\in [s,s+t_1]}{sup}|\gamma \phantom{\underset{r\in [s,s+t_1]}{sup}}\right(V_n^{ϵ}-V_n^0\left)\right(r\left){|}_E\right]}^p\right\}}^{\frac{1}{p}}\\ & \le 2c_n(s+t_1){\int }_s^{s+t_1}|u_n^{ϵ}-u_n^0{|}_{L_{s,r,2p}\left(E\right)}^2/\sqrt{ϵ}\mathrm{d}r\\ & +2|\gamma (V_n^{ϵ}-V_n^0){|}_{L_{s,r,p}\left(E\right)}\\ & \le 2c_n(s+t_1){\int }_s^{s+t_1}|c\sqrt{ϵ}{|}^2/\sqrt{ϵ}\mathrm{d}r\\ & +2c_{s,1}^{\gamma }(s+t_1)|V_n^{ϵ}-V_n^0{|}_{L_{s,r,p}\left(E\right)}\\ & =2c_n(s+t_1)c^2{t}_1\sqrt{ϵ}+2c_{s,1}^{\gamma }(s+t_1)\\ & \times |V_n^{ϵ}-V_n^0{|}_{L_{s,r,p}\left(E\right)},\end{array}$ according to inequality (17(17) $|u_n^{ϵ}-u_n^0{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)\sqrt{ϵ}$(17) ). Also, $\exp \left\{{\int }_s^{s+t_1}2c_n\right(s+t_1\left)\underset{r^{\prime }\in [s,r]}{sup}\right(F_0^{\prime }|u^0\left(r^{\prime }\right){|}_E+F_1^{\prime }\left)\mathrm{d}r\right\}$ is finitely controlled by some positive constant M from $|u_n^0{|}_E\le c_{s,2p}(s+t_1)(1+|x{|}_E)$.

Then, we take $t_2\in (0,t_1)$ such that $2c_n(s+t_1)<\frac{1}{2}$, (22) $|(V_n^{ϵ}-V_n^0){|}_{L_{s,s+t_2,p}\left(E\right)}\le 2M\cdot 2c_n(s+t_1)c^2{t}_2\sqrt{ϵ}.$(22) Then we can repeat the same arguments in the intervals $[s+t_1,s+2t_1]$, $[s+2t_1,s+3t_1]$ and so on and for any T>s, (23) $|V_n^{ϵ}-V_n^0{|}_{L_{s,T,p}\left(E\right)}\le c\sqrt{ϵ},$(23) with the help of (11(11) $|u{|}_{L_{s,T,p}\left(E\right)}\le c_{s,p}\left(T\right)(1+|x{|}_E).$(11) ).

Similarly as inequality (18(18) $\begin{array}{rl}& \mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le \underset{n\to \mathrm{\infty }}{lim}\mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le (c_{s,p}\sqrt{ϵ}{)}^p,\end{array}$(18) ) replacing $u^{ϵ}-u^0$ with $V^{ϵ}-V^0$, the proof is completed.

4. Moderate deviations

Let $Z^{ϵ}=(u^{ϵ}-u^0)/\left(\sqrt{ϵ}\lambda \right(ϵ\left)\right)$. Then $Z^{ϵ}$ satisfies the following SPDE: (24) $\begin{array}{rl}\mathrm{d}{Z}^{ϵ}\left(t\right)& =A{Z}^{ϵ}\left(t\right)\mathrm{d}t+\left[F\right(t,u^0\left(t\right)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)Z^{ϵ}\left(t\right))-F(t,u^0\left(t\right)\left)\right]\\ & /\left(\sqrt{ϵ}\lambda \right(ϵ\left)\right)\mathrm{d}t+G(t,u^0(t)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)Z^{ϵ}\left(t\right)\left)/\lambda \right(ϵ\left)Q\mathrm{d}W\right(t),\end{array}$(24) with initial value $Z^{ϵ}\left(0\right)=0$. This equation admits a unique solution $Z^{ϵ}={\mathrm{\Gamma }}^{ϵ}\left(W\right(\cdot \left)\right)$, where ${\mathrm{\Gamma }}^{ϵ}$ stands for the solution functional from $C\left(\right[0,T];H)$ into $\mathcal{E}$.

In this part, we will prove that $Z^{ϵ}$ satisfies an LDP on $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$ with $\lambda \left(ϵ\right)$ satisfying (4(4) $\lambda \left(ϵ\right)\to +\mathrm{\infty },\sqrt{ϵ}\lambda \left(ϵ\right)\to 0,\mathrm{a}\mathrm{s}ϵ\to 0,$(4) ). This special type of LDP is usually called the MDP of $u^{ϵ}$ (cf. Dembo & Zeitouni, 2000).

Firstly we recall the general criteria for an LDP given in Budhiraja and Dupuis (2000). Let $\mathcal{E}$ be a Polish space with the Borel σ-field $\mathcal{B}\left(\mathcal{E}\right)$.

