###### Abstract

In this paper, we study the large deviation principle of invariant measures of
stochastic reaction-diffusion lattice systems driven by multiplicative noise.
We first show that any limit of a sequence of invariant measures of the
stochastic system must be an invariant measure of the deterministic limiting
system as noise intensity approaches zero. We then prove the uniform Freidlin-
Wentzell large deviations of solution paths over all initial data and the
uniform Dembo-Zeitouni large deviations of solution paths over a compact set
of initial data. We finally establish the large deviations of invariant
measures by combining the idea of tail-ends estimates and the argument of
weighted spaces.

Key words. Large deviation principle, invariant measure, compactness,
exponential tightness, weighted space, stochastic lattice system.

MSC 2020. 60F10, 60H10, 37L55, 37L30.

##  1 Introduction

In this paper, we investigate the large deviation principle (LDP) of invariant
measures of the stochastic reaction-diffusion lattice system driven by
multiplicative noise defined on the $N$-dimensional integer set
$\mathbb{Z}^{N}$. Given $i\in\mathbb{Z}^{N}$, consider the Itô stochastic
system:

$du^{\varepsilon}_{i}(t)+(\lambda
u^{\varepsilon}_{i}(t)+(Au^{\varepsilon}(t))_{i}+f_{i}(u^{\varepsilon}_{i}(t)))dt=g_{i}dt+\sqrt{{\varepsilon}}\sigma_{i}(u^{\varepsilon}(t))dW,\quad
t>0,$ (1.1)

with initial data

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

where $i=(i_{1},\ldots,i_{N})\in\mathbb{Z}^{N}$,
$u=(u_{i})_{i\in\mathbb{Z}^{N}}$ is an unknown sequence, $\lambda>0$ and
${\varepsilon}\in[0,1]$ are constants, $A$ is the negative discrete
$N$-dimensional Laplace operator, $g=(g_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}$,
and $W$ is an $\ell^{2}$-cylindrical Wiener process on a complete filtered
probability space
$(\Omega,\mathcal{F},\\{{\mathcal{F}}_{t}\\}_{t\in\mathbb{R}},P)$ which
satisfies the usual condition. For every $i\in\mathbb{Z}^{N}$,
$f_{i}:\mathbb{R}\to\mathbb{R}$ is a nonlinear function with arbitrary growth
rate, and
$\sigma=(\sigma_{i})_{i\in\mathbb{Z}^{N}}:\ell^{2}\to{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})$
is a bounded Lipschitz function where ${{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})$
is the space of Hilbert-Schmidt operators from $\ell^{2}\to\ell^{2}$ with norm
$\|\cdot\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}$.

Lattice systems arise from many applications in physical and biological
systems [7, 23, 31, 33, 34]. The traveling solutions and dynamics of
deterministic lattice systems have been investigated in [1, 2, 3, 4, 8, 20,
21, 22, 26, 27, 30, 32, 39, 42, 47] and the references therein. Recently, the
dynamics of random lattice systems has been extensively studied, see, e.g.,
[5, 6, 12, 13, 29] for random attractors; and [17, 18, 19, 35, 36, 43, 45, 46]
for invariant measures and periodic measures. The goal of this paper is to
prove the LDP of invariant measures of system (1.1)-(1.2) as ${\varepsilon}\to
0$, which is stated below.

######  Theorem 1.1.

If (2.1), (2.2)-(2.5) and (4.2) given in Sections 2 and 4 are fulfilled, then
the family $\\{\mu^{\varepsilon}\\}$ of invariant measures of (1.1)-(1.2)
satisfies the LDP in $\ell^{2}$ as ${\varepsilon}\to 0$.

The LDP of distributions of solutions  has been established for a variety of
stochastic differential equations in the literature, including the stochastic
ordinary differential equations as well as the stochastic partial differential
equations. However, the LDP of invariant measures of stochastic systems is not
well understood, and so far there are only a few publications in this area.
The reader is referred to [28] for the LDP of invariant measures of finite-
dimensional stochastic systems, and to [9, 14, 15, 37, 38, 41] for that of
infinite-dimensional stochastic systems. In particular, in the infinite-
dimensional case, the LDP of invariant measures for additive noise has been
investigated in [9, 15, 37, 38], and for multiplicative noise in [14, 41]. The
stochastic partial differential equations in all these papers are defined in a
bounded spatial domain where the Sobolev embeddings are compact. As far as the
author is aware, there is no result reported in the literature on the LDP of
invariant measures of infinite-dimensional lattice systems like (1.1).

Note that the lattice system (1.1) can be considered as an infinite-
dimensional stochastic system defined in $\ell^{2}$ which has distinct
features from the finite-dimensional systems associated with stochastic
ordinary differential equations as well as infinite-dimensional systems
associated with stochastic partial differential equations in bounded domains.
For example, every bounded set in a finite-dimensional space is precompact,
but that is false for the infinite-dimensional space $\ell^{2}$. On the other
hand, since the Sobolev embedding $H^{1}({{\mathcal{O}}})\hookrightarrow
L^{2}({{\mathcal{O}}})$ is compact for a bounded domain ${{\mathcal{O}}}$, the
solution operator generated by the Laplacian in $L^{2}({{\mathcal{O}}})$ is
compact. However, since there is no such compact embedding available in the
discrete case, the solution operator generated by the discrete Laplacian is no
longer compact in $\ell^{2}$.

For ordinary differential equations and partial differential equations in
bounded domains, the compactness of solution operators is a key for proving
the tightness of distributions of a family of solutions, the compactness of
the level sets of rate functions as well as the LDP of solutions and invariant
measures. Since the solution operator of the lattice system (1.1) is not
compact, there is an essential difficulty to obtain the existence and the LDP
of invariant measures in this case, including proving the exponential
tightness of invariant measures over compact sets.

In order to overcome the non-compactness of solutions in $\ell^{2}$, we will
apply the idea of uniform tail-ends estimates to prove the tightness of
distributions of a family of solutions and hence the existence of invariant
measures of (1.1) as in [43]. The uniform tail-ends estimates (see Lemma 2.2)
indicate that all the tails of solutions of (1.1) are uniformly small which
along with the precompactness of bounded sets in a finite-dimensional space
yields the tightness of distributions of solutions.

In order to prove the LDP of invariant measures of (1.1) in $\ell^{2}$, we
will first establish the uniform Freidlin-Wentzell LDP [28] of solutions over
all initial data (see Theorem 3.7) by the weak convergence method as developed
in [10, 11, 25, 40]. We will then establish the uniform Dembo-Zeitouni LDP
[24] of solutions with respect to a compact set of initial data. We remark
that both the uniform Freidlin-Wentzell LDP and the uniform Dembo-Zeitouni LDP
of solutions are needed for proving the LDP of invariant measures of (1.1).

As proved by Salins in [40], the uniform Freidlin-Wentzell LDP and the uniform
Dembo-Zeitouni LDP of solutions are not equivalent over a non-compact set of
initial data, and hence the uniform Dembo-Zeitouni LDP of solutions over a
non-compact set of initial data does not follow from the uniform Freidlin-
Wentzell LDP as obtained in Theorem 3.7. Nevertheless, in light of [40], we
can show that Theorem 3.7 does imply the uniform Dembo-Zeitouni LDP of (1.1)
over a compact set of initial data (see Theorem 3.9). In order to apply
Theorem 3.9 to establish the LDP of invariant measures of (1.1), we must prove
the exponential tightness of of invariant measures over compact subsets in
$\ell^{2}$. In the finite-dimensional case, such a compact set can be chosen
as a bounded closed subsets of the space. However, a bounded closed subset is
not compact in $\ell^{2}$. To solve this problem, in the present paper, we
will apply the idea of weighted $\ell^{2}$ space and construct a compact
subset of $\ell^{2}$ from the weighted space with appropriate weight (see
Lemma 4.5).

It is worth mentioning that the stochastic system (1.1) is driven by a
multiplicative noise, and the uniform Dembo-Zeitouni LDP of solutions of (1.1)
over a bounded set of initial data remains open. For the stochastic reaction-
diffusion equation driven by multiplicative noise in a bounded domain, the
uniform Dembo-Zeitouni LDP of solutions was used in [41] over bounded initial
data and in [14] over all initial data. However, there was no proof given in
both papers. It seems that it is nontrivial to prove the uniform Dembo-
Zeitouni LDP of solutions for multiplicative noise over bounded or entire
initial data, and we do not know how to establish the uniform Dembo-Zeitouni
LDP of the reaction-diffusion equation with multiplicative noise over non-
compact  initial data. That is why in this paper, we prove the uniform Dembo-
Zeitouni LDP only over a compact set of initial data but not over a bounded
set of initial data. Because the uniform Dembo-Zeitouni LDP of solutions is
available only over compact initial data, we must establish the exponential
tightness of invariant measures over compact sets in order to prove the LDP of
invariant measures. Note that the uniform exponential probability estimates of
solutions over bounded  subsets are not sufficient to prove the LDP of
invariant measures when the solutions satisfy the uniform Dembo-Zeitouni LDP
only over compact subsets of initial data. Of course, the uniform exponential
probability estimates of solutions over bounded  subsets can be used to prove
the LDP of invariant measures when the solutions satisfy the uniform Dembo-
Zeitouni LDP over bounded subsets of initial data. Since bounded subsets of
$\ell^{2}$ are not precompact, we have to derive the exponential estimates of
solutions in a weighted $\ell^{2}$ space, and use the compactness of embedding
from a weighted space to the standard $\ell^{2}$ space to obtain the
exponential tightness of invariant measures. For that purpose, the diffusion
term must satisfy certain conditions in a weighted $\ell^{2}$ space, see,
e.g., (2.5).

This paper is organized as follows. In the next section, we prove the
existence of invariant measures of (1.1)-(1.2) under some conditions on the
nonlinear drift and diffusion terms. We also prove that any limit of a
sequence of invariant measures of the stochastic system must be an invariant
measure of the limiting system with ${\varepsilon}=0$. In Section 3, we prove
the uniform Freidlin-Wentzell LDP and the uniform Dembo-Zeitouni LDP of
solutions. Section 4 is devoted to the rate functions of invariant measures,
energy estimates and the exponential tightness of solutions. In the last two
sections, we prove the LDP lower bound and the LDP upper bound of invariant
measures, respectively, by combining the ideas of [38, 41] with weighted
spaces.

##  2 Existence and limits of invariant measures

This section is devoted to the existence and the limiting behavior of
invariant measures of (1.1)-(1.2) under appropriate conditions. We first
discuss the assumptions on the nonlinear terms in (1.1), and then show the
existence of invariant measures of the system. We finally prove that any limit
of a sequence of invariant measures of the stochastic system must be an
invariant measure of the limiting system with ${\varepsilon}=0$.

Let $\ell^{2}$ be the Hilbert space of square-summable sequences of real
numbers on $\mathbb{Z}^{N}$ with norm $\|\cdot\|$ and inner product
$(\cdot,\cdot)$. For every $j=1,\ldots,N$, define the operators
$A_{j},B_{j},B^{*}_{j}:\ell^{2}\to\ell^{2}$ by, for
$u=(u_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}$ and
$i=(i_{1},i_{2},\dots,i_{N})\in\mathbb{Z}^{N}$,

$(A_{j}u)_{i}=-u_{(i_{1},\dots,i_{j}+1,\dots,i_{N})}+2u_{(i_{1},\dots,i_{j},\dots,i_{N})}-u_{(i_{1},\dots,i_{j}-1,\dots,i_{N})},$
$(B_{j}u)_{i}=u_{(i_{1},\dots,i_{j}+1,\dots,i_{N})}-u_{(i_{1},\dots,i_{j},\dots,i_{N})},$

and

$(B_{j}^{*}u)_{i}=u_{(i_{1},\dots,i_{j}-1,\dots,i_{N})}-u_{(i_{1},\dots,i_{j},\dots,i_{N})}.$

The negative discrete Laplace operator $A:\ell^{2}\to\ell^{2}$ is given by,
for $u=(u_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}$ and
$i=(i_{1},i_{2},\dots,i_{N})\in\mathbb{Z}^{N}$,

$(Au)_{i}=-u_{(i_{1}-1,i_{2},\dots,i_{N})}-u_{(i_{1},i_{2}-1,\dots,i_{N})}-\dots-
u_{(i_{1},i_{2},\dots,i_{N}-1)}$
$+2Nu_{(i_{1},i_{2},\dots,i_{N})}-u_{(i_{1}+1,i_{2},\dots,i_{N})}-u_{(i_{1},i_{2}+1,\dots,i_{N})}-\dots-
u_{(i_{1},i_{2},\dots,i_{N}+1)}\big{)}.$

Then we have
$A=\sum_{j=1}^{N}A_{j}=\sum_{j=1}^{N}B_{j}B_{j}^{*}=\sum_{j=1}^{N}B_{j}^{*}B_{j}$.

For the nonlinear drift $f=(f_{i})_{i\in\mathbb{Z}^{N}}$, we assume that for
every $i\in\mathbb{Z}^{N}$, $f_{i}\in C^{1}(\mathbb{R},\mathbb{R})$ such that
for $s\in\mathbb{R}$,

$f_{i}(0)=0,\quad f_{i}(s)s\geq 0,\quad\text{and}\quad
f_{i}^{\prime}(s)\geq-\gamma,$ (2.1)

where $\gamma>0$ is a constant.

In the sequel, we assume that
$\sigma:\ell^{2}\to{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})$ is a bounded
Lipschitz map; that is, there exists $L_{\sigma}>0$ such that

$\sup_{u\in\ell^{2}}\|\sigma(u)\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\leq
L_{\sigma},$ (2.2)

and

$\|\sigma(u_{1})-\sigma(u_{2})\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\leq
L_{\sigma}\|u_{1}-u_{2}\|,\quad\forall\ u_{1},u_{2}\in\ell^{2}.$ (2.3)

For the existence of invariant measures, we also assume

$\lim_{k\to\infty}\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}=0,$ (2.4)

where $1_{\\{|\cdot|\geq k\\}}\sigma(u)$ is the tail of $\sigma(u)$ as defined
by, for every $v\in\ell^{2}$ and $i\in\mathbb{Z}^{N}$,

$\left((1_{\\{|\cdot|\geq
k\\}}\sigma(u))(v)\right)_{i}=1_{[k,\infty)}(|i|)(\sigma(u)(v))_{i}=\left\\{\begin{array}[]{ll}\left(\sigma(u)(v)\right)_{i},&\
\ \text{if }\ |i|\geq k\\\ 0,&\ \ \text{if }\ |i|<k.\end{array}\right.$

We remark that condition (2.4) is needed when deriving the uniform tail-ends
estimates of solutions (see Lemmas 2.2 and 3.2), which are necessary for
proving the tightness of distribution laws of a family of solutions and hence
the existence of invariant measures.

