Large deviations for stochastic models of two-dimensional second grade fluids driven by L\'evy noise

TL;DR

This paper proves a large deviation principle for 2D second grade fluid models influenced by Le9vy noise, using the weak convergence method, advancing understanding of stochastic fluid dynamics under jump noise.

Contribution

It introduces a large deviation principle for stochastic second grade fluids driven by Le9vy noise, employing the weak convergence approach for the first time in this context.

Findings

Established a large deviation principle for the model.

Applied the weak convergence method successfully.

Enhanced theoretical understanding of stochastic fluid models.

Abstract

In this paper, we establish a large deviation principle for stochastic models of two-dimensional second grade fluids driven by L\'evy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas in [5] plays a key role.

Equations

$$\displaystyle\left\{\begin{aligned} &d(X^{\varepsilon}(t)-\alpha\Delta X^{\varepsilon}(t))+\Big{(}-\kappa\Delta X^{\varepsilon}(t)+{\rm curl}(X^{\varepsilon}(t)-\alpha\Delta X^{\varepsilon}(t))\times X^{\varepsilon}(t)+\nabla\mathfrak{P}\Big{)}\,dt\\ &=F(X^{\varepsilon}(t),t)\,dt+\sqrt{\varepsilon}G(X^{\varepsilon}(t),t)\,dW(t)+\varepsilon\int_{\mathbb{Z}}\sigma(t,X^{\varepsilon}(t-),z)\widetilde{N}^{\varepsilon^{-1}}(dzdt),\quad\rm{in}\ \mathcal{O}\times(0,T],\\ &\begin{aligned} &{\rm{div}}\,X^{\varepsilon}=0\quad&&\rm{in}\ \mathcal{O}\times(0,T];\\ &X^{\varepsilon}=0&&\rm{in}\ \partial\mathcal{O}\times[0,T];\\ &X^{\varepsilon}(0)=X_{0}&&\rm{in}\ \mathcal{O},&\end{aligned}\end{aligned}\right.$$

$$\sup_{0<\varepsilon<\varepsilon_{0}}E\Big{[}\sup_{t\in[0,T]}\|\widetilde{X}^{\varepsilon}(t)\|_{H^{3}(\mathcal{O})\times H^{3}(\mathcal{O})}^{2}\Big{]}<\infty.$$

$$((u,v))={\int }_{\mathcal{O}}\nabla u\cdot \nabla vdx=\sum _{i=1}^2{\int }_{\mathcal{O}}\frac{\partial u}{\partial x_i}\frac{\partial v}{\partial x_i}dx,$$

$$\displaystyle\mathcal{C}=\Big{\{}u\in[\mathcal{C}^{\infty}_{c}(\mathcal{O})]^{2}:{\rm div}\ u=0\Big{\}},$$

$$\mathbb{V}=\mathrm{the}\mathrm{closure}\mathrm{of}\mathcal{C}\mathrm{in}{\mathbb{H}}^1(\mathcal{O}),$$

$$\mathbb{H}=\mathrm{the}\mathrm{closure}\mathrm{of}\mathcal{C}\mathrm{in}{\mathbb{L}}^2(\mathcal{O}).$$

$$(u,v{)}_{\mathbb{V}}:=(u,v)+\alpha ((u,v)),\text{for any }u,v\in \mathbb{V},$$

$$({\mathcal{P}}^2+\alpha {)}^{-1}\Vert v{\Vert }_{\mathbb{V}}^2\le \Vert v{\Vert }^2\le {\alpha }^{-1}\Vert v{\Vert }_{\mathbb{V}}^2,\text{for any }v\in \mathbb{V}.$$

$$\mathbb{W}=\big{\{}u\in\mathbb{V}:{\rm curl}(u-\alpha\Delta u)\in L^{2}(\mathcal{O})\big{\}},$$

$$\displaystyle(u,v)_{\mathbb{W}}:=\big{(}{\rm curl}(u-\alpha\Delta u),{\rm curl}(v-\alpha\Delta v)\big{)}.$$

$$\displaystyle\mathbb{W}=\big{\{}v\in\mathbb{H}^{3}(\mathcal{O}):{\rm div}\,v=0\ {\rm and}\ v|_{\partial\mathcal{O}}=0\big{\}},$$

$$|v{|}_{{\mathbb{H}}^3(\mathcal{O})}\le C\Vert v{\Vert }_{\mathbb{W}},\forall v\in \mathbb{W}.$$

$$\mathbb{W}\subset \mathbb{V}\subset {\mathbb{W}}^{\ast }.$$

$$(v,w{)}_{\mathbb{V}}=⟨v,w⟩,\forall v\in \mathbb{V},\forall w\in \mathbb{W}.$$

$$(v,e_i{)}_{\mathbb{W}}={\lambda }_i(v,e_i{)}_{\mathbb{V}},\text{for any }v\in \mathbb{W},$$

$$e_i\in {\mathbb{H}}^4(\mathcal{O}),\forall i\in \mathbb{N}.$$

$$\begin{array}{rl}v-\alpha \Delta v& =f\mathrm{in}\mathcal{O},\\ \mathrm{div}v& =0\mathrm{in}\mathcal{O},\\ v& =0\mathrm{on}\partial \mathcal{O}.\end{array}$$

$$(v,g{)}_{\mathbb{V}}=(f,g),\forall g\in \mathbb{V},$$

$$|v{|}_{{\mathbb{H}}^{l+2}}\le C|f{|}_{{\mathbb{H}}^l}.$$

$$Au:=-\mathbb{P}\Delta u,\forall u\in D(A)={\mathbb{H}}^2(\mathcal{O})\cap \mathbb{V},$$

$$((I+\alpha A{)}^{-1}f,g{)}_{\mathbb{V}}=(f,g),$$

$$\Vert (I+\alpha A{)}^{-1}f{\Vert }_{\mathbb{W}}\le C_A\Vert f{\Vert }_{\mathbb{V}}.$$