Definition 4.1

Rate function

A function $I:\mathcal{E}\to [0,\mathrm{\infty }]$ is called a rate function on $\mathcal{E}$, if for each $M<\mathrm{\infty }$, the level set $\{x\in \mathcal{E}:I(x)\le M\}$ is a compact subset of $\mathcal{E}$.

Definition 4.2

LDP

Let I be a rate function on $\mathcal{E}$. A family $\left\{Z^{ϵ}\right\}$ of $\mathcal{E}$-valued random elements is said to satisfy the LDP on $\mathcal{E}$ with rate function I, if the following two conditions hold.

1. (Large deviation upper bound) For each closed subset F of $\mathcal{E}$, $\begin{array}{r}\underset{ϵ\to 0}{lim sup}ϵ\log \mathbb{P}(Z^{ϵ}\in F)\le -\underset{x\in F}{inf}I\left(x\right).\end{array}$

2. (Large deviation lower bound) For each open subset G of $\mathcal{E}$, $\begin{array}{r}\underset{ϵ\to 0}{lim inf}ϵ\log \mathbb{P}(Z^{ϵ}\in G)\ge -\underset{x\in G}{inf}I\left(x\right).\end{array}$

Next, we introduce the Skeleton Equations. The Cameron-Martin space associated with the Wiener process $\left\{W\right(t),t\in [0,T\left]\right\}$ is given by $\begin{array}{rl}{\mathcal{H}}_0& :=\{\phantom{{\int }_0^T}h:[0,T]\to H_0:h\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{l}\mathrm{y}\\ & \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{u}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}{\int }_0^T|\dot{h}\left(s\right){|}_0^2\mathrm{d}s<+\mathrm{\infty }\}.\end{array}$ The space ${\mathcal{H}}_0$ is a Hilbert space with inner product $〈h_1,h_2{〉}_{{\mathcal{H}}_0}:={\int }_0^T〈\dot{\phantom{a}}{h}_1\left(s\right),{\dot{h}}_2\left(s\right){〉}_0\mathrm{d}s.$ Let $\mathcal{A}$ denote the class of $H_0$-valued $\left\{{\mathcal{F}}_t\right\}$-predictable processes ϕ belonging to ${\mathcal{H}}_0$ a.s.. Let $S_N=\{h\in {\mathcal{H}}_0:{\int }_0^T|\dot{h}(s){|}_0^2\mathrm{d}s\le N\}$. The set $S_N$ endowed with the weak topology is a Polish space. Define ${\mathcal{A}}_N=\{\varphi \in \mathcal{A}:\varphi (\omega )\in S_N,\mathbb{P}-\mathrm{a}.\mathrm{s}.\}$.

For any $h\in {\mathcal{H}}_0$, consider the deterministic equation (25) $\begin{array}{rl}\mathrm{d}{X}^h\left(t\right)& =A{X}^h\left(t\right)\mathrm{d}t+F^{\prime }\left(u^0\right(t\left)\right)X^h\left(t\right)\mathrm{d}t\\ & +G(t,u^0(t\left)\right)Q\dot{h}\left(t\right)\mathrm{d}t,\end{array}$(25) with initial value $X^h\left(0\right)=0$ and for any ${\varphi }^{ϵ}\in \mathcal{A}$, consider (26) $\begin{array}{rl}\mathrm{d}{X}^{ϵ}\left(t\right)& =A{X}^{ϵ}\left(t\right)\mathrm{d}t+\left[F\right(t,u^0\left(t\right)+\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(t\right))\\ & -F(t,u^0(t\left)\right)\left]/\right(\sqrt{ϵ}\lambda \left(ϵ\right))\mathrm{d}t\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X^{ϵ}\right(t\left)\right)/\lambda \left(ϵ\right)Q\mathrm{d}W\left(t\right)\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X^{ϵ}\right(t\left)\right)Q{\dot{\varphi }}^{ϵ}\left(t\right)\mathrm{d}t,\\ X^{ϵ}\left(0\right)& =0.\end{array}$(26) Now we are ready to state the second main result.

Theorem 4.3

Moderate Deviation Principle

Under Hypotheses 2.1–3.1, $Z^{ϵ}$ obeys an LDP on $L^p(\mathrm{\Omega };C([0,T];E)$ with speed ${\lambda }^2\left(ϵ\right)$ and with rate function I given by $\begin{array}{rl}I\left(g\right)& :=\underset{\{h\in {\mathcal{H}}_0:g=X^h\}}{inf}\left\{\frac{1}{2}{\int }_0^T|\dot{h}\right(s\left){|}_0^2\mathrm{d}s\right\},\\ & \mathrm{\forall }g\in L^p(\mathrm{\Omega };C([0,T];E\left)\right),\end{array}$ with the convention $inf\left\{\mathrm{\varnothing }\right\}=\mathrm{\infty }$.

We will adopt the following weak convergence method to prove the MDP.