We now introduce a weight function ${\kappa}:\mathbb{R}\to\mathbb{R}$ as given
by

${\kappa}(s)=\left(1+s^{2}\right)^{\frac{1}{2}},\quad\forall\ s\in\mathbb{R}.$

Let $\ell^{2}_{\kappa}$ be the weighted space defined by

$l^{2}_{\kappa}=\left\\{u\in\ell^{2}:\
\|u\|_{\ell^{2}_{\kappa}}=\left(\sum_{i\in\mathbb{Z}^{N}}{\kappa}^{2}(|i|)|u_{i}|^{2}\right)^{\frac{1}{2}}<\infty\right\\}.$

For the LDP of invariant measures, we further assume
$\sigma:\ell^{2}\to{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2}_{\kappa})$ is
bounded:

$\sup_{u\in\ell^{2}}\|\sigma(u)\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2}_{\kappa})}\leq
L_{\sigma}.$ (2.5)

Since bounded subsets of $\ell^{2}$ are not precompact, we need to use
condition (2.5) to derive the exponential estimates of solutions in the
weighted space $\ell^{2}_{\kappa}$, and then use the compactness of embedding
$\ell^{2}_{\kappa}\hookrightarrow\ell^{2}$ to obtain the exponential tightness
of invariant measures in $\ell^{2}$ (see Lemmas 4.4 and 4.5), which is
necessary to establish the LDP of invariant measures.

Next, we discuss an example where $\sigma$ satisfies (2.2)-(2.5). Given
$i\in\mathbb{Z}^{N}$, let $e_{i}$ be the element in $\ell^{2}$ which has an
entry $1$ at position $i$; and $0$ otherwise. Then the sequence
$\\{e_{i}\\}_{i\in\mathbb{Z}^{N}}$ is an orthonormal basis of $\ell^{2}$. Let
$\\{a_{i}\\}_{i\in\mathbb{Z}^{N}}$ be an element in $\ell^{2}_{\kappa}$; that
is,

$\|a\|^{2}_{\ell^{2}_{\kappa}}=\sum_{i\in\mathbb{Z}^{N}}(1+|i|^{2})a_{i}^{2}<\infty.$
(2.6)

Given $u=(u_{i})_{i\in\mathbb{Z}^{N}}$,
$v=(v_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}$, denote by

$\sigma(u)(v)=(1+\|u\|)^{-1}\sum_{i\in\mathbb{Z}^{N}}a_{i}v_{i}e_{i}.$ (2.7)

Then $\sigma(u):\ell^{2}\to\ell^{2}$ is a Hilbert-Schmidt operator with norm

$\|\sigma(u)\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}=\|a\|(1+\|u\|)^{-1}\leq\|a\|,\quad\forall\
u\in\ell^{2}.$ (2.8)

In addition, we have for all $u,u_{1},u_{2}\in\ell^{2}$,

$\|\sigma(u_{1})-\sigma(u_{2})\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}=\|a\|\|u_{1}-u_{2}\|,$
(2.9) $\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}=(1+\|u\|)^{-2}\sum_{|i|\geq
k}a_{i}^{2}\leq\sum_{|i|\geq k}a_{i}^{2},$ (2.10)

and

$\|\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2}_{\kappa})}=(1+\|u\|)^{-2}\|a\|^{2}_{\ell^{2}_{\kappa}}\leq\|a\|^{2}_{\ell^{2}_{\kappa}}.$
(2.11)

It follows from (2.6)-(2.11) that all conditions (2.2)-(2.5) are satisfied for
$\sigma$ given by (2.7) with $L_{\sigma}=\|a\|_{\ell^{2}_{\kappa}}$.

We now rewrite system (1.1)-(1.2) as the stochastic equation in $\ell^{2}$:

$du^{\varepsilon}(t)+\lambda u^{\varepsilon}(t)dt+\
Au^{\varepsilon}(t)dt+f(u^{\varepsilon}(t))dt=g+\sqrt{{\varepsilon}}\sigma(u^{\varepsilon}(t))dW,\quad
u^{\varepsilon}(0)=u_{0}\in\ell^{2}.$ (2.12)

For ${\varepsilon}=0$, the deterministic limiting system corresponding to
(2.12) is given by

${\frac{du(t)}{dt}}+\lambda u(t)+\ Au(t)+f(u(t))=g,\quad
u(0)=u_{0}\in\ell^{2}.$ (2.13)

Given ${\varepsilon}\in[0,1]$ and $u_{0}\in
L^{2}(\Omega,{{\mathcal{F}}}_{0};\ell^{2})$, by a solution $u^{\varepsilon}$
of (2.12), we mean that $u^{\varepsilon}$ is a continuous $\ell^{2}$-valued
${{\mathcal{F}}}_{t}$-adapted process such that $u^{\varepsilon}\in
L^{2}(\Omega,C([0,T],\ell^{2}))$ for each $T>0$, and for all $t\geq 0$,

$u^{\varepsilon}(t)+\int_{0}^{t}\left(\lambda
u^{\varepsilon}(s)+Au^{\varepsilon}(s)+f(u^{\varepsilon}(s))\right)ds=u_{0}+gt+\sqrt{{\varepsilon}}\int_{0}^{t}\sigma(u^{\varepsilon}(s))dW(s)$

in $\ell^{2}$, $P$-almost surely.

If (2.1) and (2.2)-(2.3) are fulfilled, then by the argument of [43, Theorem
2.5], one can verify that for each $u_{0}\in
L^{2}(\Omega,{{\mathcal{F}}}_{0};\ell^{2})$, the stochastic equation (2.12)
has a unique solution $u^{\varepsilon}\in L^{2}(\Omega,C([0,T],\ell^{2}))$.

Next, we derive the uniform estimates of solutions of (2.12).

######  Lemma 2.1.

If (2.1) and (2.2)-(2.3) are fulfilled, then the solution
$u^{\varepsilon}(\cdot,0,u_{0})$ of (2.12) satisfies, for all
${\varepsilon}\in[0,1]$ and $t\geq 0$,

$\mathbb{E}(\|u^{\varepsilon}(t,0,u_{0})\|^{2})\leq e^{-\lambda
t}\mathbb{E}(\|u_{0}\|^{2})+\lambda^{-2}\|g\|^{2}+\lambda^{-1}L_{\sigma}^{2}.$

###### Proof.

By (2.1) and Itô’s formula, we obtain from (2.12) that

${\frac{d}{dt}}\mathbb{E}\left(\|u^{\varepsilon}(t)\|^{2}\right)+2\lambda\mathbb{E}\left(\|u^{\varepsilon}(t)\|^{2}\right)\leq
2\mathbb{E}\left(u^{\varepsilon}(t),g\right)+{\varepsilon}\mathbb{E}\left(\|\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\right),$

which along with (2.2) implies that

${\frac{d}{dt}}\mathbb{E}\left(\|u^{\varepsilon}(t)\|^{2}\right)+\lambda\mathbb{E}\left(\|u^{\varepsilon}(t)\|^{2}\right)\leq\lambda^{-1}\|g\|^{2}+{\varepsilon}L_{\sigma}^{2}.$

This together with the Gronwall inequality concludes the proof. ∎

The next lemma is concerned with the uniform estimates on the tails of
solutions.

######  Lemma 2.2.

Suppose (2.1) and (2.2)-(2.4) are valid. Then for any $\delta>0$ and any
compact subset ${{\mathcal{K}}}$ of $\ell^{2}$, there exists
$k_{0}=k_{0}(\delta,{{\mathcal{K}}})\geq 1$ such that for all $k\geq k_{0}$,
${\varepsilon}\in[0,1]$ and $t\geq 0$, the solution
$u^{\varepsilon}(\cdot,0,u_{0})$ of (2.12) with $u_{0}\in{{\mathcal{K}}}$
satisfies,

$\sum_{|i|\geq
k}\mathbb{E}\left(|u^{\varepsilon}_{i}(t,0,u_{0})|^{2}\right)<\delta.$

###### Proof.

Let $\rho:\mathbb{R}\to[0,1]$ be a smooth function such that

$\rho(s)=0\ \ \text{ if }\ |s|\leq 1;\ \ \rho(s)=1\ \ \text{ if }\ |s|\geq 2.$
(2.14)

Given $k\in\mathbb{N}$, we write
$\rho_{k}u=\left(\rho(\frac{|i|}{k})u_{i}\right)_{i\in\mathbb{Z}^{N}}$ for any
$u=(u_{i})_{i\in\mathbb{Z}^{N}}$. By (2.1), (2.12) and the Itô formula we get

${\frac{d}{dt}}\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})+2\sum_{j=1}^{N}\mathbb{E}(B_{j}u^{\varepsilon}(t),\
B_{j}(\rho_{k}^{2}u^{\varepsilon}(t)))+2\lambda\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})$
$\leq
2\mathbb{E}(\rho_{k}g,\rho_{k}u^{\varepsilon}(t))+{\varepsilon}\mathbb{E}(\|\rho_{k}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}).$
(2.15)

After simple calculations, we find that for all $t\geq 0$,

$-2\sum_{j=1}^{N}\mathbb{E}(B_{j}u^{\varepsilon}(t),\
B_{j}(\rho_{k}^{2}u^{\varepsilon}(t)))\leq{\frac{c_{1}}{k}}\mathbb{E}(\|u^{\varepsilon}(t)\|^{2})$
(2.16)

where $c_{1}$ is a positive number independent of $k$. By (2.16) and Lemma 2.1
we get for all $t\geq 0$,

$-2\sum_{j=1}^{N}\mathbb{E}(B_{j}u^{\varepsilon}(t),\
B_{j}(\rho_{k}^{2}u^{\varepsilon}(t)))\leq{\frac{c_{2}}{k}},$ (2.17)

where $c_{2}=c_{2}({{\mathcal{K}}})>0$ is independent of $k$.

By (2.4) we have for all ${\varepsilon}\in[0,1]$ and $t\geq 0$,

${\varepsilon}\mathbb{E}(\|\rho_{k}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})})\leq\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\to 0,\ \
\text{as }\ k\to\infty.$ (2.18)

On the other hand, by Young’s inequality we have

$2\mathbb{E}(\rho_{k}g,\rho_{k}u^{\varepsilon}(t))\leq\lambda\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})+{\frac{1}{\lambda}}\sum_{|i|\geq
k}g_{i}^{2}.$ (2.19)

It follows from (2) and (2.17)-(2.19) that for every $\delta>0$, there exists
$k_{0}=k_{0}(\delta,{{\mathcal{K}}},g)\geq 1$ such that for all $k\geq k_{0}$
and $t\geq 0$,

${\frac{d}{dt}}\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})+\lambda\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})\leq\delta.$
(2.20)

By (2.20) and Gronwall’s inequality, we infer that for all $k\geq k_{0}$ and
$t\geq 0$,

$\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})\leq e^{-\lambda
t}\mathbb{E}(\|\rho_{k}u_{0}\|^{2})+{\frac{\delta}{\lambda}}\leq\sum_{|i|\geq
k}|u_{0,i}|^{2}+{\frac{\delta}{\lambda}}.$ (2.21)

By assumption, we know that $u_{0}\in{{\mathcal{K}}}$ and ${{\mathcal{K}}}$ is
compact in $\ell^{2}$. Therefore, by (2.21) we find that there exists
$k_{1}=k_{1}(\delta,{{\mathcal{K}}},g)\geq k_{0}$ such that for all $k\geq
k_{1}$, $t\geq 0$ and $u_{0}\in{{\mathcal{K}}}$,

$\mathbb{E}\left(\sum_{|i|\geq
2k}|u^{\varepsilon}_{i}(t,0,u_{0})|^{2}\right)\leq\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t,0,u_{0})\|^{2})\leq\delta+{\frac{\delta}{\lambda}},$

as desired. ∎

By the argument of Lemma 2.2, we also obtain the following uniform tail-ends
estimates.

######  Lemma 2.3.

Suppose (2.1) and (2.2)-(2.4) are valid. Then for every $\delta>0$ and $R>0$,
there exist $T=T(\delta,R)>0$ and $k_{0}=k_{0}(\delta)\geq 1$ such that for
all $t\geq T$, $k\geq k_{0}$ and ${\varepsilon}\in(0,1)$, the solution
$u^{\varepsilon}(\cdot,0,u_{0})$ of (2.12) with
$\|u_{0}\|_{L^{2}(\Omega,\ell^{2})}\leq R$ satisfies,

$\sum_{|i|\geq
k}\mathbb{E}\left(|u^{\varepsilon}_{i}(t,0,u_{0})|^{2}\right)<\delta.$

###### Proof.

By (2.16) and Lemma 2.1 we see that for $u_{0}$ with
$\|u_{0}\|_{L^{2}(\Omega,\ell^{2})}\leq R$ and $t\geq 0$,

$-2\sum_{j=1}^{N}\mathbb{E}(B_{j}u^{\varepsilon}(t),\
B_{j}(\rho_{k}^{2}u^{\varepsilon}(t)))\leq{\frac{c_{1}}{k}}\left(e^{-\lambda
t}R+\lambda^{-2}\|g\|^{2}+\lambda^{-1}L_{\sigma}^{2}\right),$

and hence there exists $T_{1}=T_{1}(R)>0$ such that for all $t\geq T_{1}$,

$-2\sum_{j=1}^{N}\mathbb{E}(B_{j}u^{\varepsilon}(t),\
B_{j}(\rho_{k}^{2}u^{\varepsilon}(t)))\leq{\frac{c_{1}}{k}}\left(1+\lambda^{-2}\|g\|^{2}+\lambda^{-1}L_{\sigma}^{2}\right).$
(2.22)

By (2.18)-(2.19) and (2.22) we infer from (2) that for every $\delta>0$ and
$R>0$, there exist $T_{2}=T_{2}(\delta,R)\geq T_{1}$ and
$k_{0}=k_{0}(\delta,g)\geq 1$ such that for all $t\geq T_{2}$ and $k\geq
k_{0}$,

${\frac{d}{dt}}\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})+\lambda\mathbb{E}(\|\rho_{k}u^{\varepsilon}(t)\|^{2})\leq\delta.$
(2.23)

Then the desired inequality follows from (2.23) and Gronwall’s lemma. ∎

We now present the existence of invariant measures for (2.12).

######  Theorem 2.4.

If (2.1) and (2.2)-(2.4) are fulfilled, then for every
${\varepsilon}\in[0,1]$, system (2.12) has an invariant measure on $\ell^{2}$.