$$(\hat{A}u,v{)}_{\mathbb{V}}=(Au,v)=((u,v)),\forall u\in \mathbb{W},v\in \mathbb{V},$$

$$(\hat{A}u,u{)}_{\mathbb{V}}=\Vert u\Vert ,\forall u\in \mathbb{W}.$$

$$\Vert \hat{A}u{\Vert }_{{\mathbb{W}}^{\ast }}\le C\Vert \hat{A}u{\Vert }_{\mathbb{V}}\le C\Vert u{\Vert }_{\mathbb{V}}.$$

$$|(\mathrm{curl}(u-\alpha \Delta u)\times v,w)|\le C\Vert u{\Vert }_{\mathbb{W}}\Vert v{\Vert }_{\mathbb{V}}\Vert w{\Vert }_{\mathbb{W}},$$

$$|(\mathrm{curl}(u-\alpha \Delta u)\times u,w)|\le C\Vert u{\Vert }_{\mathbb{V}}^2\Vert w{\Vert }_{\mathbb{W}}.$$

$$\displaystyle\widehat{B}(u,v):=(I+\alpha A)^{-1}\mathbb{P}\big{(}{\rm curl}(u-\alpha\Delta u)\times v\big{)}.$$

$$\Vert \hat{B}(u,v){\Vert }_{{\mathbb{W}}^{\ast }}\le C\Vert u{\Vert }_{\mathbb{W}}|v{|}_{\mathbb{V}},$$

$$\Vert \hat{B}(u,u){\Vert }_{{\mathbb{W}}^{\ast }}\le C_B\Vert u{\Vert }_{\mathbb{V}}^2.$$

$$⟨\hat{B}(u,v),v⟩=0,\forall u,v\in \mathbb{W},$$

$$⟨\hat{B}(u,v),w⟩=-⟨\hat{B}(u,w),v⟩,\forall u,v,w\in \mathbb{W}.$$

Full text

Large deviations for stochastic models of two-dimensional second grade fluids driven by Lévy noise

Jianliang Zhai 1,,  Tusheng Zhang2,,  Wuting Zheng1,

1* School of Mathematical Sciences,

University of Science and Technology of China,

Hefei, 230026, China

2 School of Mathematics, University of Manchester,

Oxford Road, Manchester, M13 9PL, UK

zhaijl@ustc.edu.cnTusheng.Zhang@manchester.ac.ukzwtzjr@mail.ustc.edu.cn

Abstract: In this paper, we establish a large deviation principle for stochastic models of two-dimensional second grade fluids driven by Lévy noise. The weak convergence method introduced by Budhiraja, Dupuis and Maroulas in [5] plays a key role.

Key Words: Large deviations; Second grade fluids; Lévy process; Weak convergence method.

1 Introduction

The second grade fluids is an admissible model of slow flow fluids, which contains industrial fluids, slurries, polymer melts, etc.. It has attracted much attention from a theoretical point of view, since it has properties of boundedness, stability and exponential decay, and has interesting connections with many other fluid models, see e.g. [6], [14], [15], [27] and references therein.

Recently, taking into account random enviroment, the stochastic models of two-dimensional second grade fluids have been studied. For the case of Wiener noises, we refer to [9, 22, 23, 24, 33, 36], where the authors obtained the existence and uniqueness of solutions, the behavior of the solutions as $\alpha\rightarrow 0$, Freidlin-Wentzell’s large deviation principles and exponential mixing for the solutions. In the case of Lévy noises, the global existence of a martingale solution was obtained in [16], and the existence and uniqueness of strong probabilistic solutions is studied in [26].

Based on the results in [26], in this paper, we are concerned with Freidlin-Wentzell’s large deviation principle of stochastic models for the incompressible second grade fluids driven by Lévy noises, which are given as follows:

[TABLE]

where $\mathcal{O}$ is an open domain of $\mathbb{R}^{2}$; $X^{\varepsilon}=(X^{\varepsilon}_{1},X^{\varepsilon}_{2})$ and $\mathfrak{P}$ represent the random velocity and modified pressure respectively. $\mathbb{Z}$ is a locally compact Polish space. On a specified complete filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in[0,T]},P)$, $W$ is a one-dimensional standard Brownian motion, and $\widetilde{N}^{\epsilon^{-1}}$ is a compensated Poisson random measure on $[0,T]\times\mathbb{Z}$ with a $\sigma$-finite mean measure $\epsilon^{-1}\lambda_{T}\otimes\nu$, $\lambda_{T}$ is the Lebesgue measure on $[0,T]$ and $\nu$ is a $\sigma$-finite measure on $\mathbb{Z}$. The details of $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in[0,T]},P,W,\widetilde{N}^{\epsilon^{-1}})$ will be given in Section 2.

Due to the appearance of jumps in our setting, the Freidlin-Wentzell’s large deviations are distinctively different to the Wiener case in [36]. We will apply the weak convergence approach introduced in [5] and [3] for the case of Poisson random measures. This approach is mainly based on a variational representation formula for continuous time processes, and has been proved to be a powerful tool to establish the Freidlin-Wentzell large deviation principle for various finite and infinite dimensional stochastic dynamical systems with irregular coefficients driven by a non-Gaussian Lévy noise, see for example [2], [3], [34], [35], [13], [21]. Because of the nature of the second grade fluids, technical difficulties arise even in the deterministic case. In addition to the complex structure of the system (1.1), the nature of the nonlinear term {\rm curl}\big{(}X^{\varepsilon}(t)-\alpha\Delta X^{\varepsilon}(t)\big{)}\times X^{\varepsilon}(t)dt implies that the solution $X^{\varepsilon}(t)$ should be in $H^{3}(\mathcal{O})\times H^{3}(\mathcal{O})$. The chain rule or $\rm{It\hat{o}^{\prime}s}$ Formula of (1.1) shows that the linear term $-\kappa\Delta X^{\varepsilon}(t)dt$ is not acting as a smoothing term like many nonlinear evolutions such as the Navier-Stokes equations. On the other hand, when applying the weak convergence method to the system (1.1), it will be proved that the solutions of the controlled stochastic evolution equations (4.35), denoted by $\widetilde{X}^{\varepsilon},\varepsilon>0$, have the following priori estimate:

[TABLE]

This is non-trival, see Lemma 4.2.