Theorem 4.4

Budhiraja & Dupuis, 2000

For $ϵ>0$, let ${\mathrm{\Gamma }}^{ϵ}$ be a measurable mapping from $C\left(\right[0,T];H)$ into $\mathcal{E}$. Let $Y^{ϵ}:={\mathrm{\Gamma }}^{ϵ}\left(W\right(\cdot \left)\right)$. Suppose that $\{{\mathrm{\Gamma }}^{ϵ}{\}}_{ϵ>0}$ satisfies the following assumptions: there exists a measurable map ${\mathrm{\Gamma }}^0:C\left(\right[0,T];H)\to \mathcal{E}$ such that

1. for every $N<+\mathrm{\infty }$ and any family $\{h^{ϵ};ϵ>0\}\subset {\mathcal{A}}_N$ satisfying that $h^{ϵ}$ converge in distribution as $S_N$-valued random elements to h as $ϵ\to 0$, ${\mathrm{\Gamma }}^{ϵ}\left(W\right(\cdot )+{\int }_0^{\cdot }{\dot{h}}^{ϵ}(s\left)\mathrm{d}s/\sqrt{ϵ}\right)$ converges in distribution to ${\mathrm{\Gamma }}^0\left({\int }_0^{\cdot }\dot{h}\right(s\left)\mathrm{d}s\right)$ as $ϵ\to 0$;

2. for every $N<+\mathrm{\infty }$, the set $\left\{{\mathrm{\Gamma }}^0\right({\int }_0^{\cdot }\dot{h}\left(s\right)\mathrm{d}s);h\in S_N\}$ is a compact subset of $\mathcal{E}$.

Then the family $\{Y^{ϵ}{\}}_{ϵ>0}$ satisfies an LDP in $\mathcal{E}$ with the rate function I given by $\begin{array}{rl}I\left(g\right)& :=\underset{\{h\in {\mathcal{H}}_0:g={\mathrm{\Gamma }}^0({\int }_0^{\cdot }\dot{h}\left(s\right)\mathrm{d}s\left)\right\}}{inf}\left\{\frac{1}{2}{\int }_0^T|\dot{h}\right(s\left){|}_0^2\mathrm{d}s\right\},\\ & g\in \mathcal{E},\end{array}$ with the convention $inf\left\{\mathrm{\varnothing }\right\}=\mathrm{\infty }$.

For $h\in {\mathcal{H}}_0$, set ${\mathrm{\Gamma }}^0\left({\int }_0^{\cdot }\dot{\phantom{a}}h\right(s\left)\mathrm{d}s\right):=X^h$ and define ${\mathrm{\Gamma }}^{ϵ}$ satisfying ${\mathrm{\Gamma }}^{ϵ}\left(W\right(\cdot )+\lambda (ϵ\left){\int }_0^{\cdot }{\dot{\varphi }}^{ϵ}\right(s\left)\mathrm{d}s\right)=X^{ϵ}.$ To prove Theorem 4.3, we only need to verify the following two propositions according to Theorem 4.4.

Proposition 4.5

Under the same conditions as Theorem 4.3, for every fixed $N\in \mathbb{N}$, let ${\varphi }^{ϵ},\varphi \in {\mathcal{A}}_N$ be such that ${\varphi }^{ϵ}$ converges in distribution to ϕ as $ϵ\to 0$. Then ${\mathrm{\Gamma }}^{ϵ}\left(W\right(\cdot )+\lambda (ϵ\left){\int }_0^{\cdot }{\dot{\varphi }}^{ϵ}\right(s\left)\mathrm{d}s\right)\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}{\mathrm{\Gamma }}^0\left({\int }_0^{\cdot }\dot{\varphi }\right(s\left)\mathrm{d}s\right)$ in $C\left(\right[0,T];H)\cap L^2\left(\right[0,T];V)$ as $ϵ\to 0$.

Proposition 4.6

Under the same conditions as Theorem 4.3, for every positive number $N<\mathrm{\infty }$, the family $K_N:=\left\{{\mathrm{\Gamma }}^0\right({\int }_0^{\cdot }\dot{h}\left(s\right)\mathrm{d}s):h\in S_N\}$ is compact in $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$.

We start to prove Proposition 4.5 and we need the following lemma and inequalities.