###### Proof.

The proof is similar to that of [43, Theorem 4.8], by using the uniform
estimates given by Lemma 2.1 and the uniform tail-ends estimates given by
Lemma 2.2. The details are omitted here. ∎

Given ${\varepsilon}\in[0,1]$, let ${{\mathcal{M}}}^{\varepsilon}$ be the
collection of all invariant measures of (2.12) corresponding to
${\varepsilon}$. In the sequel, we will prove the set
$\bigcup\limits_{\varepsilon\in[0,1]}{{\mathcal{M}}}^{\varepsilon}$ is tight.

######  Lemma 2.5.

If (2.1) and (2.2)-(2.4) are fulfilled, then
$\bigcup\limits_{\varepsilon\in[0,1]}{{\mathcal{M}}}^{\varepsilon}$ is tight.

###### Proof.

By Lemma 2.3 we see that for every $\delta\in(0,1)$, $u_{0}\in\ell^{2}$ and
$k\in\mathbb{N}$, there exist $T_{k}=T_{k}(\delta,k,u_{0})>0$ and
$n_{k}=n_{k}(\delta,k)\geq 1$ such that for all $t\geq T_{k}$ and
${\varepsilon}\in[0,1]$,

$\sum_{|i|\geq
n_{k}}\mathbb{E}\left(|u^{\varepsilon}_{i}(t,0,u_{0})|^{2}\right)\leq\frac{\delta}{2^{4k}}.$
(2.24)

By Lemma 2.1 we know that there exists $T_{0}=T_{0}(u_{0})>0$ such that for
all $t\geq T_{0}$ and ${\varepsilon}\in[0,1]$,

$\mathbb{E}\left(\|u^{{\varepsilon}}(t,0,u_{0})\|^{2}\right)\leq c_{1},$
(2.25)

where $c_{1}=1+\lambda^{-2}\|g\|^{2}+\lambda^{-1}L_{\sigma}^{2}$. For every
$\delta>0$ and $k\in\mathbb{N}$, denote by

${{\mathcal{Y}}}_{k}^{\delta}=\left\\{u=(u_{i})_{i\in\mathbb{Z}^{N}}\in\ell^{2}:\
u_{i}=0\ \text{for}\ |i|\geq n_{k}\ \text{and}\
\|u\|\leq\frac{2^{k}\sqrt{c_{1}}}{\sqrt{\delta}}\right\\},$ (2.26)
${{\mathcal{Z}}}_{k}^{\delta}=\left\\{u\in\ell^{2}:\
\|u-v\|\leq\frac{1}{2^{k}}\ \text{for\ some}\
v\in{{\mathcal{Y}}}_{k}^{\delta}\right\\},$ (2.27)

and

${{\mathcal{K}}}^{\delta}=\bigcap_{k=1}^{\infty}{{\mathcal{Z}}}_{k}^{\delta}.$
(2.28)

By (2.26)-(2.28) we find that ${{\mathcal{K}}}^{\delta}$ is compact in
$\ell^{2}$. Furthermore, by (2.24)-(2.28) and the argument of [18, Theorem
4.4], one can verify that that for all ${\varepsilon}\in[0,1]$ and
$\mu^{\varepsilon}\in{{\mathcal{M}}}^{\varepsilon}$,

$\mu^{\varepsilon}({{\mathcal{K}}}^{\delta})>1-\delta,$

which completes the proof. ∎

Next, we investigate the limiting behavior of a family
$\\{\mu^{\varepsilon}\\}_{{\varepsilon}\in(0,1)}$ of invariant measures of
(2.12) as ${\varepsilon}\to 0$, for which the following convergence of
solutions is needed.

######  Lemma 2.6.

If (2.1) and (2.2)-(2.4) are fulfilled, then for all $t\geq 0$ and $\delta>0$,

$\lim_{{\varepsilon}\rightarrow
0}\sup_{u_{0}\in\ell^{2}}P\left(\\{\omega\in\Omega:\
\|u^{{\varepsilon}}(t,0,u_{0})-u(t,0,u_{0})\|\geq\delta\\}\right)=0,$

where $u(\cdot,0,u_{0})$ is the solution of (2.13).

###### Proof.

By (2.1), (2.12)-(2.13) and the Itô formula we get

${\frac{d}{dt}}\mathbb{E}\left(\|u^{{\varepsilon}}(t,u_{0})-u(t,u_{0})\|^{2}\right)\leq\gamma\mathbb{E}\left(\|u^{{\varepsilon}}(t,u_{0})-u(t,u_{0})\|^{2}\right)+{\varepsilon}\mathbb{E}\left(\|\sigma(u^{{\varepsilon}}(t,u_{0}))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\right),$

which along with (2.2) and Gronwall’s inequality shows that for all $t\geq 0$,

$\sup_{u_{0}\in\ell^{2}}\mathbb{E}\left(\|u^{{\varepsilon}}(t,u_{0})-u(t,u_{0})\|^{2}\right)\leq{\varepsilon}\gamma^{-1}L_{\sigma}^{2}e^{\gamma
t}.$ (2.29)

Then for every $\delta>0$, we obtain from (2.29) that

$\sup_{u_{0}\in\ell^{2}}P\left(\\{\omega\in\Omega:\
\|u^{{\varepsilon}}(t,u_{0})-u(t,u_{0})\|\geq\delta\\}\right)\leq{\varepsilon}\delta^{-2}\gamma^{-1}L_{\sigma}^{2}e^{\gamma
t}\to 0,$

as ${\varepsilon}\to 0$, as desired. ∎

Next, we show that any limit of a sequence of invariant measures of the
stochastic system (2.12) must be an invariant measure of the limiting system
(2.13) as ${\varepsilon}\to 0$.

######  Theorem 2.7.

Suppose (2.1) and (2.2)-(2.4) are valid. If ${\varepsilon}_{n}\rightarrow 0$
and $\mu^{{\varepsilon}_{n}}\in{{\mathcal{M}}}^{{\varepsilon}_{n}}$, then
there exists a subsequence ${\varepsilon}_{n_{k}}$ and an invariant measure
$\mu^{0}\in{{\mathcal{M}}}^{0}$ such that
$\mu^{{\varepsilon}_{n_{k}}}\rightarrow\mu^{0}$ weakly.

###### Proof.

It follows from Lemma 2.5 that there exists a subsequence
${\varepsilon}_{n_{k}}$ and a probability measure $\mu$ such that
$\mu^{{\varepsilon}_{n_{k}}}\to\mu$ weakly, which along with Lemma 2.6 implies
that $\mu$ must be an invariant measure of the limiting system (2.13) (see,
e.g., [35, Theorem 6.1] and [16, Theorem 2.1]). ∎

Starting from the next section, we will study the LDP of solutions and
invariant measures of (2.12) as ${\varepsilon}\to 0$.

##  3 Uniform large deviations of solutions

In this section, we prove the uniform Freidlin-Wentzell LDP of solutions as
well as the uniform Dembo-Zeitouni LDP of solutions over all initial data,
which are needed for establishing the LDP of invariant measures. We first
consider the controlled equation with control $h\in L^{2}(0,T;\ell^{2})$
corresponding to (2.12), which is given by

${\frac{du_{h}}{dt}}(t)+\lambda
u_{h}(t)+Au_{h}(t)+f(u_{h}(t))=g+\sigma(u_{h}(t))h(t),\quad
u_{h}(0)=u_{0}\in\ell^{2}.$ (3.1)

Notice that the limiting system (2.13) is a special case of (3.1) with
$h\equiv 0$.

Given $u_{0}\in\ell^{2}$ and $h\in L^{2}(0,T;\ell^{2})$, by a solution $u_{h}$
of (3.1), we mean that $u\in C([0,T],\ell^{2})$ and for all $t\in[0,T]$,

$u_{h}(t)+\lambda\int_{0}^{t}u_{h}(s)ds+\int_{0}^{t}Au_{h}(s)ds+\int_{0}^{t}f(u_{h}(s))ds=u_{0}+gt+\int_{0}^{t}\sigma(u_{h}(s)h(s)ds,$

in $\ell^{2}$. If (2.1) and (2.2)-(2.3) are fulfilled, then one can verify
that for every $u_{0}\in\ell^{2}$ and $h\in L^{2}(0,T;\ell^{2})$, system (3.1)
has a unique solution $u_{h}\in C([0,T],\ell^{2})$.

The next lemma is concerned with the uniform estimates of solutions of (3.1).

######  Lemma 3.1.

If (2.1) and (2.2)-(2.3) are fulfilled, then for every $u_{0}\in\ell^{2}$ and
$h\in L^{2}(0,T;\ell^{2})$, the solution of (3.1) satisfies, for all
$t\in[0,T]$,

$\|u_{h}(t)\|^{2}\leq e^{-\lambda
t}\|u_{0}\|^{2}+2\lambda^{-2}\|g\|^{2}+2\lambda^{-1}L^{2}_{\sigma}\int_{0}^{t}\|h(s)\|^{2}ds.$

###### Proof.

By (2.1) and (2.2), we get from (3.1) that

${\frac{d}{dt}}\|u_{h}(t)\|^{2}+2\lambda\|u_{h}(t)\|^{2}\leq
2\|u_{h}(t)\|\|g\|+2\|u_{h}(t)\|\|\sigma(u_{h}(t))\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(t)\|$
$\leq\lambda\|u_{h}(t)\|^{2}+2\lambda^{-1}\|g\|^{2}+2\lambda^{-1}L^{2}_{\sigma}\|h(t)\|^{2}$

which implies that for all $t\in[0,T]$,

$\|u_{h}(t)\|^{2}\leq e^{-\lambda
t}\|u_{0}\|^{2}+2\lambda^{-2}\|g\|^{2}+2\lambda^{-1}L^{2}_{\sigma}\int_{0}^{t}e^{\lambda(s-t)}\|h(s)\|^{2}ds.$

This completes the proof. ∎

We now derive the uniform estimates on the tails of solutions of (3.1).

######  Lemma 3.2.

If (2.1) and (2.2)-(2.4) are fulfilled, then for every $R>0$, $\delta>0$ and
compact subset ${{\mathcal{K}}}$ of $\ell^{2}$, there exists
$k_{0}=k_{0}(R,\delta,{{\mathcal{K}}})\in\mathbb{N}$ such that for all $k\geq
k_{0}$, $u_{0}\in{{\mathcal{K}}}$ and $\|h\|_{L^{2}(0,T;\ell^{2})}\leq R$ with
$T>0$, the solution $u_{h}$ of (3.1) satisfies

$\sup_{0\leq t\leq T}\sum_{|i|\geq
k}|\left(u_{h}(t,0,u_{0})\right)_{i}|^{2}<\delta.$

###### Proof.

Let $\rho$ be the smooth truncation function as given by (2.14). By (2.1) and
(3.1) we get

${\frac{d}{dt}}\|\rho_{k}u_{h}(t)\|^{2})+2\sum_{j=1}^{N}(B_{j}u_{h}(t),\
B_{j}(\rho_{k}^{2}u_{h}(t)))+2\lambda\|\rho_{k}u_{h}(t)\|^{2}$ $\leq
2(\rho_{k}g,\rho_{k}u_{h}(t))+2\left(\rho_{k}u_{h}(t),\
\rho_{k}\sigma(u_{h}(t))h(t)\right).$ (3.2)

As in (2.17), by Lemma 3.1 we have for all $u_{0}\in{{\mathcal{K}}}$ and
$\|h\|_{L^{2}(0,T;\ell^{2})}\leq R$,

$-2\sum_{j=1}^{N}(B_{j}u_{h}(t),\
B_{j}(\rho_{k}^{2}u_{h}(t)))\leq{\frac{c_{1}}{k}}\|u_{h}(t)\|^{2}\leq{\frac{c_{2}}{k}},$
(3.3)

where $c_{2}=c_{2}(R,{{\mathcal{K}}})>0$ is a constant independent of $k$. For
the last term in (3), we have

$2\left(\rho_{k}u_{h}(t),\
\rho_{k}\sigma(u_{h}(t))h(t)\right)\leq{\frac{1}{2}}\lambda\|\rho_{k}u_{h}(t)\|^{2}+2\lambda^{-1}\|\rho_{k}\sigma(u_{h}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(t)\|^{2}$
$\leq{\frac{1}{2}}\lambda\|\rho_{k}u_{h}(t)\|^{2}+2\lambda^{-1}\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(t)\|^{2}.$
(3.4)

By Young’s inequality we have

$2(\rho_{k}g,\rho_{k}u_{h}(t))\leq{\frac{1}{2}}\lambda\
\|\rho_{k}u_{h}(t)\|^{2}+2\lambda^{-1}\sum_{|i|\geq k}g_{i}^{2}.$ (3.5)

It follows from (3)-(3.5) that

${\frac{d}{dt}}\|\rho_{k}u_{h}(t)\|^{2}+\lambda\|\rho_{k}u_{h}(t)\|^{2}$
$\leq{\frac{c_{2}}{k}}+2\lambda^{-1}\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(t)\|^{2}+2\lambda^{-1}\sum_{|i|\geq
k}g_{i}^{2}.$ (3.6)

By (3.6) and Gronwall’s inequality, we obtain that for all $t\in[0,T]$,

$\|\rho_{k}u_{h}(t)\|^{2}\leq e^{-\lambda
t}\|\rho_{k}u_{0}\|^{2}+c_{2}\lambda^{-1}k^{-1}+2\lambda^{-2}\sum_{|i|\geq
k}g_{i}^{2}$ $+2\lambda^{-1}\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\int_{0}^{t}e^{\lambda(s-t)}\|h(s)\|^{2}ds$
$\leq\sum_{|i|\geq
k}|u_{0,i}|^{2}+c_{2}\lambda^{-1}k^{-1}+2\lambda^{-2}\sum_{|i|\geq
k}g_{i}^{2}+2R^{2}\lambda^{-1}\sup_{u\in\ell^{2}}\|1_{\\{|\cdot|\geq
k\\}}\sigma(u)\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}.$ (3.7)

Note that $u_{0}\in{{\mathcal{K}}}$ and ${{\mathcal{K}}}$ is compact, which
along with (2.4) and (3.7) implies that for every $R>0$ and $\delta>0$, there
exists $k_{0}=k_{0}(R,\delta,{{\mathcal{K}}})\in\mathbb{N}$ such that for all
$k\geq k_{0}$ and $t\in[0,T]$,

$\sum_{|i|\geq 2k}|(u_{h}(t))_{i}|^{2}\leq\|\rho_{k}u_{h}(t)\|^{2}<\delta.$

This completes the proof. ∎

The following uniform tail-ends estimates are also useful when studying the
long term dynamics of the deterministic system (2.13).