We organize this paper as follows. In Section 2, we introduce some functional spaces and $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in[0,T]},\\ P,W,\widetilde{N}^{\epsilon^{-1}})$ appeared in the system (1.1). In Section 3, we formulate the hypotheses. In section 4, we establish the large deviation principle.

2 Notations and Preliminaries

In this section, we first introduce some functional spaces and preliminaries that are needed in the paper, and then specify $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\in[0,T]},P,W,\widetilde{N}^{\epsilon^{-1}})$ in the system (1.1).

In this paper, we assume that $\mathcal{O}$ is a simply connected and bounded open domain of $\mathbb{R}^{2}$ with boundary $\partial\mathcal{O}$ of class $\mathcal{C}^{3,1}$. For $p\geq 1$ and $k\in\mathbb{N}$, we denote by $L^{p}(\mathcal{O})$ and $W^{k,2}(\mathcal{O})$ the usual $L^{p}$ and Sobolev spaces over $\mathcal{O}$ respectively. Let $W^{k,2}_{0}(\mathcal{O})$ be the closure in $W^{k,2}(\mathcal{O})$ of $\mathcal{C}^{\infty}_{c}(\mathcal{O})$ the space of infinitely differentiable functions with compact supports in $\mathcal{O}$. For simplicity, we write $H^{k}(\mathcal{O}):=W^{k,2}(\mathcal{O})$ and $H_{0}^{k}(\mathcal{O}):=W^{k,2}_{0}(\mathcal{O})$. We equip $H^{1}_{0}(\mathcal{O})$ with the scalar product

[TABLE]

where $\nabla$ is the gradient operator. It is well known that the norm $\|\cdot\|$ generated by this scalar product is equivalent to the usual norm of $H^{1}(\mathcal{O})$.

Throughout this paper, we set $\mathbb{Y}=Y\times Y$ for any Banach space $Y$. Set

[TABLE]

We denote by $(\cdot,\cdot)$ and $|\cdot|$ the inner product in $\mathbb{L}^{2}(\mathcal{O})$(in $\mathbb{H}$) and the induced norm, respectively. The inner product and the norm of $\mathbb{H}^{1}_{0}(\mathcal{O})$ are denoted respectively by $((\cdot,\cdot))$ and $\|\cdot\|$. We endow the space $\mathbb{V}$ with the norm generated by the following inner product

[TABLE]

and the norm in $\mathbb{V}$ is denoted by $\|\cdot\|_{\mathbb{V}}$. The $\rm Poincar\acute{e}$’s inequality implies that there exists a constant $\mathcal{P}>0$ such that the following inequalities holds

[TABLE]

We also introduce the following space

[TABLE]

and endow it with the norm generated by the scalar product

[TABLE]

The norm in $\mathbb{W}$ is denoted by $\|\cdot\|_{\mathbb{W}}$. It has been proved that, see e.g. [10, 12], the following (algebraic and topological) identity holds:

[TABLE]

moreover, there exists a constant $C>0$ such that

[TABLE]

This result states that the norm $\|\cdot\|_{\mathbb{W}}$ is equivalent to the usual norm in $\mathbb{H}^{3}(\mathcal{O})$.

Identifying the Hilbert space $\mathbb{V}$ with its dual space $\mathbb{V}^{*}$ by the Riesz representation, we get a Gelfand triple

[TABLE]

We denote by $\langle f,v\rangle$ the dual relation between $f\in\mathbb{W}^{*}$ and $v\in\mathbb{W}$ from now on. It is easy to see

[TABLE]

Note that the injection of $\mathbb{W}$ into $\mathbb{V}$ is compact, thus there exists a sequence $\{e_{i}\}$ of elements of $\mathbb{W}$ which forms an orthonormal basis in $\mathbb{W}$, and an orthogonal system in $\mathbb{V}$, moreover this sequence verifies:

[TABLE]

where $0<\lambda_{i}\uparrow\infty$. From Lemma 4.1 in [10] we have

[TABLE]

Consider the following “generalized Stokes equations”:

[TABLE]

The following result can be derived from [28] and also can be found in [23, 24].

Lemma 2.1

Set $l=1,2,3$. Let $f$ be a function in $\mathbb{H}^{l}$, then the system (2.6) has a unique solution $v$. Moreover if $f$ is an element of $\mathbb{H}^{l}\cap\mathbb{V}$, then $v\in\mathbb{H}^{l+2}\cap\mathbb{V}$, and the following relations hold

[TABLE]

Define the Stokes operator by

[TABLE]

here the mapping $\mathbb{P}:\mathbb{L}^{2}(\mathcal{O})\longrightarrow\mathbb{H}$ is the usual Helmholtz-Leray projection. It follows from Lemma 2.1 that the operator $(I+\alpha A)^{-1}$ defines an isomorphism from $\mathbb{H}^{l}(\mathcal{O})\cap\mathbb{H}$ into $\mathbb{H}^{l+2}(\mathcal{O})\cap\mathbb{V}$ for $l=1,2,3$. Moreover, for any $f\in\mathbb{H}^{l}(\mathcal{O})\cap\mathbb{V}$ and $g\in\mathbb{V}$, the following properties hold

[TABLE]

Let $\widehat{A}:=(I+\alpha A)^{-1}A$ , then $\widehat{A}$ is a continuous linear operator from $\mathbb{H}^{l}(\mathcal{O})\cap\mathbb{V}$ onto itself for $l=2,3$, and satisfies

[TABLE]

hence

[TABLE]

We also have

[TABLE]

We recall the following estimates which can be found in [24].

Lemma 2.2

For any $u,v,w\in\mathbb{W}$, we have

[TABLE]

and

[TABLE]

Defining the bilinear operator $\widehat{B}(\cdot,\cdot):\ \mathbb{W}\times\mathbb{V}\longrightarrow\mathbb{W}^{*}$ by

[TABLE]

We have the following consequence from this lemma.