Lemma 4.1

Under the same conditions as Theorem 4.3, for any $\varphi \in {\mathcal{H}}_0$ and ${\varphi }^{ϵ}\in \mathcal{A}$, (25(25) $\begin{array}{rl}\mathrm{d}{X}^h\left(t\right)& =A{X}^h\left(t\right)\mathrm{d}t+F^{\prime }\left(u^0\right(t\left)\right)X^h\left(t\right)\mathrm{d}t\\ & +G(t,u^0(t\left)\right)Q\dot{h}\left(t\right)\mathrm{d}t,\end{array}$(25) ) and (26(26) $\begin{array}{rl}\mathrm{d}{X}^{ϵ}\left(t\right)& =A{X}^{ϵ}\left(t\right)\mathrm{d}t+\left[F\right(t,u^0\left(t\right)+\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(t\right))\\ & -F(t,u^0(t\left)\right)\left]/\right(\sqrt{ϵ}\lambda \left(ϵ\right))\mathrm{d}t\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X^{ϵ}\right(t\left)\right)/\lambda \left(ϵ\right)Q\mathrm{d}W\left(t\right)\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X^{ϵ}\right(t\left)\right)Q{\dot{\varphi }}^{ϵ}\left(t\right)\mathrm{d}t,\\ X^{ϵ}\left(0\right)& =0.\end{array}$(26) ) replacing F and $F^{\prime }$ with $F_n$ and $F_n^{\prime }$ respectively, admit the unique solutions, respectively, $X_n^{\varphi }$, $X_n^{ϵ}$ in $L^p(\mathrm{\Omega };C([0,T];E\left)\right)$. Moreover, for any N>0 and $p\ge 2$, there exist constants $c_{n,N,T}$ and ${ϵ}_0>0$ such that for any $\varphi \in S_N$, ${\varphi }^{ϵ}\in {\mathcal{A}}_N$, (27) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}$(27) and (28) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{\varphi }{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(28)

Proof.

Similarly as Theorem 2.4 in Chueshov and Millet (2010), the existence and uniqueness of the solution can be proved. Here, we will prove inequalities (27(27) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}$(27) ) and (28(28) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{\varphi }{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(28) ).

Define the following mappings $\begin{array}{rl}\psi \left(X^{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}L{X}^{ϵ}\left(r\right)\mathrm{d}r;\\ {\mathrm{\Lambda }}_n\left(X^{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\\ & \times \left[F_n\right(r,u^0\left(r\right)+\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right))\\ & -F_n(r,u^0(r\left)\right)\left]/\right(\sqrt{ϵ}\lambda \left(ϵ\right))\mathrm{d}r;\\ \gamma \left(X^{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}G(r,u^0(r)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right)\left)/\lambda \right(ϵ)Q\mathrm{d}{W}_r;\\ {\gamma }^{\varphi }\left(X^{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}G(r,u^0(r)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right)\left)Q{\dot{\varphi }}^{ϵ}\right(r)\mathrm{d}r.\end{array}$ Then $\begin{array}{rl}{X}_n^{ϵ}\left(t\right)& =\psi \left(X_n^{ϵ}\right)\left(t\right)+{\mathrm{\Lambda }}_n\left(X_n^{ϵ}\right)\left(t\right)+\gamma \left(X_n^{ϵ}\right)\left(t\right)\\ & +{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(t\right).\end{array}$ In view of the Lipschitz continuity of $F_n$ and inequality (16(16) $\begin{array}{rl}|\psi (u^{ϵ}-u^0){|}_E& \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\\ & \times |(u^{ϵ}-u^0)\left(r\right){|}_E\mathrm{d}r\\ & \le C{\int }_s^t\left[\right(t-r)\wedge 1{]}^{-\frac{1+ϵ}{2}}\mathrm{d}r\\ & \times \underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|(u^{ϵ}-u^0)\left(r\right){|}_E,\end{array}$(16) ), one obtains $\begin{array}{rl}\underset{r\in [s,t]}{sup}|X_n^{ϵ}\left(r\right){|}_E& \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|X_n^{ϵ}\left(r\right){|}_E\\ & +c_n\left(T\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|X_n^{ϵ}\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +\underset{r\in [s,t]}{sup}|\gamma \left(X_n^{ϵ}\right)\left(r\right){|}_E\mathrm{d}r\\ & +\underset{r\in [s,t]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E\mathrm{d}r.\end{array}$ If we take $t_3>0$ such that $c_s^{\psi }(s+t_3)<\frac{1}{2}$, this yields $\begin{array}{rl}\underset{r\in [s,s+t_3]}{sup}|X_n^{ϵ}\left(r\right){|}_E& \le 2c_n\left(T\right){\int }_s^{s+t_3}\underset{r^{\prime }\in [s,r]}{sup}|X_n^{ϵ}\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & +2\underset{r\in [s,s+t_3]}{sup}|\gamma \left(X_n^{ϵ}\right)\left(r\right){|}_E\\ & +2\underset{r\in [s,s+t_3]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E.\end{array}$ Gronwall's inequality implies $\begin{array}{rl}& \underset{r\in [s,s+t_3]}{sup}|X_n^{ϵ}\left(r\right){|}_E\le 2\left[\underset{r\in [s,s+t_3]}{sup}|\gamma \right(X_n^{ϵ}\left)\right(r){|}_E\\ & +\underset{r\in [s,s+t_3]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E]{\mathrm{e}}^{2c_n\left(T\right)t_3}.\end{array}$ Then, $\begin{array}{rl}|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}& =\left[\mathbb{E}\underset{r\in [s,s+t_3]}{sup}|X_n^{ϵ}\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & \le 2{\mathrm{e}}^{2c_n\left(T\right)t_3}\left[|\gamma \right(X_n^{ϵ}){|}_{L_{s,s+t_3,p}\left(E\right)}\\ & +|{\gamma }^{\varphi }\left(X_n^{ϵ}\right){|}_{L_{s,s+t_3,p}\left(E\right)}].\end{array}$ Furthermore, $\begin{array}{rl}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right){|}_{L_{s,s+t_3,p}\left(E\right)}& \le c_{s,p}^{\gamma }(s+t_3)|u_n^0\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\cdot N\end{array}$ as similar proof as Theorem 4.2 in Cerrai (2003); due to (4.6) and the proof in Theorem 5.3 in Cerrai (2003), it yields (29) $\begin{array}{rl}& |\gamma \left(X_n^{ϵ}\right){|}_{L_{s,s+t_3,p}\left(E\right)}\\ & \le c_{s,p}^{\gamma }(s+t_3)|u_n^0+\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}/\lambda \left(ϵ\right)\\ & \le c_{s,p}^{\gamma }(s+t_3)[|u_n^0{|}_E+\sqrt{ϵ}\lambda (ϵ\left)|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\right]/\lambda \left(ϵ\right)\\ & \le c_{s,p}^{\gamma }(s+t_3)\left[c_{s,2p}\right(s+t_3\left)\right(1+|x{|}_E)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\left]/\lambda \right(ϵ).\end{array}$(29) Therefore, for some constant $c_{n,N,s+t_3}$, $|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\le c_{n,N,s+t_3}.$ Next we can repeat the same arguments in the intervals $[s+t_3,s+2t_3]$, $[s+2t_3,s+3t_3]$ and so on and for any T>s, there exists some constant $c_{n,N,T}$, (30) $|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(30) Here choosing ε small enough, we get (27(27) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}$(27) ).