######  Lemma 3.3.

Suppose (2.1) and (2.2)-(2.4) are valid. Then for every $\delta>0$ and $R>0$,
there exist $T=T(\delta,R)>0$ and $k_{0}=k_{0}(\delta)\geq 1$ such that for
all $t\geq T$ and $k\geq k_{0}$, the solution $u(\cdot,0,u_{0})$ of (2.13)
with $\|u_{0}\|\leq R$ satisfies

$\sum_{|i|\geq k}|u_{i}(t,0,u_{0})|^{2}<\delta.$

###### Proof.

Note that the solution $u(\cdot,0,u_{0})$ of (2.13) is also the solution of
(3.1) with $h\equiv 0$. By (3.3) and Lemma 3.1 we find that for all
$\|u_{0}\|\leq R$ and $t\geq 0$,

$-2\sum_{j=1}^{N}(B_{j}u(t),\
B_{j}(\rho_{k}^{2}u(t)))\leq{\frac{c_{1}}{k}}\left(e^{-\lambda
t}R+\lambda^{-2}\|g\|^{2}\right),$

and hence there exists $T_{1}=T_{1}(R)>0$ such that for all $t\geq T_{1}$,

$-2\sum_{j=1}^{N}(B_{j}u(t),\
B_{j}(\rho_{k}^{2}u(t)))\leq{\frac{c_{1}}{k}}\left(1+\lambda^{-2}\|g\|^{2}\right).$
(3.8)

Similar to (3.6), by (3.8) we find that for every $\delta>0$ and $R>0$, there
exist $T_{2}=T_{2}(\delta,R)\geq T_{1}$ and $k_{0}=k_{0}(\delta)\geq 1$ such
that for all $t\geq T_{2}$ and $k\geq k_{0}$,

${\frac{d}{dt}}\|\rho_{k}u(t)\|^{2}+\lambda\|\rho_{k}u(t)\|^{2}<\delta,$

which implies the desired estimates. ∎

We now prove the continuity of solutions of (2.12) in initial data.

######  Lemma 3.4.

If (2.1) and (2.2)-(2.3) are fulfilled, then the solutions
$u_{h_{i}}(\cdot,0,u_{0,i})$ of (3.1) with $i=1,2$, satisfy for all
$t\in[0,T]$,

$\|u_{h_{1}}(t,0,u_{0,1})-u_{h_{2}}(t,0,u_{0,2})\|^{2}\leq
L_{1}\left(\|u_{0,1}-u_{0,2}\|^{2}+\|h_{1}-h_{2}\|^{2}_{L^{2}(0,T;\ell^{2})}\right),$
(3.9)

where $L_{1}=L_{1}(T,R)>0$ when $\|h_{1}\|_{L^{2}(0,T;\ell^{2})}\leq R$.

In addition, if $\lambda>\gamma$, then we have for all $t\in[0,T]$,

$\|u_{h_{1}}(t,0,u_{0,1})-u_{h_{2}}(t,0,u_{0,2})\|^{2}\leq
e^{-\int_{0}^{t}(\lambda-\gamma-2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(r)\|^{2})dr}\|u_{0,1}-u_{0,2}\|^{2}$
$+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\int_{0}^{t}e^{\int_{t}^{s}(\lambda-\gamma-2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(r)\|^{2})dr}\|h_{1}(s)-h_{2}(s)\|^{2}ds.$
(3.10)

###### Proof.

Let $v(t)=u_{h_{1}}(t,0,u_{0,1})-u_{h_{2}}(t,0,u_{0,2})$. By (3.1) we have

${\frac{d}{dt}}\|v(t)\|^{2}+2\lambda\|v(t)\|^{2}+2\sum_{j=1}^{N}\|B_{j}v(t)\|^{2}+2\left(f(u_{h_{1}}(t,u_{0,1}))-f(u_{h_{2}}(t,u_{0,2})),v(t)\right)$
$=2\left(\sigma(u_{h_{1}}(t,u_{0,1}))h_{1}(t)-\sigma(u_{h_{2}}(t,u_{0,2}))h_{2}(t),\
v(t)\right).$ (3.11)

By (2.1) we get

$2\left(f(u_{h_{1}}(t,u_{0,1}))-f(u_{h_{2}}(t,u_{0,2})),v(t)\right)\geq-2\gamma\|v(t)\|^{2}.$
(3.12)

For the last term on the right-hand side of (3.11), by (2.2)-(2.3) we obtain

$2\left(\sigma(u_{h_{1}}(t,u_{0,1}))h_{1}(t)-\sigma(u_{h_{2}}(t,u_{0,2}))h_{2}(t),\
v(t)\right)$
$=2\left((\sigma(u_{h_{1}}(t,u_{0,1}))-\sigma(u_{h_{2}}(t,u_{0,2})))h_{1}(t),\
v(t)\right)$ $+2\left(\sigma(u_{h_{2}}(t,u_{0,2}))(h_{1}(t)-h_{2}(t)),\
v(t)\right)$ $\leq
2L_{\sigma}\|h_{1}(t)\|\|v(t)\|^{2}+2L_{\sigma}\|h_{1}(t)-h_{2}(t)\|\|v(t)\|.$
(3.13)

By Young’s inequality we have

$2L_{\sigma}\|h_{1}(t)-h_{2}(t)\|\|v(t)\|\leq\lambda\|v(t)\|^{2}+\lambda^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2},$

which together with (3.13) shows that

$2\left(\sigma(u_{h_{1}}(t,u_{0,1}))h_{1}(t)-\sigma(u_{h_{2}}(t,u_{0,2}))h_{2}(t),\
v(t)\right)$
$\leq(\lambda+2L_{\sigma}\|h_{1}(t)\|)\|v(t)\|^{2}+\lambda^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2}.$
(3.14)

It follows from (3.11)-(3.12) and (3.14) that

${\frac{d}{dt}}\|v(t)\|^{2}-2\left(\gamma+L_{\sigma}\|h_{1}(t)\|\right)\|v(t)\|^{2}\leq\lambda^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2},$

and hence for all $t\in[0,T]$,

$\|v(t)\|^{2}\leq
e^{2\int_{0}^{t}(\gamma+L_{\sigma}\|h_{1}(r)\|)dr}\|v(0)\|^{2}$
$+\lambda^{-1}L^{2}_{\sigma}\int_{0}^{t}e^{2\int^{t}_{s}(\gamma+L_{\sigma}\|h_{1}(r)\|)dr}\|h_{1}(s)-h_{2}(s)\|^{2}ds,$

which implies (3.9).

If $\lambda>\gamma$, then we have

$2L_{\sigma}\|h_{1}(t)-h_{2}(t)\|\|v(t)\|\leq{\frac{1}{2}}(\lambda-\gamma)\|v(t)\|^{2}+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2},$

which together with (3.13) shows that

$2\left(\sigma(u_{h_{1}}(t,u_{0,1}))h_{1}(t)-\sigma(u_{h_{2}}(t,u_{0,2}))h_{2}(t),\
v(t)\right)$
$\leq({\frac{1}{2}}(\lambda-\gamma)+2L_{\sigma}\|h_{1}(t)\|)\|v(t)\|^{2}+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2}$
$\leq((\lambda-\gamma)+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)\|^{2})\|v(t)\|^{2}+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2}.$
(3.15)

It follows from (3.11)-(3.12) and (3.15) that

${\frac{d}{dt}}\|v(t)\|^{2}+\left(\lambda-\gamma-2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)\|^{2}\right)\|v(t)\|^{2}\leq
2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(t)-h_{2}(t)\|^{2},$

and hence for all $t\in[0,T]$,

$\|v(t)\|^{2}\leq
e^{-\int_{0}^{t}(\lambda-\gamma-2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(r)\|^{2})dr}\|v(0)\|^{2}$
$+2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\int_{0}^{t}e^{\int_{t}^{s}(\lambda-\gamma-2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\|h_{1}(r)\|^{2})dr}\|h_{1}(s)-h_{2}(s)\|^{2}ds.$

which implies (3.10) and thus completes the proof. ∎

In terms of (3.1), for every $T>0$ and $u_{0}\in\ell^{2}$, we define a rate
function $I_{T,u_{0}}:C([0,T],\ell^{2})\to[0,+\infty]$ by:

$I_{T,u_{0}}(u)=\inf\left\\{{\frac{1}{2}}\int_{0}^{T}\|h(t)\|^{2}dt:\ h\in
L^{2}(0,T;\ell^{2}),\ u(0)=u_{0},\ \ u=u_{h}\right\\},$ (3.16)

where $u_{h}$ is the solution of (3.1). Note that the infimum of the empty set
is $+\infty$.

By the argument of [44], one can verify that if $h_{n}\to h$ weakly in
$L^{2}(0,T;\ell^{2})$, then $u_{h_{n}}(\cdot,0,u_{0})\to u_{h}(\cdot,0,u_{0})$
strongly in $C([0,T],\ell^{2})$ for every $u_{0}\in\ell^{2}$, which implies
that for every $T>0$, $N>0$ and $u_{0}\in\ell^{2}$, the set

$\left\\{u_{h}(\cdot,0,u_{0}):\|h\|_{L^{2}(0,T;\ell^{2})}\leq N\right\\}\
\text{ is a compact set in }\ C([0,T],\ell^{2}).$ (3.17)

For each $s\geq 0$, the $s$-level set of $I_{T,u_{0}}$ is given by

$I_{T,u_{0}}^{s}=\left\\{u\in C([0,T],\ell^{2}):I_{T,u_{0}}(u)\leq
s\right\\}.$ (3.18)

It follows from (3.17) that $I_{T,u_{0}}^{s}$ is compact for all $s\geq 0$,
and hence $I_{T,u_{0}}$ is a good rate function.

Next, we will prove the solutions of the stochastic system (2.12) satisfies
the Freidlin-Wentzell uniform LDP with respect to all initial data in
$\ell^{2}$, for which we need to consider the following stochastic system with
$h\in L^{2}(0,T;\ell^{2})$:

$du_{h}^{\varepsilon}+\left(\lambda
u_{h}^{\varepsilon}+Au^{\varepsilon}_{h}+f(u^{\varepsilon}_{h})\right)dt=\left(g+\sigma(u^{\varepsilon}_{h})h\right)dt+\sqrt{{\varepsilon}}\sigma(u^{\varepsilon}_{h})dW,$
(3.19)

with initial condition $u^{\varepsilon}_{h}(0)=u_{0}\in\ell^{2}.$

Note that system (3.19) is obtained from (2.12) by replacing $W$ by
$W+{\varepsilon}^{-\frac{1}{2}}\int_{0}^{\cdot}h(t)dt$, which is also an
$\ell^{2}$-cylindrical Wiener process by Girsanov’s theorem. Consequently, the
existence and uniqueness of solutions of (3.19) follows from that of system
(2.12).

If ${\varepsilon}=0$, then system (3.19) reduces to the control system (3.1).
The relation between the solutions of (3.1) and (3.19) is given below.

######  Lemma 3.5.

If (2.1) and (2.2)-(2.3) hold, then for every $N>0$ and $T>0$, the solutions
of (3.1) and (3.19) satisfy

$\lim_{{\varepsilon}\to
0}\sup_{u_{0}\in\ell^{2}}\sup_{h\in{{\mathcal{A}}}_{N}}\mathbb{E}\left(\|u_{h}^{\varepsilon}(\cdot,0,u_{0})-u_{h}(\cdot,0,u_{0})\|^{2}_{C([0,T],\ell^{2})}\right)=0,$

where ${{\mathcal{A}}}_{N}$ is the set of all $\ell^{2}$-valued processes $h$
which are progressively measurable and $\|h\|_{L^{2}(0,T;\ell^{2})}\leq N$,
$P$-almost surely.

###### Proof.

By (3.1) and (3.19) we have

$d(u_{h}^{\varepsilon}-u_{h})++\lambda(u^{\varepsilon}_{h}-u_{h})dt+A(u^{\varepsilon}_{h}-u_{h})dt+(f(u^{\varepsilon}_{h})-f(u_{h}))dt$
$=\left(\sigma(u^{\varepsilon}_{h})h-\sigma(u_{h})h\right)dt+\sqrt{{\varepsilon}}\sigma(u^{\varepsilon}_{h})dW,$
(3.20)

with $u_{h}^{\varepsilon}(0)-u_{h}(0)=0$. By (2.1), (3.20) and Itô’s formula
we obtain

$\|u_{h}^{\varepsilon}(t)-u_{h}(t)\|^{2}+2(\lambda-\gamma)\int_{0}^{t}\|u^{\varepsilon}_{h}(s)-u_{h}(s)\|^{2}ds$
$\leq
2\int_{0}^{t}\left(\sigma(u^{\varepsilon}_{h}(s))h(s)-\sigma(u_{h}(s))h(s),\
u^{\varepsilon}_{h}(s)-u_{h}(s)\right)dt$
$+{\varepsilon}\int_{0}^{t}\|\sigma(u^{\varepsilon}_{h}(s))\|_{{{\mathcal{L}}}_{2}(H,\ell^{2})}^{2}ds+2\sqrt{{\varepsilon}}\int_{0}^{t}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right),$

and hence for all $t\in[0,T]$,

$\sup_{0\leq r\leq t}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}\leq
2\gamma\int_{0}^{t}\|u^{\varepsilon}_{h}(s)-u_{h}(s)\|^{2}ds$
$+2\int_{0}^{t}\|\sigma(u^{\varepsilon}_{h}(s))-\sigma(u_{h}(s))\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(s)\|\|u^{\varepsilon}_{h}(s)-u_{h}(s)\|dt$
$+{\varepsilon}\int_{0}^{t}\|\sigma(u^{\varepsilon}_{h}(s))\|_{{{\mathcal{L}}}_{2}(H,\ell^{2})}^{2}ds+2\sqrt{{\varepsilon}}\sup_{0\leq
r\leq t}\left|\int_{0}^{r}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right)\right|.$ (3.21)

By (2.3) we have

$2\int_{0}^{t}\|\sigma(u^{\varepsilon}_{h}(s))-\sigma(u_{h}(s))\|_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|h(s)\|\|u^{\varepsilon}_{h}(s)-u_{h}(s)\|dt$
$\leq 2L_{\sigma}\int_{0}^{t}\|h(s)\|\sup_{0\leq r\leq
s}\|u^{\varepsilon}_{h}(r)-u_{h}(r)\|^{2}ds.$ (3.22)

By (2.2) we get for all $t\in[0,T]$, ${\varepsilon}\in[0,1]$ and
$h\in{{\mathcal{A}}}_{N}$,

${\varepsilon}\int_{0}^{t}\|\sigma(u^{\varepsilon}_{h}(s))\|_{{{\mathcal{L}}}_{2}(H,\ell^{2})}^{2}ds\leq{\varepsilon}L_{\sigma}^{2}T.$
(3.23)

By (3.21)-(3.23) we get for all $t\in[0,T]$, ${\varepsilon}\in[0,1]$ and
$h\in{{\mathcal{A}}}_{N}$,