Lemma 2.3

For any $u\in\mathbb{W}$ and $v\in\mathbb{V}$, it holds that

[TABLE]

and

[TABLE]

In addition

[TABLE]

which implies

[TABLE]

We are now to introduce $(\Omega,\mathcal{F},\mathbb{F}:=\{\mathcal{F}_{t}\}_{t\in[0,T]},P,W,\widetilde{N}^{\epsilon^{-1}})$.

Set $S$ be a locally compact Polish space. We put $M_{FC}(S)$ be the space of all Borel measures $\vartheta$ on $S$ such that $\vartheta(K)<\infty$ for each compact set $K\subseteq S$. Endow $M_{FC}(S)$ with the weakest topology, denoted it by $\mathcal{T}(M_{FC}(S))$, such that for each $f\in C_{c}(S)$ the mapping $\vartheta\in M_{FC}(S)\rightarrow\int_{S}f(s)\vartheta(ds)$ is continuous. This topology is metrizable such that $M_{FC}(S)$ is a Polish space, see [5] for more details.

Recall that $\mathbb{Z}$ is a locally compact Polish space, and in this paper, we assume that $\nu$ is a given element of $M_{FC}(\mathbb{Z})$. We specify the underlying probability space $(\Omega,\mathcal{F},{\mathbb{F}}:=\{\mathcal{F}_{t}\}_{t\in[0,T]},P)$ in the following way:

[TABLE]

We introduce the functions

[TABLE]

Define for each $t\in[0,T]$ the $\sigma$-algebra

[TABLE]

Let $\lambda_{T}$ and $\lambda_{\infty}$ be Lebesgue measure on $[0,T]$ and $[0,\infty)$ respectively. It follows from [17, Sec.I.8] that there exists a unique probability measure $P$ on $(\Omega,\mathcal{F})$ such that:

1. (a)

$W$ is one-dimension standard Brownian motion; 2. (b)

$N$ is a Poisson random measure on $\Omega$ with intensity measure $\lambda_{T}\otimes\nu\otimes\lambda_{\infty}$; 3. (c)

$W$ and $N$ are independent.

We denote by $\mathbb{F}:=\{{\mathcal{F}}_{t}\}_{t\in[0,T]}$ the $P$-completion of $\{\mathcal{G}_{t}\}_{t\in[0,T]}$ and by $\mathcal{P}$ the $\mathbb{F}$-predictable $\sigma$-field on $[0,T]\times\Omega$. Define

[TABLE]

For $\varphi\in{\mathcal{A}}$, define a counting process $N^{\varphi}$ on $[0,T]\times{{\mathbb{Z}}}$ by

[TABLE]

for $t\in[0,T]$ and $A\in\mathcal{B}({\mathbb{Z}})$. When $\varphi(s,z,\omega)=\epsilon^{-1}$, we write $N^{\varphi}=N^{\epsilon^{-1}}$. It is easy to see that $N^{\epsilon^{-1}}$ is a Poisson random measure on $[0,T]\times\mathbb{Z}$ with a mean measure $\epsilon^{-1}\lambda_{T}\otimes\nu$. We denote $\widetilde{N}^{\epsilon^{-1}}$ the compensated Poisson random measure respect to $N^{\epsilon^{-1}}$.

At the end of this section, we introduce the following notions which will be used later.

For each $f\in L^{2}([0,T],\mathbb{R})$, we introduce the quantity

[TABLE]

and we define for each $m\in\mathbb{N}$ the space

[TABLE]

Equiped with the weak topology, $S_{1}^{m}$ is a compact subset of $L^{2}([0,T],\mathbb{R}).$ We will throughout consider $S_{1}^{m}$ endowed with this topology. By defining the function

[TABLE]

we introduce for each measurable function $g\colon[0,T]\times{\mathbb{Z}}\to[0,\infty)$ the quantity

[TABLE]

Define for each $m\in\mathbb{N}$ the space

[TABLE]

A function $g\in S_{2}^{m}$ can be identified with a measure $\nu_{T}^{g}\in M_{FC}([0,T]\times{\mathbb{Z}})$, defined by

[TABLE]

This identification induces a topology on $S_{2}^{m}$ under which $S_{2}^{m}$ is a compact space, see [3]. Throughout, we use this topology on $S_{2}^{m}$.

Define $S^{m}=S_{1}^{m}\times S_{2}^{m}$ and $\mathcal{S}=\bigcup_{m\geq 1}S^{m}$.

3 Hypotheses

In this section, we will formulate precise assumptions on coefficients.

Let

[TABLE]

be given measurable maps. We introduce the following notations:

[TABLE]

The following assumptions are from [26], which guarantee that (1.1) admits a unique solution.

Condition 3.1

There exists constants $C_{F},C_{G},C\geq 0$, such that the following conditions hold for all $u_{1},u_{2},u\in\mathbb{V}$ and $t\in[0,T]$:

(1) (Lipschitz)

[TABLE]

[TABLE]

[TABLE]

(2) (Growth)

[TABLE]

Let

[TABLE]

[TABLE]

To study large deviation principle of (1.1), besides Condition 3.1, we further need

Condition 3.2

(1)($L^{2}$-integrability) For $i=0,1$, $\|\sigma(\cdot,\cdot)\|_{i,\mathbb{V}}\in L^{2}(\lambda_{T}\otimes\nu)$, i.e.

[TABLE]

(2)(Exponential integrability) For $i=0,1$, there exists $\delta_{1}^{i}>0$ such that for all $E\in{\mathcal{B}}([0,T]\times\mathbb{Z})$ satisfying $\lambda_{T}\otimes\nu(E)<\infty$, the following holds

[TABLE]

Remark 1

Condition 3.2 implies that, for every $\delta_{2}^{i}>0$ and for all $E\in{\mathcal{B}}([0,T]\times\mathbb{Z})$ satisfying $\lambda_{T}\otimes\nu(E)<\infty$,

[TABLE]

The following lemma was proved in Budhiraja, Chen and Dupuis [3]. For the second part of this lemma, the case $i=0$ can be found in Remark 2 of Yang, Zhai and Zhang [34], and the case $i=1$ can be proved similarly. We omit its proof.