The proof of (28(28) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{\varphi }{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(28) ) is very similar to that of (27(27) $\underset{ϵ\in (0,{ϵ}_0]}{sup}|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}$(27) ), we omit it here. The proof of this lemma is complete.

Under the conditions of Lemma 4.1, we have the following three inequalities. Let $G_M^{ϵ}\left(t\right):=\{\omega :(\underset{0\le s\le t}{sup}|X_n^{ϵ}\left(s\right){|}^2)\le M\}$. Then, from inequality (29(29) $\begin{array}{rl}& |\gamma \left(X_n^{ϵ}\right){|}_{L_{s,s+t_3,p}\left(E\right)}\\ & \le c_{s,p}^{\gamma }(s+t_3)|u_n^0+\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}/\lambda \left(ϵ\right)\\ & \le c_{s,p}^{\gamma }(s+t_3)[|u_n^0{|}_E+\sqrt{ϵ}\lambda (ϵ\left)|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\right]/\lambda \left(ϵ\right)\\ & \le c_{s,p}^{\gamma }(s+t_3)\left[c_{s,2p}\right(s+t_3\left)\right(1+|x{|}_E)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)|X_n^{ϵ}{|}_{L_{s,s+t_3,p}\left(E\right)}\left]/\lambda \right(ϵ).\end{array}$(29) ), (31) $\begin{array}{rl}& \left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(t\right)}\underset{r\in [s,t]}{sup}|\gamma \right(X_n^{ϵ}\left)\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & \le {\mathbf{1}}_{G_M^{ϵ}\left(t\right)}{c}_{s,p}^{\gamma }\left(t\right)[|u_n^0{|}_E+\sqrt{ϵ}\lambda (ϵ\left)|X_n^{ϵ}{|}_{L_{s,t,p}\left(E\right)}\right]/\lambda \left(ϵ\right)\\ & \le c_{s,p}^{\gamma }\left(t\right)\left[c_{s,2p}\right(t\left)\right(1+|x{|}_E)+\sqrt{ϵ}\lambda (ϵ\left)M\right]/\lambda \left(ϵ\right);\end{array}$(31) (32) $\begin{array}{rl}& \left|{\mathbf{1}}_{G_M^{ϵ}\left(t\right)}{\int }_s^t{\mathrm{e}}^{(t-r)C}[G\right(u_n^0+\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ})\\ & -{G\left(u_n^0\right)]Q{\dot{\varphi }}^{ϵ}\left(r\right)\mathrm{d}r\phantom{{\int }_s^t}|}_{L_{s,t,p}\left(E\right)}\\ & \le {\mathbf{1}}_{G_M^{ϵ}\left(t\right)}{c}_{s,p}^{\gamma }\left(t\right)\cdot \sqrt{ϵ}\lambda \left(ϵ\right)|X_n^{ϵ}{|}_{L_{s,t,p}\left(E\right)}\cdot N\\ & \le c_{s,p}^{\gamma }\left(t\right)\cdot \sqrt{ϵ}\lambda \left(ϵ\right)MN\end{array}$(32) and (33) $\begin{array}{rl}& [\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(t\right)}\underset{r\in [s,T]}{sup}\left|{\int }_s^t{\mathrm{e}}^{(t-r)C}G\right(u_n^0\left)Q\right[{\dot{\varphi }}^{ϵ}\left(r\right)\\ & {-{\dot{\varphi }\left(r\right)\left]\mathrm{d}r\phantom{{\int }_s^t}\right|}_E^p\phantom{{\int }_0^T\underset{r\in [s,T]}{sup}}]}^{\frac{1}{p}}\\ & \le c_{s,p}^{\gamma }\left(t\right)|u_n^0{|}_{L_{s,t,p}\left(E\right)}/\lambda \left(ϵ\right)\\ & \times {\left[{\int }_s^t|{\dot{\varphi }}^{ϵ}\right(r)-\dot{\varphi }(r\left){|}_0^2\mathrm{d}r\right]}^{\frac{1}{2}}\to 0,\mathrm{a}\mathrm{s}ϵ\to 0.\end{array}$(33)