$\sup_{0\leq r\leq
t}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}\leq\int_{0}^{t}\left(2\gamma+2L_{\sigma}\|h(s)\|\right)\sup_{0\leq
r\leq s}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}ds$
$+{\varepsilon}L_{\sigma}^{2}T+2\sqrt{{\varepsilon}}\sup_{0\leq r\leq
T}\left|\int_{0}^{r}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right)\right|.$ (3.24)

By (3.24) and Gronwall’s lemma, we get for all $t\in[0,T]$,
${\varepsilon}\in[0,1]$ and $h\in{{\mathcal{A}}}_{N}$, $P$-almost surely,

$\sup_{0\leq r\leq
t}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}\leq{\varepsilon}L_{\sigma}^{2}Tc_{1}+2\sqrt{{\varepsilon}}c_{1}\sup_{0\leq
r\leq T}\left|\int_{0}^{r}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right)\right|$ (3.25)

where $c_{1}=e^{2\gamma T+2L_{\sigma}T^{\frac{1}{2}}N}$. It follows from
(3.25) that for all ${\varepsilon}\in[0,1]$ and $h\in{{\mathcal{A}}}_{N}$,

$\mathbb{E}\left(\sup_{0\leq r\leq
T}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}\right)$
$\leq{\varepsilon}L_{\sigma}^{2}Tc_{1}+2\sqrt{{\varepsilon}}c_{1}\mathbb{E}\left(\sup_{0\leq
r\leq T}\left|\int_{0}^{r}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right)\right|\right).$ (3.26)

By (2.2) and the Burkholder inequality we have

$2\sqrt{{\varepsilon}}c_{1}\mathbb{E}\left(\sup_{0\leq r\leq
T}\left|\int_{0}^{r}\left(u^{\varepsilon}_{h}(s)-u_{h}(s),\
\sigma(u^{\varepsilon}_{h}(s))dW\right)\right|\right)$ $\leq
6\sqrt{{\varepsilon}}c_{1}\mathbb{E}\left(\left(\int_{0}^{T}\|u^{\varepsilon}_{h}(s)-u_{h}(s)\|^{2}\|\sigma(u^{\varepsilon}_{h}(s))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}ds\right)^{\frac{1}{2}}\right)$
$\leq 6\sqrt{{\varepsilon}}c_{1}\mathbb{E}\left(\sup_{0\leq r\leq
T}\|u^{\varepsilon}_{h}(r)-u_{h}(r)\|\left(\int_{0}^{T}\|\sigma(u^{\varepsilon}_{h}(s))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}ds\right)^{\frac{1}{2}}\right)$
$\leq{\frac{1}{2}}\mathbb{E}\left(\sup_{0\leq r\leq
T}\|u^{\varepsilon}_{h}(r)-u_{h}(r)\|^{2}\right)+18{\varepsilon}c_{1}^{2}\mathbb{E}\left(\int_{0}^{T}\|\sigma(u^{\varepsilon}_{h}(s))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}ds\right)$
$\leq{\frac{1}{2}}\mathbb{E}\left(\sup_{0\leq r\leq
T}\|u^{\varepsilon}_{h}(r)-u_{h}(r)\|^{2}\right)+18{\varepsilon}c_{1}^{2}TL^{2}_{\sigma},$

which along with (3.26) shows that for all ${\varepsilon}\in[0,1]$,

$\sup_{u_{0}\in\ell^{2}}\sup_{h\in{{\mathcal{A}}}_{N}}\mathbb{E}\left(\sup_{0\leq
r\leq T}\|u_{h}^{\varepsilon}(r)-u_{h}(r)\|^{2}\right)\leq
2{\varepsilon}L_{\sigma}^{2}Tc_{1}+36{\varepsilon}c_{1}^{2}TL^{2}_{\sigma}.$

This completes the proof. ∎

As an immediate consequence of Lemma 3.5, we have the following convergence of
solutions in probability.

######  Corollary 3.6.

If (2.1) and (2.2)-(2.3) hold, then for every $N>0$, $T>0$ and $\delta>0$, the
solutions of (3.1) and (3.19) satisfy

$\lim_{{\varepsilon}\to 0}\sup_{u_{0}\in\ell^{2}}\
\sup_{h\in{{\mathcal{A}}}_{N}}P\left(\|u^{\varepsilon}_{h}(\cdot,0,u_{0})-u_{h}(\cdot,0,u_{0})\|_{C([0,T],\ell^{2})}>\delta\right)=0.$

From now on, for every ${\varepsilon}\in[0,1]$, $T>0$ and $u_{0}\in\ell^{2}$,
we denote by $\nu^{\varepsilon}_{u_{0}}$ the distribution law of the solution
$u^{\varepsilon}(\cdot,0,u_{0})$ of (2.12) in $C([0,T],\ell^{2})$. Then by
(3.17) and Corollary 3.6 we infer from [40, Theorem 2.13] that the family
$\\{\nu^{\varepsilon}_{u_{0}}\\}_{{\varepsilon}\in(0,1)}$ satisfies the
Freidlin-Wentzell uniform LDP on $C([0,T],\ell^{2})$ with rate function
$I_{T,u_{0}}$ uniformly for all $u_{0}\in\ell^{2}$.

######  Theorem 3.7.

If (2.1) and (2.2)-(2.3) hold, then for every $T>0$, the family
$\\{u^{\varepsilon}(\cdot,0,u_{0})\\}_{0<{\varepsilon}<1}$ of the solutions of
(2.12) satisfies the Freidlin-Wentzell uniform LDP on $C([0,T],\ell^{2})$ with
rate function $I_{T,u_{0}}$, uniformly for all $u_{0}$ in $\ell^{2}$; more
precisely:

  1. 1.

For every $s\geq 0$, $\delta_{1}>0$ and $\delta_{2}>0$, there exists
${\varepsilon}_{0}>0$ such that

$\inf_{u_{0}\in\ell^{2}}\ \inf_{\xi\in
I_{T,u_{0}}^{s}}\left(\nu^{{\varepsilon}}_{u_{0}}({{\mathcal{N}}}(\xi,\delta_{1}))-e^{-{\frac{I_{T,u_{0}}(\xi)+\delta_{2}}{{\varepsilon}}}}\right)\geq
0,\quad\forall\ {\varepsilon}\leq{\varepsilon}_{0},$ (3.27)

where ${{\mathcal{N}}}(\xi,\delta_{1})=\left\\{u\in
C([0,T],\ell^{2}):\|u-\xi\|_{C([0,T],\ell^{2})}<\delta_{1}\right\\}$.

  2. 2.

For every $s_{0}\geq 0$, $\delta_{1}>0$ and $\delta_{2}>0$, there exists
${\varepsilon}_{0}>0$ such that

$\sup_{u_{0}\in\ell^{2}}\nu^{{\varepsilon}}_{u_{0}}\left({C([0,T],\ell^{2})}\setminus{{\mathcal{N}}}(I_{T,u_{0}}^{s},\delta_{1})\right)\leq
e^{-{\frac{s-\delta_{2}}{{\varepsilon}}}},\quad\forall\
{\varepsilon}\leq{\varepsilon}_{0},\ \forall\ s\leq s_{0},$

where
${{\mathcal{N}}}(I_{T,u_{0}}^{s},\delta_{1})=\left\\{u\in{C([0,T],\ell^{2})}:{\rm{dist}}(u,I_{T,u_{0}}^{s})<\delta_{1}\right\\}$.

The goal of the present paper is to prove the LDP of the invariant measures of
the stochastic equation (2.12), for which the Dembo-Zeitouni uniform LDP of
solutions is also needed. The following continuity of level sets of
$I_{T,u_{0}}$ will be used when proving the Dembo-Zeitouni uniform LDP of
solutions.

######  Lemma 3.8.

If (2.1) and (2.2)-(2.3) hold, and $u_{0,n}\to u_{0}$ in $\ell^{2}$, then for
every $T>0$ and $s\geq 0$,

$\lim_{n\to\infty}\max\left\\{\sup_{v\in I^{s}_{T,u_{0}}}{\rm
dist}_{C([0,T],\ell^{2})}(v,I^{s}_{T,u_{0,n}}),\ \ \sup_{v\in
I^{s}_{T,u_{0,n}}}{\rm
dist}_{C([0,T],\ell^{2})}(v,I^{s}_{T,u_{0}})\right\\}=0.$

###### Proof.

Note that if $v\in I^{s}_{T,u_{0}}$, then $v=u_{h}(\cdot,0,u_{0})$ for some
$h$ with $\|h\|_{L^{2}(0,T;\ell^{2})}^{2}\leq 2s$. Then
$u_{h}(\cdot,u_{0,n})\in I^{s}_{T,u_{0,n}}$ and by Lemma 3.4 we have

${\rm dist}_{C([0,T],\ell^{2})}(v,I^{s}_{T,u_{0,n}})\leq{\rm
dist}_{C([0,T],\ell^{2})}(u_{h}(\cdot,0,u_{0}),\ u_{h}(\cdot,0,u_{0,n}))\leq
c_{1}\|u_{0,n}-u_{0}\|,$

where $c_{1}>0$ is a constant independent of $n$, and hence

$\sup_{v\in I^{s}_{T,u_{0}}}{\rm
dist}_{C([0,T],\ell^{2})}(v,I^{s}_{T,u_{0,n}})\leq c_{1}\|u_{0,n}-u_{0}\|\to
0.$ (3.28)

By the same argument, we can also obtain

$\sup_{v\in I^{s}_{T,u_{0,n}}}{\rm
dist}_{C([0,T],\ell^{2})}(v,I^{s}_{T,u_{0}})\leq c_{1}\|u_{0,n}-u_{0}\|\to 0,$

which together with (3.28) concludes the proof. ∎

It follows from Lemma 3.8 and [40, Theorem 2.7] that the Freidlin-Wentzell and
the Dembo-Zeitouni uniform LDP of (2.12) on $C([0,T],\ell^{2})$ over a compact
subset of $\ell^{2}$ are equivalent, which along with Theorem 3.7 yields the
following result.

######  Theorem 3.9.

If (2.1) and (2.2)-(2.3) hold, then the family
$\\{u^{\varepsilon}(\cdot,0,u_{0})\\}_{0<{\varepsilon}<1}$ of the solutions of
(2.12) satisfies the Dembo-Zeitouni uniform LDP on $C([0,T],\ell^{2})$ with
rate function $I_{T,u_{0}}$, uniformly with respect to $u_{0}$ in all compact
subsets of $\ell^{2}$; more precisely:

  1. 1.

For any compact subset ${{\mathcal{K}}}$ of $\ell^{2}$ and any open subset $G$
of $C([0,T],\ell^{2})$,

$\liminf_{{\varepsilon}\to
0}\inf_{u_{0}\in{{\mathcal{K}}}}\left({\varepsilon}\ln
P(u^{\varepsilon}(\cdot,0,u_{0})\in
G)\right)\geq-\sup_{u_{0}\in{{\mathcal{K}}}}\inf_{v\in G}I_{T,u_{0}}(v).$

  2. 2.

For any compact subset ${{\mathcal{K}}}$ of $\ell^{2}$ and any closed subset
$F$ of $C([0,T],\ell^{2})$,

$\limsup_{{\varepsilon}\to
0}\sup_{u_{0}\in{{\mathcal{K}}}}\left({\varepsilon}\ln
P(u^{\varepsilon}(\cdot,0,u_{0})\in
F)\right)\leq-\inf_{u_{0}\in{{\mathcal{K}}}}\inf_{v\in F}I_{T,u_{0}}(v).$

##  4 Rate functions of invariant measures

In this section, we introduce a rate function for the invariant measures of
(2.12) and prove the exponential tightness of invariant measures, for which we
first consider the long term dynamics of the deterministic limiting system
(2.13).

Given $t\geq 0$ and $u_{0}\in\ell^{2}$, denote by $S(t)u_{0}=u(t,0,u_{0})$,
which is the solution of (2.13) with initial value $u_{0}$ at initial time
$0$. By Lemma 3.1 with $h\equiv 0$ we see that the semigroup
$\\{S(t)\\}_{t\geq 0}$ has a bounded absorbing set in $\ell^{2}$. On the other
hand, by Lemma 3.3 we know that $\\{S(t)\\}_{t\geq 0}$ is asymptotically
compact in $\ell^{2}$. Consequently, the dynamical system $\\{S(t)\\}_{t\geq
0}$ has a global attractor ${{\mathcal{A}}}$ in $\ell^{2}$ in the sense that
${{\mathcal{A}}}$ is a compact invariant set in $\ell^{2}$ and attracts every
bounded subset of $\ell^{2}$. Furthermore, by Lemma 3.1 with $h\equiv 0$, we
have

$\sup_{v\in{{\mathcal{A}}}}\|v\|^{2}\leq 2\lambda^{-2}\|g\|^{2}.$ (4.1)

If we further assume that

$\lambda>\gamma,$ (4.2)

then by (3.10) with $h_{1}\equiv h_{2}\equiv 0$, we see that any two solutions
of (2.13) converge to each other as $t\to\infty$, and hence ${{\mathcal{A}}}$
is a singleton. In this case, we write ${{\mathcal{A}}}=\\{u_{*}\\}$. By (4.1)
we know $\|u_{*}\|^{2}\leq 2\lambda^{-2}\|g\|^{2}$.

By the idea of [38], we now define a rate function $J:\ell^{2}\to[0,+\infty]$
by: for every $v\in\ell^{2}$,

$J(v)=\lim_{\delta\to 0}\ \inf\left\\{I_{r,u_{*}}(u):\ r>0,\ u\in
C([0,r],\ell^{2}),\ u(0)=u_{*},\ u(r)\in B_{\ell^{2}}(v,\delta)\right\\},$
(4.3)

where $u_{*}$ is the unique element of ${{\mathcal{A}}}$, $I_{r,u_{*}}$ is
given by (3.16), and $B_{\ell^{2}}(v,\delta)$ is the open ball in $\ell^{2}$
centered at $v$ and of radius $\delta$. Given $s\geq 0$, the $s$-level set of
$J$ is the set: $J^{s}=\left\\{v\in\ell^{2}:\ J(v)\leq s\right\\}.$

We first prove $J$ given by (4.3) is a good rate function in $\ell^{2}$ by the
uniform tail-ends estimates.

######  Lemma 4.1.

If (2.1), (2.2)-(2.4) and (4.2) are fulfilled, then for every $s\geq 0$, the
level set $J^{s}$ is compact in $\ell^{2}$.