Lemma 3.1

*Under Condition 3.1 and 3.2,

(1) For $i=0,1$ and every $m\in\mathbb{N}$,*

[TABLE]

[TABLE]

(2) For every $\eta>0$, there exist $\delta>0$ such that for any $A\subset[0,T]$ satisfying $\lambda_{T}(A)<\delta$

[TABLE]

We also need the following lemma, the proof of which can be found in Budhiraja, Chen and Dupuis [3].

Lemma 3.2

Fix $m\in\mathbb{N}$, and let $g_{n},g\in S_{2}^{m}$ be such that $g_{n}\rightarrow g$ as $n\rightarrow\infty$. Let $h:[0,T]\times\mathbb{Z}\rightarrow\mathbb{R}$ be a measurable function such that

[TABLE]

and for all $\delta\in(0,\infty)$

[TABLE]

for all $E\in{\mathcal{B}}([0,T]\times\mathbb{Z})$ satisfying $\lambda_{T}\otimes\nu(E)<\infty$. Then

[TABLE]

Let $\mathbb{K}$ be a separable Hilbert space. Given $p>1$, $\beta\in(0,1)$, let $W^{\beta,p}([0,T],\mathbb{K})$ be the space of all $u\in L^{p}([0,T],\mathbb{K})$ such that

[TABLE]

endowed with the norm

[TABLE]

The following result is a variant of the criteria for compactness proved in Lions [19] (Sect. 5, Ch. I) and Temam [32] (Sect. 13.3).

Lemma 3.3

Let $\mathbb{K}_{0}\subset\mathbb{K}\subset\mathbb{K}_{1}$ be Banach spaces, $\mathbb{K}_{0}$ and $\mathbb{K}_{1}$ reflexive, with compact embedding of $\mathbb{K}_{0}$ into $\mathbb{K}$. For $p\in(1,\infty)$ and $\beta\in(0,1)$, let $\Lambda$ be the space

[TABLE]

endowed with the natural norm. Then the embedding of $\Lambda$ into $L^{p}([0,T],\mathbb{K})$ is compact.

4 Large deviation

4.1 Skeleton equations

As a first step we show that, for every $q=(f,g)\in\mathcal{S}$ the deterministic integral equation

[TABLE]

has a unique solution. That is

Theorem 4.1

Fix $X_{0}\in\mathbb{W}$ , $q=(f,g)\in\mathcal{S}$. Suppose Condition 3.1 and 3.2 hold. Then there exists a unique $\widetilde{X}^{q}\in C([0,T],\mathbb{V})$ satisfy (4.1). Moreover, for any $m\in\mathbb{N}$, there exists a constant $C_{m}$ such that

[TABLE]

Proof.

First, let $\Phi_{n}:\mathbb{R}\rightarrow[0,1]$ be a smooth function such that $\Phi_{n}(r)=1$ if $|r|\leq n$, $\Phi_{n}(r)=0$ if $|r|>n+1$. Set $\mathcal{X}_{n}(u)=\Phi_{n}(\|u\|_{\mathbb{V}}),u\in\mathbb{V}$. Let $\mathbb{P}_{n}$ be the projection operator from $\mathbb{V}$ to $\mathbb{V}$ defined as

[TABLE]

Set

[TABLE]

[TABLE]

(2.5) and Lemma 2.3 implies that $\widehat{B}_{n}$ is a global Lipschitz operator from $\mathbb{P}_{n}\mathbb{V}$ into $\mathbb{P}_{n}\mathbb{V}$. Repeating the same arguments as in the proof of Theorem 4.1 in [35], there exists a unique $X_{n}\in C([0,T],\mathbb{P}_{n}\mathbb{V})$ satisfying the following auxiliary PDE:

[TABLE]

with the initial value $X_{n}(0)=\mathbb{P}_{n}X_{0}\in\mathbb{W}$.

The next thing is to show that

[TABLE]

and for $\alpha\in(0,\frac{1}{2})$

[TABLE]

By (4.3), we have

[TABLE]

for $i=1,2,\cdots,n$.

By (2.4), multiplying both sides of the equation (4.1) by $\lambda_{i}$, we get

[TABLE]

for $i=1,2,\cdots,n$.

By a calculation of $(X_{n}(t),e_{i})_{\mathbb{W}}^{2}$ and summing over $i$ from $1$ to $n$ yields

[TABLE]

Noticing that (see (4.11) in [26], (4.61) in [23])

[TABLE]

[TABLE]

and

[TABLE]

we have

[TABLE]

we have used Condition 3.1.

By Lemma 3.1 and Gronwall’s inequality, we get (4.4). Now we proof (4.5). By (4.3)

[TABLE]

Choose $\alpha\in(0,\frac{1}{2})$, using the similar arguments as the proof of Proposition 4.5 in Zhai and Zhang [36], we can obatin

[TABLE]

here $L_{\alpha}$ depends on $\alpha$ and $C$ from (4.4).

Using the similar arguments as (4.20) in Zhai and Zhang [35], we can obtain

[TABLE]

Combining above all inequalities, we obtain (4.5). The estimates (4.4) and (4.5) enable us to assert the existence of $X\in L^{\infty}([0,T],\mathbb{W})$ and a sub-sequence $X_{m^{\prime}}$ such that, as $m^{\prime}\rightarrow\infty$

1. $X_{m^{\prime}}\rightarrow X$ weakly in $L^{2}([0,T],\mathbb{W})$,

2. $X_{m^{\prime}}\rightarrow X$ in the weak-star topology of $L^{\infty}([0,T],\mathbb{W})$.

