Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion Equations on Unbounded Domains

Abstract

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space $\mathbb {R}^n$. Since the standard Sobolev embeddings on $\mathbb {R}^n$ are not compact and the spectrum of the Laplace operator on $\mathbb {R}^n$ are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on $\mathbb {R}^n$ is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.

1 Introduction

In this paper, we investigate the large deviation principle (LDP) of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space $\mathbb {R}^n$:

$$\begin{aligned} du (t) + ( \lambda u (t) -\Delta u (t) + F(u(t) )) dt = \sqrt{{\varepsilon }} Q {dW}, \quad t>0, \end{aligned}$$

(1.1)

with initial condition

$$\begin{aligned} u( 0, x ) = u_0 (x), \quad x\in {\mathbb {R}}^n , \end{aligned}$$

(1.2)

where $\lambda >0$ is a constant, ${\varepsilon }\in (0,1)$ is the noise intensity, $F: \mathbb {R}\rightarrow \mathbb {R}$ is a nonlinear function with polynomial growth of arbitrary order, Q is a Hilbert–Schmidt operator on $L^2(\mathbb {R}^n)$, and W is a cylindrical Wiener process in $L^2(\mathbb {R}^n)$ on a complete filtered probability space $(\Omega , \mathcal {F}, \{ { \mathcal {F}} _t\} _{t\ge 0}, P )$.

If F satisfies a dissipative condition (see (2.1)–(2.3) in the next section), then for every ${\varepsilon }>0$, the stochastic equation (1.1) possesses a unique ergodic invariant measure $\mu ^{\varepsilon }$ in $L^2(\mathbb {R}^n)$, which can be proved by the argument of [9] with minor modifications. Note that in [9], the authors deal with the fractional Laplace operator $(-\Delta )^\alpha$ with $\alpha \in (0,1)$. But all the results of [9] are also valid for the standard Laplace operator when $\alpha =1$. Actually, in the case of $\alpha =1$, the proof of those results is simpler. In particular, we find that the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ of invariant measures weakly converges to $\delta _0$ as ${\varepsilon }\rightarrow 0$, where $\delta _0$ is the probability measure in $L^2(\mathbb {R}^n)$ concentrated at zero, which is the unique invariant measure of the limiting equation corresponding to (1.1) with ${\varepsilon }=0$:

$$\begin{aligned} {\frac{du}{dt}} (t) + \lambda u (t) -\Delta u (t) + F(u(t) ) = 0, \quad t>0. \end{aligned}$$

(1.3)

In the present paper, we are interested in the asymptotic behavior of the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ as characterized by the LDP as ${\varepsilon }\rightarrow 0$; more precisely, we will prove:

Theorem 1.1

If (2.1)–(2.3) and (2.7) hold, then the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ of invariant measures of (1.1)–(1.2) enjoys the LDP in $L^2(\mathbb {R}^n)$.

The LDP of invariant measures of stochastic partial differential equations has been studied first in [20], and then in [2, 5, 7, 14, 17, 18], which include the stochastic reaction–diffusion equations [5, 20], the stochastic Navier–Stokes equations [2, 7, 18], the stochastic wave equations [17] and the singular stochastic equation [14].

Note that in all these publications, the stochastic partial differential equations are defined in a bounded spatial domain where the standard Sobolev embeddings are compact and the spectrum of the linear principal part of the equation is discrete. The compactness of Sobolev embeddings plays a crucial role for establishing the tightness of distributions of solutions, the compactness of the level sets of rate functions, the uniform LDP of solutions as well as convergence of solutions, which are all needed for proving the LDP of invariant measures of the stochastic equations. In addition, the discrete sequence of eigenvalues of the linear principal operator of the equation is often involved in the definition of the noise coefficients.

As far as the author is aware, it seems that there is no result available in the literature regarding the LDP of invariant measures of the stochastic partial differential equations defined in an unbounded domain, where neither the standard Sobolev embedding is compact, nor the spectrum of the linear principal operator of the equation is discrete. This introduces many difficulties for proving the LDP of invariant measures in the case of unbounded domains, especially for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni LDP of solution paths, and the exponential tightness of invariant measures over compact sets. We will use the arguments of [5, 18, 20] to show the LDP of invariant measures of (1.1) defined on the entire space $\mathbb {R}^n$, but we must solve the problems caused by the non-compactness of embeddings on $\mathbb {R}^n$.

For example, for partial differential equations defined in bounded domains, in order to prove the compactness of the level sets of rate functions in $L^2$ space, we only need to show the level sets are bounded in $H^1$ and then the compactness of the level sets in $L^2$ follows immediately from the compactness of the embedding $H^1 \hookrightarrow L^2$. However, in the case of unbounded domains, this argument does not work any more since the embedding is no longer compact. In order to overcome the difficulty caused by the lack of compactness of the embedding, in the present paper, we will employ the idea of uniform tail-ends estimates to establish the compactness of the level sets of rate functions. More precisely, we will show that all functions in a given level set of the rate function are uniformly small outside a sufficiently large ball in $\mathbb {R}^n$, which is established by a truncation technique for the controlled equation associated with (1.1), see Lemma 3.4. Then using the uniform smallness of the tails of the level sets and the compactness of embeddings in bounded domains, we eventually prove the compactness of the level sets in $L^2(\mathbb {R}^n)$, see Lemma 3.9.