###### Proof.

Let $s\geq 0$ be fixed. We now prove the set $J^{s}$ is precompact in
$\ell^{2}$. If $v\in J^{s}$, then for every $n\in\mathbb{N}$, by (4.3) we see
that there exist $r_{n}>0$ and $u_{n}\in C([0,r_{n}],\ell^{2})$ such that

$u_{n}(0)=u_{*},\quad\|u_{n}(r_{n})-v\|<{\frac{1}{n}},\quad
I_{r_{n},u_{*}}(u_{n})<s+{\frac{1}{n}}.$ (4.4)

By (3.16) and (4.4) we find that there exists $h_{n}\in
L^{2}(0,r_{n};\ell^{2})$ such that

$u_{n}(\cdot)=u_{h_{n}}(\cdot,0,u_{*}),\quad\text{and}\ \
{\frac{1}{2}}\int_{0}^{r_{n}}\|h_{n}(s)\|^{2}ds<s+{\frac{1}{n}},$ (4.5)

where $u_{h_{n}}(\cdot,0,u_{*})$ is the solution of (3.1) corresponding to the
initial value $u_{*}$ and control $h_{n}$. By (4.4)-(4.5) we get

$\|u_{h_{n}}(r_{n},0,u_{*})-v\|<{\frac{1}{n}}\to 0.$ (4.6)

By (4.5) and Lemma 3.1 we obtain

$\|u_{h_{n}}(r_{n},0,u_{*})\|\leq c_{1},\quad\forall\ n\in\mathbb{N},$ (4.7)

where $c_{1}=c_{1}(\lambda,L_{\sigma},g,s)>0$ is a constant independent of
$n$. By (4.6)-(4.7) we have

$\|v\|\leq c_{1},\quad\forall\ v\in J^{s}.$ (4.8)

By (4.5), we get from Lemma 3.2 that for each $\delta>0$, there exists
$k_{0}=k_{0}(\delta,s)>0$ such that for all $n\in\mathbb{N}$,

$\sum_{|i|\geq
k_{0}}|(u_{h_{n}}(r_{n},0,u_{*}))_{i}|^{2}<{\frac{1}{8}}\delta^{2}.$ (4.9)

By (4.6) and (4.9) we get

$\sum_{|i|\geq k_{0}}|v_{i}|^{2}\leq{\frac{1}{8}}\delta^{2},\quad\forall\ v\in
J^{s}.$ (4.10)

On the other hand, by (4.8) we see that the set $\\{(v_{i})_{|i|<k_{0}}:v\in
J^{s}\\}$ is bounded in a finite-dimensional space, and hence precompact,
which along with (4.10) shows that the set $J^{s}$ has a finite open cover of
radius $\delta$ in $\ell^{2}$ for every $\delta>0$; that is, $J^{s}$ is
precompact in $\ell^{2}$. In addition, one may check that the set $J^{s}$ is
also closed in $\ell^{2}$, and thus compact, which completes the proof. ∎

######  Lemma 4.2.

If (2.1), (2.2)-(2.4) and (4.2) hold, then for any $\delta_{1}>0,\delta_{2}>0$
and $s>0$, there exists $\delta>0$ such that for all $t>0$,

$\left\\{u(t):\ u\in C([0,t],\ell^{2}),\ u(0)\in{{\mathcal{A}}}_{\delta},\
I_{t,u(0)}(u)\leq s-\delta_{1}\right\\}\subseteq J^{s}_{\delta_{2}},$

where ${{\mathcal{A}}}_{\delta}$ is the $\delta$-neighborhood of
${{\mathcal{A}}}$ and $J^{s}_{\delta_{2}}$ is the $\delta_{2}$-neighborhood of
$J^{s}$.

###### Proof.

Suppose the statement is false, which implies that there exist
$\delta_{1}>0,\delta_{2}>0$ and $s>0$ such that for every $k>0$, there exist
$t_{k}>0$ and $u_{k}\in C([0,t_{k}],\ell^{2})$ such that
$u_{k}(0)\in{{\mathcal{A}}}_{\frac{1}{k}}$ and $I_{t_{k},u_{k}(0)}(u_{k})\leq
s-\delta_{1}$, but $u_{k}(t_{k})\notin J^{s}_{\delta_{2}}$. Since
$I_{t_{k},u_{k}(0)}(u_{k})\leq s-\delta_{1}$, by (3.16) we infer that there
exists $h_{k}\in L^{2}(0,t_{k};\ell^{2})$ such that

${\frac{1}{2}}\int_{0}^{t_{k}}\|h_{k}(t)\|^{2}dt<s-{\frac{1}{2}}\delta_{1},\quad\text{and
}\ u_{k}(\cdot)=u_{h_{k}}(\cdot,0,u_{k}(0)),$ (4.11)

where $u_{h_{k}}(\cdot,0,u_{k}(0))$ is the solution of (3.1) with initial
value $u_{k}(0)$ and control $h_{k}$. Since $u_{k}(t_{k})\notin
J^{s}_{\delta_{2}}$, by (4.11) we have

$u_{h_{k}}(t_{k},0,u_{k}(0))\notin J^{s}_{\delta_{2}}.$ (4.12)

Due to $u_{k}(0)\in{{\mathcal{A}}}_{\frac{1}{k}}$ and
${{\mathcal{A}}}=\\{u_{*}\\}$, we see that

$\|u_{k}(0)-u_{*}\|<{\frac{1}{k}}.$ (4.13)

If $v_{h_{k}}(\cdot,0,u_{*})$ is the solution of (3.1) with initial value
$u_{*}$ and control $h_{k}$, then (3.10), (4.11) and (4.13) imply that

$\|v_{h_{k}}(t_{k},0,u_{*})-u_{h_{k}}(t_{k},0,u_{k}(0))\|<{\frac{1}{k}}e^{(\lambda-\gamma)^{-1}L^{2}_{\sigma}\int_{0}^{t_{k}}\|h_{k}(t)\|^{2}dt}<{\frac{1}{k}}e^{(\lambda-\gamma)^{-1}L^{2}_{\sigma}(2s-\delta_{1})}.$
(4.14)

By (4.12) and (4.14) we see that there exists
$K=K(s,\delta_{1},\delta_{2})\in\mathbb{N}$ such that

$v_{h_{k}}(t_{k},0,u_{*})\notin J^{s}_{{\frac{1}{2}}\delta_{2}},\quad\forall\
k\geq K.$ (4.15)

By (4.3) and (4.11) we have for all $k\in\mathbb{N}$,

$J(v_{h_{k}}(t_{k},0,u_{*}))\leq
I_{t_{k},u_{*}}(v_{h_{k}})\leq{\frac{1}{2}}\|h_{k}\|^{2}_{L^{2}(0,t_{m};\ell^{2})}<s-{\frac{1}{2}}\delta_{1}.$
(4.16)

By (4.16) we find that $v_{h_{k}}(t_{k},0,u_{*})\in J^{s}$ for all
$k\in\mathbb{N}$, which is in contradiction with (4.15). ∎

######  Lemma 4.3.

If (2.1), (2.2)-(2.4) and (4.2) hold, then for any $R>0$ and $\delta>0$, there
exists $t_{0}=t_{0}(R,\delta)>0$ such that

$\inf\left\\{I_{t_{0},u_{0}}(u):\ \|u_{0}\|\leq R,u\in
C([0,t_{0}],\ell^{2}),u(0)=u_{0},\
u(t_{0})\notin{{\mathcal{A}}}_{\delta}\right\\}>0.$

###### Proof.

We argue by contradiction. If the statement is false, then there exist $R>0$
and $\delta>0$ such that for every $k\in\mathbb{N}$, there exist
$u_{0,k}\in\ell^{2}$ with $\|u_{0,k}\|\leq R$ and $u_{k}\in C([0,k],\ell^{2})$
such that $u_{k}(0)=u_{0,k}$, $u(k)\notin{{\mathcal{A}}}_{\delta}$ and
$I_{k,u_{0,k}}(u_{k})<{\frac{1}{k}}$. Since
$I_{k,u_{0,k}}(u_{k})<{\frac{1}{k}}$, by (3.16) we see that there exists
$h_{k}\in L^{2}(0,k;\ell^{2})$ such that

$\int_{0}^{k}\|h_{k}(t)\|^{2}dt<{\frac{4}{k}},\quad\text{and}\ \
u_{k}(\cdot)=u_{h_{k}}(\cdot,0,u_{0,k}),$ (4.17)

where $u_{h_{k}}(\cdot,0,u_{0,k})$ is the solution of (3.1) with initial
condition $u_{0,k}$ and control $h_{k}$. Since
$u(k)\notin{{\mathcal{A}}}_{\delta}$, by (4.17) we get

$u_{h_{k}}(k,0,u_{0,k})\notin{{\mathcal{A}}}_{\delta},\quad\forall\
k\in\mathbb{N}.$ (4.18)

Let $v(\cdot,0,u_{0,k)}$ be the solution of (2.13) with initial condition
$u_{0,k}$. By choosing $h_{1}\equiv 0$ and $h_{2}=h_{k}$ in (3.10) we obtain
that for all $k\in\mathbb{N}$,

$\|v(k,0,u_{0,k})-u_{h_{k}}(k,0,u_{0,k})\|^{2}\leq
2(\lambda-\gamma)^{-1}L_{\sigma}^{2}\int_{0}^{k}\|h_{k}(t)\|^{2}dt,$

which along with (4.17) shows that

$\|v(k,0,u_{0,k})-u_{h_{k}}(k,0,u_{0,k})\|^{2}\leq
8(\lambda-\gamma)^{-1}L_{\sigma}^{2}k^{-1}.$ (4.19)

By (4.18)-(4.19) we see that there exists $K=K(\delta)>0$ such that

$v(k,0,u_{0,k})\notin{{\mathcal{A}}}_{\frac{1}{2}\delta},\quad\forall\ k\geq
K.$ (4.20)

Note that ${{\mathcal{A}}}$ is the global attractor of (2.13) and
$\|u_{0,k}\|\leq R$. Therefore, there exists $K_{1}=K_{1}(R,\delta)\geq K$
such that

$v(k,0,u_{0,k})\in{{\mathcal{A}}}_{\frac{1}{2}\delta},\quad\forall\ k\geq
K_{1},$

which is in contradiction with (4.20). ∎

Note that bounded subsets of $\ell^{2}$ are not precompact in general. In
order to establish the LDP of invariant measures of (2.12) in $\ell^{2}$, we
need to consider the weighted space $\ell^{2}_{\kappa}$ with weight function
${\kappa}(s)=\left(1+s^{2}\right)^{\frac{1}{2}}$ for all $s\in\mathbb{R}$. The
following function $\kappa_{\delta}$ for $\delta>0$ is also useful in the
sequel:

${\kappa_{\delta}}(s)=\left(1+\delta^{2}s^{2}\right)^{\frac{1}{2}},\quad\forall\
s\in\mathbb{R}.$

The next lemma is concerned with the exponential probability estimates in
$\ell^{2}_{\kappa}$.

######  Lemma 4.4.

If (2.1), (2.2)-(2.5) and (4.2) are fulfilled and $g\in\ell^{2}_{\kappa}$,
then there exists $\delta\in(0,1)$ such that for all ${\varepsilon}\in(0,1]$
and $u_{0}\in\ell^{2}_{\kappa}$, the solution $u^{\varepsilon}(\cdot,0,u_{0})$
of (2.12) satisfies for all $t\geq 0$,

$\mathbb{E}\left(e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(t,0,u_{0})\|^{2}}\right)\leq
e^{-t}e^{{\frac{\delta}{{\varepsilon}}}\|\kappa_{\delta}u_{0}\|^{2}}+{\lambda}e^{(2+{\varepsilon}^{-1})\lambda^{-1}}.$

###### Proof.

By (2.12) and Itô’s formula we have

$d\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}+2\left(\lambda\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}+\sum_{j=1}^{N}(B_{j}u^{\varepsilon}(t),\
B_{j}(\kappa_{\delta}^{2}u^{\varepsilon}(t)))+2\left(\kappa_{\delta}f(u^{\varepsilon}(t)),\kappa_{\delta}u^{\varepsilon}(t)\right)\right)dt$
$=2(\kappa_{\delta}g,\kappa_{\delta}u^{\varepsilon}(t))dt+{\varepsilon}\|\kappa_{\delta}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}dt+2\sqrt{{\varepsilon}}(\kappa_{\delta}^{2}u^{\varepsilon}(t),\sigma(u^{\varepsilon}(t))dW).$
(4.21)

Let $\delta\in(0,1)$ be a number to be determined and
$\xi(t,s)=e^{t}e^{{\frac{\delta}{{\varepsilon}}}s}$. Then by (2.1) and (4.21)
we get

$d\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\leq\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left(1+{\frac{2\delta^{2}}{{\varepsilon}}}\|{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}\right)$
$+\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left(\delta\|{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}+{\frac{2\delta}{{\varepsilon}}}(\kappa_{\delta}g,\kappa_{\delta}u^{\varepsilon}(t))\right)dt$
$-{\frac{2\delta}{{\varepsilon}}}\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left(\lambda\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}+\sum_{j=1}^{N}(B_{j}u^{\varepsilon}(t),\
B_{j}(\kappa_{\delta}^{2}u^{\varepsilon}(t)))\right)dt$
$+{\frac{2\delta}{\sqrt{{\varepsilon}}}}\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})({\kappa_{\delta}}u^{\varepsilon}(t),{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))dW).$
(4.22)

Next, we estimate the terms in (4.22). Note that

$-\sum_{j=1}^{N}(B_{j}u^{\varepsilon}(t),\
B_{j}(\kappa_{\delta}^{2}u^{\varepsilon}(t)))=-\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}(B_{j}u^{\varepsilon}(t))_{i}\
(B_{j}(\kappa_{\delta}^{2}u^{\varepsilon}(t)))_{i}$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}\left((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})}\right)$
$\displaystyle\times\left((\kappa_{\delta}^{2}u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(\kappa_{\delta}^{2}u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})}\right)$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}\left((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})}\right)$
$\displaystyle\times\left((1+\delta^{2}(|i|^{2}+2i_{j}+1))(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(1+\delta^{2}|i|^{2})(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})}\right)$
$\displaystyle=$
$\displaystyle-\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}(2i_{j}+1)(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})$
$-\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}(1+\delta^{2}|i|^{2})((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})^{2}.$
(4.23)