Lemma 3.3 has been used to obtain claim 3:

3. $X_{m^{\prime}}\rightarrow X$ strongly in $L^{2}([0,T],\mathbb{V})$. Finally, we use the similar argument as in the proof of Theorem 4.1 in [35], we can conclude that $X$ is a solution of (4.1) and refer to Tenam [31] Chapter 3, $X\in C([0,T],\mathbb{V})$. (4.4) also implies that

[TABLE]

(Uniqueness) Let us assume that $X$ and $Y$ are two solutions of (4.1), and let $Z=X-Y$. We have

[TABLE]

By Lemma 2.3, we get

[TABLE]

By Condition 3.1, we get

[TABLE]

and

[TABLE]

For $I_{4}$, we have

[TABLE]

Setting

[TABLE]

we have

[TABLE]

By Lemma 3.1 and Gronwall’s equality, we can conclude $X=Y$.

The proof is complete. ∎

4.2 The main result

We are now ready to state the main result. Recall that for $q=(f,g)\in\mathcal{S},\nu_{T}^{g}(dsdz)=g(s,z)\nu(dz)ds$, define

[TABLE]

The next theorem is contained in Theorem 3.2 in Shang, Zhai and Zhang [26].

Theorem 4.2

Assume Condition 3.1, if $X_{0}\in\mathbb{W}$, there exists a unique $\mathbb{V}$-valued progressively measurable process $X^{\varepsilon}\in L^{\infty}([0,T],\mathbb{W})\cap D([0,T],\mathbb{V})$ such that for any $t>0$

[TABLE]

Theorem 4.2 shows that the above equation admits a strong solution in the probabilistic sense. In particular, for every $\varepsilon>0$, there exists a measurable map $\mathcal{G}^{\varepsilon}:C([0,T],\mathbb{R})\times M_{FC}([0,T]\times\mathbb{Z})\rightarrow D([0,T],\mathbb{V})$ such that, for any Poisson random measures $n^{\varepsilon^{-1}}$ on $[0,T]\times\mathbb{Z}$ with intensity measure $\varepsilon^{-1}\lambda_{T}\otimes\nu$, $\mathcal{G}^{\varepsilon}(\sqrt{\varepsilon}W,\varepsilon n^{\varepsilon^{-1}})$ is the unique solution of (4.2) with $\widetilde{N}^{\varepsilon^{-1}}$ replaced by $\widetilde{n}^{\varepsilon^{-1}}$.

Theorem 4.3

Suppose that Condition 3.1 and 3.2 hold. Then the family $\{X^{\varepsilon}\}_{\varepsilon>0}$ satisfies a large deviation principle on $D([0,T],\mathbb{V})$ with a good rate function $I:D([0,T],\mathbb{V})\rightarrow[0,\infty]$, defined by

[TABLE]

Proof.

Theorem 4.2 implies that for each $\varepsilon>0$ there exists a mapping $\mathcal{G}^{\varepsilon}$ such that

[TABLE]

where $X^{\varepsilon}$ is the solution of (1.1) and $\stackrel{{\scriptstyle\mathcal{D}}}{{=}}$ denotes equality in distribution.

Define for each $m\in\mathbb{N}$ a space of stochastic processes on $\Omega$ by

[TABLE]

Let $(K_{n})_{n\in\mathbb{N}}$ be a sequence of compact sets $K_{n}\subseteq\mathbb{Z}$ with $K_{n}\nearrow\mathbb{Z}$. For each $n\in\mathbb{N}$, let

[TABLE]

and let $\mathcal{A}_{b}=\bigcup_{n=1}^{\infty}\mathcal{A}_{b,n}$. Define for each $m\in\mathbb{N}$ a space of stochastic process on $\Omega$ by

[TABLE]

According to Theorem 2.4 in [3], our claim is established once we have proved:

1. (C1)

if $(f_{n})_{n\in\mathbb{N}}\subseteq S_{1}^{m}$ converges to $f\in S_{1}^{m}$ and $(g_{n})_{n\in\mathbb{N}}\subseteq S_{2}^{m}$ converges to $g\in S_{2}^{m}$ for some $m\in\mathbb{N}$, then

[TABLE] 2. (C2)

if $(\psi_{\varepsilon})_{\varepsilon>0}\subseteq\mathcal{S}_{1}^{m}$ converges in distribution to $\psi\in\mathcal{S}_{1}^{m}$ and $(\varphi_{\varepsilon})_{\varepsilon>0}\subseteq\mathcal{S}_{2}^{m}$ converges in distribution to $\varphi\in\mathcal{S}_{2}^{m}$, then

[TABLE]

We give the details of the proof in the next section. (C1) will be given by Proposition 4.4. (C2) will be established by Proposition 4.5.

∎

4.3 The proofs

Proposition 4.4

Fix $m\in\mathbb{N}$, and let $q_{n}=(f_{n},g_{n}),q=(f,g)\in S^{m}$ be such that $q_{n}\rightarrow q$ as $n\rightarrow\infty$. Then

[TABLE]

Proof.

Recall $\mathcal{G}^{0}(\int_{0}^{\cdot}f_{n}(s)ds,\nu_{T}^{g_{n}})=\widetilde{X}^{q_{n}}$. For simplicity we denote $X_{n}=\widetilde{X}^{q_{n}}$. Using similar arguments as (4.4), (4.5), we can prove that there exists $C_{m}$ and $C_{\alpha,m}$ such that

[TABLE]

and for $\alpha\in(0,\frac{1}{2})$

[TABLE]

Hence, we can assert the existence of an element $X\in L^{2}([0,T],\mathbb{W})\cap L^{\infty}([0,T],\mathbb{V})$ and a sub-sequence $X_{m^{\prime}}$ such that, as $m^{\prime}\rightarrow\infty$

(a) $\sup_{s\in[0,T]}\|X(s)\|_{\mathbb{W}}^{2}\leq C_{m}$,

(b) $X_{m^{\prime}}\rightarrow X$ in $L^{2}([0,T],\mathbb{W})$ weakly,

(c) $X_{m^{\prime}}\rightarrow X$ in $L^{\infty}([0,T],\mathbb{V})$ weak-star.