Proof of Proposition 4.5.

Let $Z_{ϵ}:=X^{ϵ}-X^{\varphi }$, where $X^{\varphi }$ is the solution of (25(25) $\begin{array}{rl}\mathrm{d}{X}^h\left(t\right)& =A{X}^h\left(t\right)\mathrm{d}t+F^{\prime }\left(u^0\right(t\left)\right)X^h\left(t\right)\mathrm{d}t\\ & +G(t,u^0(t\left)\right)Q\dot{h}\left(t\right)\mathrm{d}t,\end{array}$(25) ) replaced h by ϕ. Then $Z_{ϵ}\left(0\right)=0$ and $Z_{ϵ}^n=X_n^{ϵ}-X_n^{\varphi }$, where $X_n^{\varphi }$, $X_n^{ϵ}$ satisfy, respectively, $\begin{array}{rl}\mathrm{d}{X}_n^{\varphi }\left(t\right)& =A{X}_n^{\varphi }\left(t\right)\mathrm{d}t+F_n^{\prime }\left(u^0\right(t\left)\right)X_n^{\varphi }\left(t\right)\mathrm{d}t\\ & +\sigma (t,u^0(t\left)\right)Q\dot{\varphi }\left(t\right)\mathrm{d}t,\end{array}$ with initial value $X_n^{\varphi }\left(0\right)=0$ and $\begin{array}{rl}\mathrm{d}{X}_n^{ϵ}\left(t\right)& =A{X}_n^{ϵ}\left(t\right)\mathrm{d}t\\ & +\left[F_n\right(t,u^0\left(t\right)+\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ}\left(t\right))\\ & -F_n(t,u^0(t\left)\right)\left]/\sqrt{ϵ}\lambda \right(ϵ)\mathrm{d}t\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X_n^{ϵ}\right(t\left)\right)/\lambda \left(ϵ\right)Q\mathrm{d}W\left(t\right)\\ & +G(t,u^0(t)+\sqrt{ϵ}\lambda (ϵ\left)X_n^{ϵ}\right(t\left)\right)Q{\varphi }^{ϵ}\left(t\right)\mathrm{d}t,\end{array}$ with initial value $X_n^{ϵ}\left(0\right)=0$. Define the following mappings $\begin{array}{rl}\psi \left(Z_{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}L\left(Z_{ϵ}\right)\left(r\right)\mathrm{d}r;\\ {\mathrm{\Lambda }}_n\left(Z_{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\left\{\right[F_n^{(}r,u^0\left(r\right)+\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right))\\ & -F_n^{(}r,u^0\left(r\right)\left)\right]/\left(\sqrt{ϵ}\lambda \right(ϵ\left)\right)\\ & -F_n^{\prime }(r,u^0(r\left)\right)X^{\varphi }\left(r\right)\}\mathrm{d}r;\\ \gamma \left(X_{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}G\left(u^0\right(r)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right)\left)/\lambda \right(ϵ)Q\mathrm{d}{W}_r;\\ {\gamma }^{\varphi }\left(Z_{ϵ}\right)\left(t\right)& :={\int }_s^t{\mathrm{e}}^{(t-r)C}\left[G\right(u^0\left(r\right)\\ & +\sqrt{ϵ}\lambda \left(ϵ\right)X^{ϵ}\left(r\right)\left)Q{\dot{\varphi }}^{ϵ}\right(r)\\ & -G\left(u^0\right(r\left)\right)Q\dot{\varphi }\left(r\right)]\mathrm{d}r.\end{array}$ Then $\begin{array}{rl}{Z}_{ϵ}^n\left(t\right)& =\psi \left(Z_{ϵ}^n\right)\left(t\right)+{\mathrm{\Lambda }}_n\left(Z_{ϵ}^n\right)\left(t\right)\\ & +\gamma \left(X_n^{ϵ}\right)\left(t\right)+{\gamma }^{\varphi }\left(Z_{ϵ}^n\right)\left(t\right).\end{array}$ Thus, $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|Z_{ϵ}^n\left(r\right){|}_E\\ & \le c_s^{\psi }\left(t\right)\underset{r\in [s,t]}{sup}|Z_{ϵ}^n\left(r\right){|}_E+\underset{r\in [s,t]}{sup}|{\mathrm{\Lambda }}_n\left(Z_{ϵ}^n\right)\left(r\right){|}_E\\ & +\underset{r\in [s,t]}{sup}|\gamma \left(X_n^{ϵ}\right)\left(r\right){|}_E+\underset{r\in [s,t]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E.