For the first term on the right-hand side of (4.23) we have

$-\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}(2i_{j}+1)(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})$
$\leq\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}2|i_{j}|\left|(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})\right|$
$+\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}\left|(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})\right|$
$\leq\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}|i_{j}|^{2}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})^{2}$
$+\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}\left({\frac{5}{2}}|(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}|^{2}+{\frac{1}{2}}|(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})}|^{2}\right)$
$\leq\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}|i|^{2}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})^{2}+3N\delta^{2}\|u^{\varepsilon}(t)\|^{2}$
$\leq\delta^{2}\sum_{j=1}^{N}\sum_{i\in\mathbb{Z}^{N}}|i|^{2}((u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j}+1,\ldots,i_{N})}-(u^{\varepsilon}(t))_{(i_{1},\ldots,i_{j},\ldots,i_{N})})^{2}+3N\delta^{2}\|\kappa_{\delta}u^{\varepsilon}(t)\|^{2},$

which along with (4.23) shows that

$-\sum_{j=1}^{N}(B_{j}u^{\varepsilon}(t),\
B_{j}(\kappa_{\delta}^{2}u^{\varepsilon}(t)))\leq
3N\delta^{2}\|\kappa_{\delta}u^{\varepsilon}(t)\|^{2}.$ (4.24)

On the other hand, by (2.5) we have for all $\delta\in(0,1)$,

$\|{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\leq\|(1+|\cdot|^{2})^{\frac{1}{2}}\sigma(u^{\varepsilon}(t))\|^{2}_{{{\mathcal{L}}}_{2}(\ell^{2},\ell^{2})}\leq
L_{\sigma}^{2}.$ (4.25)

By Young’s inequality we have for all $\delta\in(0,1)$,

$2(\kappa_{\delta}g,\kappa_{\delta}u^{\varepsilon}(t))\leq{\frac{1}{2}}\lambda\|\kappa_{\delta}u^{\varepsilon}(t)\|^{2}+2\lambda^{-1}\|\kappa_{\delta}g\|^{2}\leq{\frac{1}{2}}\lambda\|\kappa_{\delta}u^{\varepsilon}(t)\|^{2}+2\lambda^{-1}\|g\|_{\kappa}^{2}.$
(4.26)

It follows from (4.22), (4.24)-(4.26) that

$d\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\leq\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left(1+\delta
L_{\sigma}^{2}+{\frac{2\delta}{{\varepsilon}\lambda}}\|g\|^{2}_{\kappa}\right)dt$
$-{\frac{\delta}{{\varepsilon}}}\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left(2\lambda-2\delta
L_{\sigma}^{2}-{\frac{1}{2}}\lambda-6N\delta^{2}\right)\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}dt$
$+{\frac{2\delta}{\sqrt{{\varepsilon}}}}\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})({\kappa_{\delta}}u^{\varepsilon}(t),{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))dW).$
(4.27)

Fix a positive number $\delta$ such that

$0<\delta<\min\\{1,{\frac{\lambda}{1+2\|g\|^{2}_{\kappa}}},L^{-2}_{\sigma}\\}\quad\text{and}\
\ (L^{2}_{\sigma}+3N\delta)\delta<{\frac{1}{4}}\lambda.$

Then by (4.27) we get

$d\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\leq\lambda\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})\left((2+{\varepsilon}^{-1})\lambda^{-1}-{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}\right)dt$
$+{\frac{2\delta}{\sqrt{{\varepsilon}}}}\xi(t,\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2})({\kappa_{\delta}}u^{\varepsilon}(t),{\kappa_{\delta}}\sigma(u^{\varepsilon}(t))dW).$
(4.28)

Using a stopping time if necessary, by (4.28) we get

$\mathbb{E}\left(e^{t}e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}}\right)$
$\leq\mathbb{E}\left(e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(0)\|^{2}}\right)+{\lambda}\mathbb{E}\left(\int_{0}^{t}e^{s}e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(s)\|^{2}}\left((2+{\varepsilon}^{-1})\lambda^{-1}-{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(s)\|^{2}\right)ds\right).$
(4.29)

Note that $e^{r}(a-r)\leq e^{a-1}\leq e^{a}$ for all $a,r\in\mathbb{R}$. We
obtain from (4.29) that

$\mathbb{E}\left(e^{t}e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}}\right)\leq\mathbb{E}\left(e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(0)\|^{2}}\right)+{\lambda}e^{t+(2+{\varepsilon}^{-1})\lambda^{-1}},$

which shows that for all $t\geq 0$,

$\mathbb{E}\left(e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(t)\|^{2}}\right)\leq
e^{-t}e^{{\frac{\delta}{{\varepsilon}}}\|{\kappa_{\delta}}u^{\varepsilon}(0)\|^{2}}+{\lambda}e^{(2+{\varepsilon}^{-1})\lambda^{-1}}.$

This completes the proof. ∎

If (4.2) holds, then for any ${\varepsilon}<(\lambda-\gamma)L^{-2}_{\sigma}$,
by (2.1) and (2.3) we obtain from (2.12) that for all
$u_{0,1},u_{0,2}\in\ell^{2}$ and $t\geq 0$,

$\mathbb{E}\left(\|u^{\varepsilon}(t,0,u_{0,1})-u^{\varepsilon}(t,0,u_{0,2})\|^{2}\right)\leq
e^{-(\lambda-\gamma)t}\|u_{0,1}-u_{0,2}\|^{2},$

and hence the invariant measure $\mu^{\varepsilon}$ of (2.12) is unique in
this case, which is the weak limit of the sequence
${\frac{1}{k}}\int_{0}^{k}{{\mathcal{L}}}(u^{\varepsilon}(t,0,0))dt$ as
$k\to\infty$, where $u^{\varepsilon}(\cdot,0,0)$ is the solution of (2.12)
with initial value $0$ at initial time $0$, and
${{\mathcal{L}}}(u^{\varepsilon}(t,0,0))$ is the distribution of
$u^{\varepsilon}(t,0,0)$. Then by Lemma 4.4 with $u_{0}=0$ we infer that the
invariant measure of (2.12) is supported on $\ell^{2}_{\kappa}$.

######  Lemma 4.5.

If (2.1), (2.2)-(2.5) and (4.2) are fulfilled and $g\in\ell^{2}_{\kappa}$,
then

$\lim_{R\to\infty}\limsup_{{\varepsilon}\to
0}{\varepsilon}\ln\mu^{\varepsilon}(\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R))=-\infty,$

where ${\overline{B}}_{\ell^{2}_{\kappa}}(0,R))$ is the closed ball in
$\ell^{2}_{\kappa}$ centered at $0$ and of radius $R$.

###### Proof.

Let $\delta\in(0,1)$ be the small number as given by Lemma 4.4. Given
$u_{0}\in\ell^{2}$ and $t>0$, we have

$P\left(u^{\varepsilon}(t,0,u_{0})\notin{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)=P\left(\|u^{\varepsilon}(t,0,u_{0})\|^{2}_{\ell^{2}_{\kappa}}>R^{2}\right)$
$\leq
P\left(\|\kappa_{\delta}u^{\varepsilon}(t,0,u_{0})\|^{2}>R^{2}\delta^{2}\right)=P\left(e^{{\frac{\delta}{{\varepsilon}}}\|\kappa_{\delta}u^{\varepsilon}(t,0,u_{0})\|^{2}}>e^{{\frac{R^{2}\delta^{3}}{{\varepsilon}}}}\right)$
$\leq
e^{-{\frac{R^{2}\delta^{3}}{{\varepsilon}}}}\mathbb{E}\left(e^{{\frac{\delta}{{\varepsilon}}}\|\kappa_{\delta}u^{\varepsilon}(t,0,u_{0})\|^{2}}\right).$
(4.30)

By Lemma 4.4 and (4.30) we get

$P\left(u^{\varepsilon}(t,0,u_{0})\notin{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)\leq
e^{-t}e^{-{\frac{R^{2}\delta^{3}}{{\varepsilon}}}}e^{{\frac{\delta}{{\varepsilon}}}\|\kappa_{\delta}u_{0}\|^{2}}+{\lambda}e^{(2+{\varepsilon}^{-1})\lambda^{-1}}e^{-{\frac{R^{2}\delta^{3}}{{\varepsilon}}}}.$
(4.31)

It follows from (4.31) and the invariance of $\mu^{\varepsilon}$ that for all
$t\geq 0$,

$\mu^{\varepsilon}((\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)))=\int_{\ell^{2}}P\left(u^{\varepsilon}(t,0,u_{0})\notin{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)d\mu^{\varepsilon}(u_{0})$
$=\int_{\ell^{2}_{\kappa}}P\left(u^{\varepsilon}(t,0,u_{0})\notin{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)d\mu^{\varepsilon}(u_{0}),$
(4.32)

The last equality follows from the fact that $\mu^{\varepsilon}$ is supported
on $\ell^{2}_{\kappa}$. By (4.31)-(4.32) and the Fatou theorem we get

$\mu^{\varepsilon}((\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)))\leq\int_{\ell^{2}_{\kappa}}\limsup_{t\to\infty}P\left(u^{\varepsilon}(t,0,u_{0})\notin{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)d\mu^{\varepsilon}(u_{0})$
$\leq{\lambda}e^{(2+{\varepsilon}^{-1})\lambda^{-1}}e^{-{\frac{R^{2}\delta^{3}}{{\varepsilon}}}}.$
(4.33)

Then the desired limit follows from (4.33) immediately. ∎

In the next two sections, we prove the LDP of the family
$\\{\mu^{\varepsilon}\\}$ of invariant measures of (2.12) as ${\varepsilon}\to
0$, based on the results obtained in the previous sections.

##  5 LDP lower bound of invariant measures

This section is devoted to the LDP lower bound of the family
$\\{\mu^{\varepsilon}\\}$ of invariant measures of (2.12) as ${\varepsilon}\to
0$. The argument is standard (see, e.g., [38, 41]). We here sketch the idea of
the proof only for the sake of reader’s convenience.

We will show that for any $z\in\ell^{2}$, $s_{1}>0$ and $s_{2}>0$, there
exists ${\varepsilon}_{0}$ such that

$\mu^{\varepsilon}(B_{\ell^{2}}(z,s_{1}))\geq
e^{-{\frac{J(z)+s_{2}}{{\varepsilon}}}},\quad\forall\
{\varepsilon}\leq{\varepsilon}_{0},$ (5.1)

where $B_{\ell^{2}}(z,s_{1})$ is the open ball in $\ell^{2}$ centered at $z$
and of radius $s_{1}$. Without loss of generality, we may assume that
$J(z)<\infty$. In this case, by (4.3) we see that there exist $r_{0}>0$ and
$\widetilde{u}\in C([0,r_{0}],\ell^{2})$ such that $\widetilde{u}(0)=u_{*}$,
$\widetilde{u}(r_{0})\in B_{\ell^{2}}(z,{\frac{1}{4}}s_{1})$, and
$I_{r_{0},u_{*}}(\widetilde{u})<J(z)+{\frac{1}{6}}s_{2}.$ By (3.16) we infer
that there exists $h\in L^{2}(0,r_{0};\ell^{2})$ such that

${\frac{1}{2}}\int_{0}^{r_{0}}\|h(t)\|^{2}dt<I_{r_{0},u_{*}}(\widetilde{u})+{\frac{1}{6}}s_{2},\quad\text{and}\quad\widetilde{u}(\cdot)=u_{h}(\cdot,0,u_{*}),$
(5.2)

where $u_{h}(\cdot,0,u_{*})$ is the solution of (3.1) with $u_{0}=u_{*}$. By
(5.2) we get

${\frac{1}{2}}\int_{0}^{r_{0}}\|h(t)\|^{2}dt<J(z)+{\frac{1}{3}}s_{2},\quad\text{and}\quad\|u_{h}(r_{0},0,u_{*})-z\|<{\frac{1}{4}}s_{1}.$
(5.3)

By (3.10) and (5.3) we find that for any $y\in\ell^{2}$,

$\|u_{h}(r_{0},0,y)-u_{h}(r_{0},0,u_{*})\|^{2}\leq
e^{2(\lambda-\gamma)^{-1}L^{2}_{\sigma}\int_{0}^{r_{0}}\|h(t)\|^{2}dt}\|y-u_{*}\|^{2}$
$\leq
e^{4(\lambda-\gamma)^{-1}L^{2}_{\sigma}(J(z)+{\frac{1}{3}s_{2}})}\|y-u_{*}\|^{2},$

and hence there exists $\eta=\eta(z,s_{1},s_{2})>0$ such that for all
$\|y-u_{*}\|<\eta$,

$\|u_{h}(r_{0},0,y)-u_{h}(r_{0},0,u_{*})\|<{\frac{1}{4}}s_{1}.$ (5.4)

By (5.3)-(5.4) we have

$\|u_{h}(r_{0},0,y)-z\|<{\frac{1}{2}}s_{1},\quad\text{for all }y\in\ell^{2}\
\text{with}\ \|y-u_{*}\|<\eta,$

which implies that if $\|y-u_{*}\|<\eta$, then

$\|u^{\varepsilon}(r_{0},0,y)-z\|<\|u^{\varepsilon}(r_{0},0,y)-u_{h}(r_{0},0,y)\|+{\frac{1}{2}}s_{1},$

and hence

$P\left(\|u^{\varepsilon}(r_{0},0,y)-u_{h}(r_{0},0,y)\|<{\frac{1}{2}}s_{1}\right)\leq
P\left(\|u^{\varepsilon}(r_{0},0,y)-z\|<s_{1}\right).$ (5.5)

By (3.27) and (5.3) we infer that

$P\left(\|u^{\varepsilon}(\cdot,0,y)-u_{h}(\cdot,0,y)\|_{C([0,r_{0}],\ell^{2})}<{\frac{1}{2}}s_{1}\right)\geq
e^{-{\frac{I_{r_{0},y}(u_{h}(\cdot,0,y))+{\frac{1}{3}}s_{2}}{{\varepsilon}}}}\geq
e^{-{\frac{J(z)+{\frac{2}{3}}s_{2}}{{\varepsilon}}}}.$ (5.6)

By (5.5)-(5.6) we see that if $\|y-u_{*}\|<\eta$, then

$P\left(\|u^{\varepsilon}(r_{0},0,y)-z\|<s_{1}\right)\geq
e^{\frac{s_{2}}{3{\varepsilon}}}e^{-{\frac{J(z)+s_{2}}{{\varepsilon}}}}.$
(5.7)

By (5.7) and the invariance of $\mu^{\varepsilon}$ we get

$\mu^{\varepsilon}(B_{\ell^{2}}(z,s_{1}))=\int_{\ell^{2}}P(u^{\varepsilon}(r_{0},0,y)\in
B_{\ell^{2}}(z,s_{1}))\mu^{\varepsilon}(dy)$
$\geq\int_{B_{\ell^{2}}(u_{*},\eta)}P(u^{\varepsilon}(r_{0},0,y)\in
B_{\ell^{2}}(z,s_{1}))\mu^{\varepsilon}(dy)\geq\mu^{\varepsilon}(B_{\ell^{2}}(u_{*},\eta))e^{\frac{s_{2}}{3{\varepsilon}}}e^{-{\frac{J(z)+s_{2}}{{\varepsilon}}}}.$
(5.8)

Since ${{\mathcal{A}}}=\\{u_{*}\\}$, by Theorem 2.7 we have
$\mu^{\varepsilon}\to\delta_{u_{*}}$ weakly, and thus,

$\liminf_{{\varepsilon}\to
0}\left(e^{\frac{s_{2}}{3{\varepsilon}}}\mu^{\varepsilon}(B_{\ell^{2}}(u_{*},\eta))\right)\geq\lim_{{\varepsilon}\to
0}e^{\frac{s_{2}}{3{\varepsilon}}}=\infty,$

which along with (5.8) yields (5.1), and thus completes the proof of the LDP
lower bound of $\\{\mu^{\varepsilon}\\}$.