Combining Lemma 3.3, we have

(d) $X_{m^{\prime}}\rightarrow X$ in $L^{2}([0,T],\mathbb{V})$ strongly. We will prove $X=\widetilde{X}^{q}$.

Let $\psi$ be a differentiable function on $[0,T]$ with $\psi(T)=0$. We multiply $X_{m^{\prime}}(t)$ by $\psi(t)e_{j}$, and then integrate by parts to obtain

[TABLE]

Set

[TABLE]

[TABLE]

[TABLE]

Using $X_{m^{\prime}}\rightarrow X$ in $L^{2}([0,T],\mathbb{V})$ strongly, we can easily get

[TABLE]

Since $f_{n}\rightarrow f$ in $S_{1}^{m}$ and the linear mapping :h\mapsto\int_{0}^{T}\big{(}\widehat{G}(X(t),t)h(t),\psi(t)e_{j}\big{)}_{\mathbb{V}}dt is strong continuous, we have

[TABLE]

Combining (4.21), (4.22), we get

[TABLE]

Set

[TABLE]

[TABLE]

[TABLE]

Similarly as the proof of (4.25)( see (4.26) and (4.29)) in Zhai and Zhang [35], we can get

[TABLE]

Then, we can easily get

[TABLE]

We know that

[TABLE]

combining this with Remark 1, we now get from Lemma 3.2 that

[TABLE]

Combining (4.24),(4.25), we obtain

[TABLE]

By (4.20), (4.23), (4.26) and claim (a),(b),(c),(d), we can prove that $X$ satisfies

[TABLE]

and using the same argument as in the proof of Theorem 3.1 in Temam [31], Section 3, Chapter 3, we can conclude $X=\widetilde{X}^{q}$.

Next, we prove $X_{m^{\prime}}\rightarrow X$ in $C([0,T],\mathbb{V})$. Let $Z_{m^{\prime}}=X_{m^{\prime}}-X$. Then

[TABLE]

By Lemma 2.3 and claim (a), we have

[TABLE]

By Condition 3.1, we have

[TABLE]

and

[TABLE]

where \Upsilon_{m^{\prime}}^{1}(T)=C_{G}\big{(}\sup_{s\in[0,T]}\|X(s)\|_{\mathbb{V}}\big{)}\big{(}\int_{0}^{T}|f_{m^{\prime}}(s)-f(s)|^{2}ds\big{)}^{\frac{1}{2}}\big{(}\int_{0}^{T}\|X_{m^{\prime}}(s)-X(s)\|_{\mathbb{V}}^{2}ds\big{)}^{\frac{1}{2}}, claim (d) and $f_{m^{\prime}},f\in S_{1}^{m}$ imply that

[TABLE]

For $I^{4}_{m^{\prime}}(s)$, we have

[TABLE]

Together with (4.28), (4.29), (4.30), (4.31), (4.33), we obtain

[TABLE]

where $\Psi_{m^{\prime}}(s)=C+C(1+f_{m^{\prime}}^{2}(s))$, satisfying $\sup_{m^{\prime}}\int_{0}^{T}\Psi_{m^{\prime}}(s)ds<\infty.$

Then by Gronwall’s inequality, we get

[TABLE]

By (4.29) in Zhai and Zhang [35], we can easily get $\Upsilon_{m^{\prime}}^{2}(T)\rightarrow 0$ as $m^{\prime}\rightarrow\infty$, combining (4.32), we get

[TABLE]

The proof is completed. ∎

Let $\phi_{\varepsilon}=(\psi_{\varepsilon},\varphi_{\varepsilon})\in\mathcal{S}_{1}^{m}\times\mathcal{S}_{2}^{m}$ and $\vartheta_{\varepsilon}=\frac{1}{\varphi_{\varepsilon}}$. The following lemma was stated in Budhiraja, Dupuis and Maroulas [5]( see Lemma 2.3 there).

Lemma 4.1

[TABLE]

and

[TABLE]

are $\mathbb{F}$-martingales. Set

[TABLE]

Then

[TABLE]

defines a probability measure on $C([0,T],\mathbb{R})\times M_{FC}([0,T]\times\mathbb{Z})$.

Since $(\sqrt{\varepsilon}W(\cdot)+\int_{0}^{\cdot}\psi_{\varepsilon}(s)ds,\varepsilon N^{{\varepsilon}^{-1}\varphi_{\varepsilon}})$ under $Q_{T}^{\varepsilon}$ has the same law as that of $(\sqrt{\varepsilon}W,\varepsilon N^{\varepsilon^{-1}})$ under $P$, by Theorem 4.2 it follows that there exists a unique solution to the following controlled stochastic evolution equation, denoted by $\widetilde{X}^{\varepsilon}$:

[TABLE]

and we have

[TABLE]

The following estimates will be used later.

Lemma 4.2

There exists $\varepsilon_{0}>0$ such that

[TABLE]

Proof.

Define

[TABLE]

Multiplying $\lambda_{i}$ at both sides of the equation (4.35), we can use (2.4) to obtain

[TABLE]

for any $i\in\mathbb{N}$.

Applying $It\hat{o}$ formula to \big{(}\widetilde{X}^{\varepsilon}(s),e_{i}\big{)}_{\mathbb{W}}^{2} and then summing over $i$ from $1$ to $\infty$ yields

[TABLE]

By a simple calculus, refer to (4.1), we get

[TABLE]

where

[TABLE]

By Condition 3.1 and (2), (4.8), we have

[TABLE]

By B-D-G and Young’s inequalities and Lemma 3.1, we get

[TABLE]

and

[TABLE]

By Condition 3.1 and Lemma 3.1, we get

[TABLE]

[TABLE]

and

[TABLE]

Combining the estimates (4.41), (4.44) and (4.45), we have

[TABLE]

By Lemma 3.1 and Gronwall’s inequality, we get

[TABLE]

Set C_{0}=e^{C\big{(}1+m+C_{0,1}^{m}\big{)}}. By (4.42), (4.43) and (4.46), we have

[TABLE]

Since $C_{0},C,C_{0,1}^{m},C_{0,2}^{m}$ are constant independent of $\varepsilon$, we can select $\varepsilon_{0}$ small enough, such that

[TABLE]

then letting $M\rightarrow\infty$, we get (4.37).