\end{array}$ There exists a random field ${\eta }^{ϵ}(r,x)$ taking values in $(0,1)$ such that $\begin{array}{rl}& \left[F_n\right(u^0+\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ})-F_n(u^0\left)\right]/\left(\sqrt{ϵ}\lambda \right(ϵ\left)\right)\\ & -F_n^{\prime }\left(u_n^0\right)X_n^{\varphi }\\ & =F_n^{\prime }(u_n^0+{\eta }^{ϵ}\sqrt{ϵ}\lambda (ϵ\left)X_n^{ϵ}\right)\cdot X_n^{ϵ}-F_n^{\prime }\left(u_n^0\right)X_n^{\varphi }\\ & =\left[F_n^{\prime }\right(u_n^0+{\eta }^{ϵ}\sqrt{ϵ}\lambda \left(ϵ\right)X_n^{ϵ})-F_n^{\prime }(u_n^0\left)\right]\cdot X_n^{ϵ}\\ & +F_n^{\prime }\left(u_n^0\right)Z_{ϵ}^n,\end{array}$ which yields $\begin{array}{rl}& \underset{r\in [s,t]}{sup}|{\mathrm{\Lambda }}_n\left(Z_{ϵ}^n\right)\left(r\right){|}_E\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|X_n^{ϵ}\left(r^{\prime }\right){|}_E^2\cdot \sqrt{ϵ}\lambda \left(ϵ\right)\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|F_n^{\prime }\left(u^0\right(r^{\prime }\left)\right)|\cdot |Z_{ϵ}^n\left(r^{\prime }\right){|}_E\mathrm{d}r\\ & \le c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}|X_n^{ϵ}\left(r^{\prime }\right){|}_E^2\cdot \sqrt{ϵ}\lambda \left(ϵ\right)\mathrm{d}r\\ & +c_n\left(t\right){\int }_s^t\underset{r^{\prime }\in [s,r]}{sup}\left(F_0^{\prime }|u^0\right(r^{\prime }){|}_E+F_1^{\prime })|Z_{ϵ}^n\left(r^{\prime }\right){|}_E\mathrm{d}r.\end{array}$ We take $t_4>0$ such that $c_s^{\psi }(s+t_4)<\frac{1}{2}$. One obtains $\begin{array}{rl}& \underset{r\in [s,s+t_4]}{sup}|Z_{ϵ}^n\left(r\right){|}_E\\ & \le 2\underset{r\in [s,s+t_4]}{sup}|{\mathrm{\Lambda }}_n\left(Z_{ϵ}^n\right)\left(r\right){|}_E+2\underset{r\in [s,s+t_4]}{sup}|\gamma \left(X_n^{ϵ}\right)\left(r\right){|}_E\\ & +2\underset{r\in [s,s+t_4]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E.\end{array}$ Gronwall's inequality implies $\begin{array}{rl}& \underset{r\in [s,s+t_4]}{sup}|Z_{ϵ}^n\left(r\right){|}_E\\ & \le 2\left[c_n\right(s+t_4\left){\int }_s^{s+t_4}\underset{r^{\prime }\in [s,r]}{sup}|X_n^{ϵ}\right(r^{\prime }){|}_E^2\cdot \sqrt{ϵ}\lambda (ϵ)\mathrm{d}r\\ & +\underset{r\in [s,s+t_4]}{sup}|\gamma \left(X_n^{ϵ}\right)\left(r\right){|}_E\\ & +\underset{r\in [s,s+t_4]}{sup}|{\gamma }^{\varphi }\left(X_n^{ϵ}\right)\left(r\right){|}_E]\cdot \exp \left\{{\int }_s^{s+t_4}2c_n\right(s+t_4)\\ & \times \underset{r^{\prime }\in [s,r]}{sup}\left(F_0^{\prime }|u^0\right(r^{\prime }){|}_E+F_1^{\prime })\mathrm{d}r\}.\end{array}$ Thus, $\begin{array}{rl}& |Z_{ϵ}^n{|}_{L_{s,s+t_4,p}\left(E\right)}\\ & =[\mathbb{E}\underset{r\in [s,s+t_4]}{sup}|Z_{ϵ}^n{|}_E^p{]}^{\frac{1}{p}}\\ & \le 2\left[c_n\right(s+t_4){\int }_s^{s+t_4}|X_n^{ϵ}{|}_{L_{s,s+t_4,p}\left(E\right)}^2\cdot \sqrt{ϵ}\lambda (ϵ)\mathrm{d}r\\ & +|\gamma \left(X_n^{ϵ}\right){|}_{L_{s,s+t_4,p}\left(E\right)}\\ & +|{\gamma }^{\varphi }\left(X_n^{ϵ}\right){|}_{L_{s,s+t_4,p}\left(E\right)}]\\ & \times \exp \left\{{\int }_s^{s+t_4}2c_n\right(s+t_4\left)\underset{r^{\prime }\in [s,r]}{sup}\right(F_0^{\prime }|u^0\left(r^{\prime }\right){|}_E+F_1^{\prime }\left)\right\}.