##  6 LDP upper bound of invariant measures

This section is devoted to the LDP upper bound of the family
$\\{\mu^{\varepsilon}\\}$ of invariant measures of (2.12) as ${\varepsilon}\to
0$. Again, the argument follows from [38, 41], and we just sketch the proof
for reader’s convenience.

We will show that for any $s_{1}>0$, $s_{2}>0$ and $s>0$, there exists
${\varepsilon}_{0}>0$ such that

$\mu^{\varepsilon}(\ell^{2}\setminus J^{s}_{s_{1}})\leq
e^{-{\frac{s-s_{2}}{{\varepsilon}}}},\quad\forall\
{\varepsilon}\leq{\varepsilon}_{0},$ (6.1)

where $J^{s}_{s_{1}}$ is the $s_{1}$-neighborhood of $J^{s}$.

By Lemma 4.2, there exists $\eta>0$ such that for all $t>0$,

$\left\\{u(t):\ u\in C([0,t],\ell^{2}),\ u(0)\in{{\mathcal{A}}}_{\eta},\
I_{t,u(0)}(u)\leq s-{\frac{1}{4}}s_{2}\right\\}\subseteq
J^{s}_{{\frac{1}{2}}s_{1}}.$ (6.2)

By Lemma 4.5, there exist $R=R(s)>0$ and
${\varepsilon}_{1}={\varepsilon}_{1}(s)>0$ such that for all
${\varepsilon}\leq{\varepsilon}_{1}$,

$\mu^{\varepsilon}\left(\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)\right)\leq
e^{-{\frac{s}{{\varepsilon}}}}.$ (6.3)

By Lemma 4.3, we infer that there exists $t_{0}=t_{0}(R,\eta)>0$ such that

$\delta=\inf\left\\{I_{t_{0},u_{0}}(u):\ \|u_{0}\|\leq R,\ u\in
C([0,t_{0}],\ell^{2}),\ u(0)=u_{0},\
u(t_{0})\notin{{\mathcal{A}}}_{\eta}\right\\}>0.$ (6.4)

For every $n\in\mathbb{N}$, denote by

${{\mathcal{Y}}}_{n}=\left\\{u\in C([0,nt_{0}],\ell^{2}):\ \|u(0)\|\leq R,\
u(kt_{0})\notin{{\mathcal{A}}}_{\eta},\ \|u(kt_{0})\|\leq R,\ \forall\
k=1,\ldots,n\right\\}.$ (6.5)

By (6.4) and (6.5) one can verify that if $n>{\frac{s}{\delta}}$, then

$\inf\left\\{I_{nt_{0},u_{0}}(u):\ \|u_{0}\|\leq R,\ u\in{{\mathcal{Y}}}_{n},\
u(0)=u_{0}\right\\}>s.$ (6.6)

Let $n_{0}$ be a fixed positive integer such that $n_{0}>{\frac{s}{\delta}}$,
which along with (6.6) implies that

$\inf\left\\{I_{n_{0}t_{0},u_{0}}(u):\ \|u_{0}\|\leq R,\
u\in{{\mathcal{Y}}}_{n_{0}},\ u(0)=u_{0}\right\\}>s.$ (6.7)

Note that if $\|u_{0}\|_{\ell^{2}_{\kappa}}\leq R$, then $\|u_{0}\|\leq R$,
and hence by (6.7) we have

$\inf\left\\{I_{n_{0}t_{0},u_{0}}(u):\ \|u_{0}\|_{\ell^{2}_{\kappa}}\leq R,\
u\in{{\mathcal{Y}}}_{n_{0}},\ u(0)=u_{0}\right\\}>s.$ (6.8)

It is easy to verify that ${\overline{B}}_{\ell^{2}_{\kappa}}(0,R)$ is a
compact subset of $\ell^{2}$. Then applying Proposition 3.9 to the compact set
${\overline{B}}_{\ell^{2}_{\kappa}}(0,R)$ and the closed subset
${{\mathcal{Y}}}_{n_{0}}$ of $C([0,n_{0}t_{0}],\ell^{2})$, by (6.8) we infer
that there exists ${\varepsilon}_{2}\in(0,{\varepsilon}_{1})$ such that for
all ${\varepsilon}\leq{\varepsilon}_{2}$,

$\sup_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P\left(u^{\varepsilon}(\cdot,0,u_{0})\in{{\mathcal{Y}}}_{n_{0}}\right)\leq
e^{-{\frac{s}{{\varepsilon}}}}.$ (6.9)

For convenience, we set $r_{0}=(1+n_{0})t_{0}$. Since $\mu^{\varepsilon}$ is
an invariant measure, we have

$\mu^{\varepsilon}(\ell^{2}\setminus
J^{s}_{s_{1}})=\int_{\ell^{2}}P(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}})\mu^{\varepsilon}(du_{0})$
$=\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}>R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}})\mu^{\varepsilon}(du_{0})+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}})\mu^{\varepsilon}(du_{0})$
$=\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}>R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}})\mu^{\varepsilon}(du_{0})$
$+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\ \
u^{\varepsilon}(\cdot,0,u_{0})\in{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0})$
$+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\ \
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0})$
$\leq\mu^{\varepsilon}(\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R))+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(\
u^{\varepsilon}(\cdot,0,u_{0})\in{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0})$
$+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\ \
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0})$
$\leq\mu^{\varepsilon}(\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R))+\sup_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(\cdot,0,u_{0})\in{{\mathcal{Y}}}_{n_{0}})$
$+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\ \
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0}).$
(6.10)

By (6.3) (6.9) and (6.10) we get

$\mu^{\varepsilon}(\ell^{2}\setminus J^{s}_{s_{1}})\leq
2e^{-{\frac{s}{{\varepsilon}}}}+\int_{\|u_{0}\|_{\ell^{2}_{\kappa}}\leq
R}P(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}})\mu^{\varepsilon}(du_{0}).$
(6.11)

By (6.5) we have

$\int_{\|u_{0}\|\leq R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}},\
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}}\right)\
\mu^{\varepsilon}(du_{0})$ $\leq\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
u^{\varepsilon}(kt_{0},0,u_{0})\in{{\mathcal{A}}}_{\eta}\right)\
\mu^{\varepsilon}(du_{0})$ $+\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
\|u^{\varepsilon}(kt_{0},0,u_{0})\|>R\right)\ \mu^{\varepsilon}(du_{0}).$
(6.12)

By the invariance of $\mu^{\varepsilon}$, we have

$\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
\|u^{\varepsilon}(kt_{0},0,u_{0})\|>R\right)\ \mu^{\varepsilon}(du_{0})$
$\leq\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(\|u^{\varepsilon}(kt_{0},,0,u_{0})\|>R\right)\
\mu^{\varepsilon}(du_{0})\leq\sum_{k=1}^{n_{0}}\int_{\ell^{2}}P\left(\|u^{\varepsilon}(kt_{0},0,u_{0})\|>R\right)\
\mu^{\varepsilon}(du_{0})$
$=n_{0}\mu^{\varepsilon}\left(\ell^{2}\setminus{\overline{B}}_{\ell^{2}}(0,R)\right)\leq
n_{0}\mu^{\varepsilon}(\ell^{2}\setminus{\overline{B}}_{\ell^{2}_{\kappa}}(0,R)),$

which together with (6.3) shows that

$\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
\|u^{\varepsilon}(kt_{0},0,u_{0})\|>R\right)\ \mu^{\varepsilon}(du_{0})\leq
n_{0}e^{-{\frac{s}{{\varepsilon}}}}.$ (6.13)

We now write the transition probability of $u^{\varepsilon}(t,0,u_{0})$ as
$P(0,u_{0};t,\cdot)$. By the Markov property, we find that the first term on
the right-hand side of (6.12) satisfies

$\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
u^{\varepsilon}(kt_{0},0,u_{0})\in{{\mathcal{A}}}_{\eta}\right)\
\mu^{\varepsilon}(du_{0})$ $=\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}\left(\int_{y\in{{\mathcal{A}}}_{\eta}}P\left(u^{\varepsilon}(r_{0}-kt_{0},y)\notin
J^{s}_{s_{1}}\right)p(0,u_{0};kt_{0},dy)\right)\ \mu^{\varepsilon}(du_{0})$
$\leq\sum_{k=1}^{n_{0}}\left(\sup_{y\in{{\mathcal{A}}}_{\eta}}P\left(u^{\varepsilon}(r_{0}-kt_{0},y)\notin
J^{s}_{s_{1}}\right)\right).$ (6.14)

By (6.2) we see that

$\left\\{u(r_{0}-kt_{0}):\ u\in C([0,r_{0}-kt_{0}],\ell^{2}),\
u(0)\in{{\mathcal{A}}}_{\eta},\ I_{r_{0}-kt_{0},u(0)}(u)\leq
s-{\frac{1}{4}}s_{2}\right\\}\subseteq J^{s}_{{\frac{1}{2}}s_{1}}.$ (6.15)

By (6.15) we find that for any $y\in{{\mathcal{A}}}_{\eta}$,

$\left\\{\omega\in\Omega:\
u^{\varepsilon}(\cdot,0,y)\in{{\mathcal{N}}}\left(I^{s-{\frac{1}{4}}s_{2}}_{r_{0}-kt_{0},y},\
{\frac{1}{2}}s_{1}\right)\right\\}\subseteq\left\\{\omega\in\Omega:\
u^{\varepsilon}(r_{0}-kt_{0},0,y)\in J^{s}_{s_{1}}\right\\}.$ (6.16)

Actually, for every
$u^{\varepsilon}(\cdot,y)\in{{\mathcal{N}}}\left(I^{s-{\frac{1}{4}}\delta_{2}}_{r_{0}-kt_{0},y},\
{\frac{1}{2}}s_{1}\right)$, there exists $\phi\in
I^{s-{\frac{1}{4}}s_{2}}_{r_{0}-kt_{0},y}$ such that

$\sup_{t\in[0,r_{0}-kt_{0}]}\|u^{\varepsilon}(t,0,y)-\phi(t)\|<{\frac{1}{2}}s_{1},$

and hence by (6.15) we get

${\rm dist}_{\ell^{2}}\left(u^{\varepsilon}(r_{0}-kt_{0},0,y),\
J^{s}\right)\leq\|u^{\varepsilon}(r_{0}-kt_{0},0,y)-\phi(r_{0}-kt_{0})\|+{\rm
dist}_{\ell^{2}}\left(\phi(r_{0}-kt_{0}),\ J^{s}\right)<s_{1},$

and thus (6.16) is valid. By (6.16) we get

$P\left(u^{\varepsilon}(r_{0}-kt_{0},0,y)\notin J^{s}_{s_{1}}\right)\leq
P\left(u^{\varepsilon}(\cdot,0,y)\notin{{\mathcal{N}}}\left(I^{s-{\frac{1}{4}}\delta_{2}}_{r_{0}-kt_{0},y},\
{\frac{1}{2}}s_{1}\right)\right).$ (6.17)

By (6.17) and Theorem 3.7 we see that there exists
${\varepsilon}_{3}\in(0,{\varepsilon}_{2})$ such that for all
${\varepsilon}\leq{\varepsilon}_{3}$,

$\sup_{y\in{{\mathcal{A}}}_{\delta}}P\left(u^{\varepsilon}(r_{0}-kt_{0},0,y)\notin
J^{s}_{s_{1}}\right)\leq
e^{-{\frac{1}{{\varepsilon}}}(s-{\frac{1}{2}}s_{2})}.$ (6.18)

It follows from (6.14) and (6.18) that for all
${\varepsilon}\leq{\varepsilon}_{3}$,

$\sum_{k=1}^{n_{0}}\int_{\|u_{0}\|\leq
R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin J^{s}_{s_{1}},\
u^{\varepsilon}(kt_{0},0,u_{0})\in{{\mathcal{A}}}_{\eta}\right)\
\mu^{\varepsilon}(du_{0})\leq
n_{0}e^{-{\frac{1}{{\varepsilon}}}(s-{\frac{1}{2}}s_{2})}.$ (6.19)

By (6.12), (6.13) and (6.19) we obtain that for all
${\varepsilon}\leq{\varepsilon}_{3}$,

$\int_{\|u_{0}\|\leq R}P\left(u^{\varepsilon}(r_{0},0,u_{0})\notin
J^{s}_{s_{1}},\
u^{\varepsilon}(\cdot,0,u_{0})\notin{{\mathcal{Y}}}_{n_{0}}\right)\
\mu^{\varepsilon}(du_{0})\leq
n_{0}e^{-{\frac{s}{{\varepsilon}}}}+n_{0}e^{-{\frac{s-{\frac{1}{2}}s_{2}}{{\varepsilon}}}}.$
(6.20)

By (6.11) and (6.20) we get for all ${\varepsilon}\leq{\varepsilon}_{3}$,

$\mu^{\varepsilon}(\ell^{2}\setminus
J^{s}_{s_{1}})\leq(2+n_{0})e^{-{\frac{s}{{\varepsilon}}}}+n_{0}e^{-{\frac{s-{\frac{1}{2}}s_{2}}{{\varepsilon}}}}\leq\left((2+n_{0})e^{-{\frac{s_{2}}{{\varepsilon}}}}+n_{0}e^{-{\frac{s_{2}}{2{\varepsilon}}}}\right)e^{-{\frac{s-s_{2}}{{\varepsilon}}}}.$

Note that
$(2+n_{0})e^{-{\frac{s_{2}}{{\varepsilon}}}}+n_{0}e^{-{\frac{s_{2}}{2{\varepsilon}}}}\to
0$ as ${\varepsilon}\to 0$ and hence (6.1) is valid. In other words, the
family $\\{\mu^{\varepsilon}\\}_{{\varepsilon}\in(0,1)}$ satisfies the LDP
upper bound.

##  7 Statements and Declarations

###  7.1 Funding

No funding was received for conducting this study.

###  7.2 Competing Interests

The author has no relevant financial or non-financial interests to disclose.

###  7.3 Author contributions

I was the contributor in writing the manuscript.

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