∎

By Proposition 3.1 in [25], there exists a unique solution $\widetilde{Y}^{\varepsilon}(t),t\geq 0$ to the followig equation:

[TABLE]

with initial value $\widetilde{Y}^{\varepsilon}(0)=0$. Moreover, $\widetilde{Y}^{\varepsilon}\in D([0,T],\mathbb{V})\cap L^{2}([0,T],\mathbb{W})$, P-a.s. and we have the following estimates.

Lemma 4.3

There exists $C>0$ such that

[TABLE]

Proof.

Define

[TABLE]

By (4.48), we have

[TABLE]

for any $i\in\mathbb{N}$.

Multiplying both sides of the equation (4.51) by $\lambda_{i}$, we can use (2.4) to obtain

[TABLE]

for any $i\in\mathbb{N}$.

Applying $It\hat{o}$ formula to \big{(}\widetilde{Y}^{\varepsilon}(t),e_{i}\big{)}_{\mathbb{W}}^{2} and summing over $i$ from $1$ to $\infty$ yields

[TABLE]

Similar to (4.1) and (4.40), we get

[TABLE]

Taking the sup over $t\leq T\wedge\tau_{M}$, then taking expectations we get

[TABLE]

By Condition 3.1 and Lemma 3.1, we get

[TABLE]

and

[TABLE]

By (4.8), we have

[TABLE]

By B-D-G inequality and Young inequality and Lemma 3.1, we get

[TABLE]

and

[TABLE]

Combining above all inequalities, we get

[TABLE]

By (4.37) and Gronwall’s inequality , then letting $M\rightarrow\infty$, we get (4.49). ∎

Proposition 4.5

Fix $m\in\mathbb{N}$, and let $\phi_{\varepsilon}=(\psi_{\varepsilon},\varphi_{\varepsilon}),\phi=(\psi,\varphi)\in{{\mathcal{S}}}_{1}^{m}\times{{\mathcal{S}}}_{2}^{m}$ be such that $\phi_{\varepsilon}$ converges in distribution to $\phi$ as $n\rightarrow\infty$. Then

[TABLE]

Proof.

Set $\widetilde{Z}^{\varepsilon}=\widetilde{X}^{\varepsilon}-\widetilde{Y}^{\varepsilon}$, which satisfies

[TABLE]

Set

[TABLE]

Let \big{(}(\psi,\varphi),0\big{)} be any limit point of the tight family \big{\{}\big{(}(\psi_{\varepsilon},\varphi_{\varepsilon}),\widetilde{Y}^{\varepsilon}\big{)},\varepsilon\in(0,\varepsilon_{0})\big{\}} in $\Pi$. By Skorokhod representation theorem, there exists a probability space \big{(}\Omega^{1},\mathcal{F}^{1},P^{1}\big{)} and on this basis, there exist $\Pi$-valued random variables \big{(}(\psi^{1},\varphi^{1}),0\big{)}, \big{\{}\big{(}(\psi_{\varepsilon}^{1},\varphi_{\varepsilon}^{1}),\widetilde{Y}^{\varepsilon}_{1}\big{)},\varepsilon\in(0,\varepsilon_{0})\big{\}} such that \big{(}(\psi_{\varepsilon}^{1},\varphi_{\varepsilon}^{1}),\widetilde{Y}^{\varepsilon}_{1}\big{)} (respectively \big{(}(\psi^{1},\varphi^{1}),0\big{)}) has the same law as \big{(}(\psi_{\varepsilon},\varphi_{\varepsilon}),\widetilde{Y}^{\varepsilon}\big{)} (respectively \big{(}(\psi,\varphi),0\big{)}) and \big{(}(\psi_{\varepsilon}^{1},\varphi_{\varepsilon}^{1}),\widetilde{Y}^{\varepsilon}_{1}\big{)}\rightarrow\big{(}(\psi^{1},\varphi^{1}),0\big{)} $P^{1}$-a.s. in $\Pi$.

Since

[TABLE]

we get

(1)$\sup_{t\in[0,T]}\|\widetilde{Y}^{\varepsilon}_{1}(t)\|_{\mathbb{W}}^{2}<\infty,\ \ P^{1}-a.s.$

(2)there exists a sub-sequence $\varepsilon_{k}$ and a subset $\Omega^{1}_{0}$ of $\Omega^{1}$, such that P^{1}\big{(}\Omega^{1}_{0}\big{)}=1 and $\forall\omega^{1}\in\Omega^{1}_{0}$

[TABLE]

Let $\bar{Z}^{\varepsilon}$ be the solution of the following equation,

[TABLE]

Keep in mind claim (1), similarly as the proof of Theorem 4.1, we can prove (4.63) has a unique solution.

Comparing (4.60) and (4.63), and \big{(}(\psi_{\varepsilon}^{1},\varphi_{\varepsilon}^{1}),\widetilde{Y}^{\varepsilon}_{1}\big{)} has the same law as \big{(}(\psi_{\varepsilon},\varphi_{\varepsilon}),\widetilde{Y}^{\varepsilon}\big{)}, we can conclude that \big{(}\bar{Z}^{\varepsilon},\widetilde{Y}^{\varepsilon}_{1}\big{)} has the same law as \big{(}\widetilde{Z}^{\varepsilon},\widetilde{Y}^{\varepsilon}\big{)}, hence

[TABLE]

By (4.62) and a similar argument as in the proof of Proposition 4.4, we can show that

[TABLE]

where

[TABLE]

Actually $\widehat{X}$ has the same law of $\mathcal{G}^{0}(\int_{0}^{\cdot}\psi(s)ds,\nu_{T}^{\varphi})$. Using (4.62) and (4.65) to obtain

[TABLE]

Combining (4.64), we yield

[TABLE]

The proof is completed. ∎