\end{array}$ With the help of inequalities (30(30) $|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(30) )–(33(33) $\begin{array}{rl}& [\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(t\right)}\underset{r\in [s,T]}{sup}\left|{\int }_s^t{\mathrm{e}}^{(t-r)C}G\right(u_n^0\left)Q\right[{\dot{\varphi }}^{ϵ}\left(r\right)\\ & {-{\dot{\varphi }\left(r\right)\left]\mathrm{d}r\phantom{{\int }_s^t}\right|}_E^p\phantom{{\int }_0^T\underset{r\in [s,T]}{sup}}]}^{\frac{1}{p}}\\ & \le c_{s,p}^{\gamma }\left(t\right)|u_n^0{|}_{L_{s,t,p}\left(E\right)}/\lambda \left(ϵ\right)\\ & \times {\left[{\int }_s^t|{\dot{\varphi }}^{ϵ}\right(r)-\dot{\varphi }(r\left){|}_0^2\mathrm{d}r\right]}^{\frac{1}{2}}\to 0,\mathrm{a}\mathrm{s}ϵ\to 0.\end{array}$(33) ), we get (34) $\left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}(s+t_4)}\underset{r\in [s,s+t_4)]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\to 0,\mathrm{a}\mathrm{s}ϵ\to 0.$(34) Then we can repeat the same arguments in the intervals $[s+t_4,s+2t_4]$, $[s+2t_4,s+3t_4]$ and so on and for any T>s, (35) $\left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(T\right)}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\to 0,\mathrm{a}\mathrm{s}ϵ\to 0.$(35) Furthermore, by Chebysev inequality and (30(30) $|X_n^{ϵ}{|}_{L_{s,T,p}\left(E\right)}\le c_{n,N,T}.$(30) ), (36) $\mathbb{P}(\left(G_M^{ϵ}\right(T){)}^c)\le c_{n,N,T}/M.$(36) Finally, $\begin{array}{rl}|Z_{ϵ}^n{|}_{L_{s,T,p}\left(E\right)}& \le \left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(T\right)}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & +\left[\mathbb{E}{\mathbf{1}}_{\left(G_M^{ϵ}\right(T){)}^c}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & \le \left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(T\right)}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & +\left[\mathbb{E}\right({\mathbf{1}}_{\left(G_M^{ϵ}\right(T){)}^c}{)}^p{]}^{\frac{1}{p}}\cdot |Z_{ϵ}^n{|}_{L_{s,T,p}\left(E\right)}\\ & =\left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(T\right)}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\\ & +\mathbb{P}(\left(G_M^{ϵ}\right(T){)}^c)\cdot |Z_{ϵ}^n{|}_{L_{s,T,p}\left(E\right)}.\end{array}$ Taking $M_0>0$ and for $M>M_0$, $\mathbb{P}\left(\right(G_M^{ϵ}\left(T\right){)}^c)\le \frac{1}{2}$. Then, (37) $\begin{array}{rl}& |Z_{ϵ}^n{|}_{L_{s,T,p}\left(E\right)}\le 2\left[\mathbb{E}{\mathbf{1}}_{G_M^{ϵ}\left(T\right)}\underset{r\in [s,T]}{sup}|Z_{ϵ}^n\right(r){|}_E^p{]}^{\frac{1}{p}}\to 0,\\ & \mathrm{a}\mathrm{s}ϵ\to 0.\end{array}$(37) Similarly as inequality (18(18) $\begin{array}{rl}& \mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u^{ϵ}-u^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & =\mathbb{E}\left[\underset{n\to \mathrm{\infty }}{lim}\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le \underset{n\to \mathrm{\infty }}{lim}\mathbb{E}\left[\underset{r\in [s,T]}{sup}|\right(u_n^{ϵ}-u_n^0\left)\right(r\left){|}_E^p{\mathbf{1}}_{\{T\le {\tau }_n\}}\right]\\ & \le (c_{s,p}\sqrt{ϵ}{)}^p,\end{array}$(18) ), replacing $u^{ϵ}-u^0$ with $Z_{ϵ}^n$, the proof of the proposition is completed.

Proof of Proposition 4.6

The proof is similar as that of Proposition 4.5 and easier. The proof will be omitted.

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Additional information

Funding

This work was supported by Fundamental Research Funds for the Central University [grant number 2019XD-A11] and (National Natural Science Foundation of China) NSFC [grant number 11871010].