To prove the LDP of invariant measures of (1.1), we need the uniform Freidlin–Wentzell LDP of solution paths and the uniform Dembo–Zeitouni LDP of solution paths with respect to a set of initial data. The uniform Freidlin–Wentzell LDP of solution paths of (1.1) can be established as in [24] (see also [23] for regular additive noise) for a bounded set of initial data in $L^2(\mathbb {R}^n)$ by the weak convergence method as developed in [3, 4, 12]. However the uniform Freidlin–Wentzell LDP and the uniform Dembo–Zeitouni LDP over a bounded set of initial data are not equivalent as demonstrated in [19]. As such, we cannot obtain the uniform Dembo–Zeitouni LDP of (1.1) for a bounded set of initial data in $L^2(\mathbb {R}^n)$ from that of the uniform Freidlin–Wentzell LDP. According to [19], under certain circumstances, the two types of uniform LDP are equivalent over a compact subset of initial data instead of a bounded subset of initial data. To apply the uniform Dembo–Zeitouni LDP for compact subsets of $L^2(\mathbb {R}^n)$ to prove the LDP of $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ of invariant measures, we need to establish the exponential tightness of $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ over compact subsets of $L^2(\mathbb {R}^n)$. In particular, for every $l>0$, we must find a compact subset ${{\mathcal {Z}}}$ in $L^2(\mathbb {R}^n)$ and ${\varepsilon }_0>0$ such that

$$\begin{aligned} \mu ^{\varepsilon }\left( L^2(\mathbb {R}^n) {\setminus } {{\mathcal {Z}}}\right)< e^{-{\frac{l}{{\varepsilon }}}}, \quad \forall \ {\varepsilon }<{\varepsilon }_0. \end{aligned}$$

(1.4)

In the case of bounded domains, the compact set ${{\mathcal {Z}}}$ in (1.4) can be chosen as a closed ball in $H^1$, and the compactness of the embedding $H^1\hookrightarrow L^2$ yields the compactness of ${{\mathcal {Z}}}$ in $L^2$. Again, due to the non-compactness of the embedding $H^1(\mathbb {R}^n) \hookrightarrow L^2(\mathbb {R}^n)$, we cannot choose the set ${{\mathcal {Z}}}$ in (1.4) simply as a closed ball in $H^1(\mathbb {R}^n)$. Unlike the proof of compactness of the level sets of rate functions in $L^2(\mathbb {R}^n)$, it is hard to construct a compact set Z satisfying (1.4) by the idea of uniform tail-ends estimates, and hence in the present case, a different approach has to be employed to overcome the non-compactness of standard embeddings on unbounded domains. Indeed, we will construct the compact set ${{\mathcal {Z}}}$ in $L^2(\mathbb {R}^n)$ by the idea of weighted Sobolev spaces and use the compactness of embeddings of certain weighted spaces with appropriate weight to conclude the compactness of ${{\mathcal {Z}}}$ in $L^2(\mathbb {R}^n)$, see Lemma 3.15. Of course, for this approach, we also need to establish the uniform exponential estimates of solutions in the weighted spaces, see Lemma 3.14.

We remark that the reaction–diffusion equation defined on an unbounded domain in $\mathbb {R}^n$ has many physical applications. For example, when $n=1$, the reaction–diffusion equation on $\mathbb {R}$ is a model of heat flow in an infinitely long bar, see, e.g., [16, p. 50]. On the other hand, when the reaction–diffusion equation is defined in a large bounded domain and the boundary conditions have little impact on the solutions in the region far away from the boundary, the solutions of the reaction–diffusion equation can be considered as defined on the entire space $\mathbb {R}^n$. Since physical systems are often subject to noise fluctuations, it is interesting to investigate the solutions and their long term dynamics of the stochastic reaction–diffusion (1.1) on $\mathbb {R}^n$, including the LDP of the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ of invariant measures. Let J be the rate function as defined in (3.36). Suppose G is an open subset of $L^2(\mathbb {R}^n)$ such that

$$ \inf _{v\in G} J (v) = \inf _{v\in \overline{G} } J (v), $$

where $\overline{G}$ is the closure of G. Then by [10, p. 341] we find that the LDP of $\{\mu ^{\varepsilon }\}_{{\varepsilon }>0}$ of invariant measures of (1.1) implies that

$$ \lim _{{\varepsilon }\rightarrow 0} {\varepsilon }\ln \mu ^{\varepsilon }(G) =- \inf _{v\in G} J (v). $$

In particular, if $\inf \limits _{v\in G} J (v) >0$, then $\mu ^{\varepsilon }(G)$ converges to zero exponentially fast as ${\varepsilon }\rightarrow 0$.

The paper is organized as follows. In the next section, we recall the existence and uniqueness of solutions of (1.1)–(1.2) under certain conditions on the nonlinear term F. Section 3 is devoted to the uniform estimates of solutions for the controlled equation associated with (1.1), including the uniform exponential estimates and the tail-ends estimates in both standard and weighted spaces. In the last two sections, we prove the LDP lower bound and the LDP upper bound, respectively, by combining the ideas of [5, 18, 20] with that of uniform tail-ends estimates and the embeddings of weighted spaces on unbounded domains.

2 Existence of Solutions of Stochastic Equations

In this section, we discuss the assumptions on the nonlinear term in (1.1) as well as the existence and uniqueness of solutions of the equation.

In the sequel, we will write $H= L^2(\mathbb {R}^n)$ and $V= H^1(\mathbb {R}^n)$. Then we have $V \hookrightarrow H = H^* \hookrightarrow V^*$ where $H^*$ and $V^*$ are the dual spaces of H and V, respectively. The norm of H is denoted by $\Vert \cdot \Vert$ with inner product $(\cdot , \cdot )$. Given $u\in H$ and $R>0$, let $B_H(u, R)$ be the open ball in H with radius R centered at u, and $\overline{B} _H(u,R)$ be the closed ball. Similarly, $B_V(u,R)$ and $\overline{B} _V (u, R)$ are the open and closed balls in V with radius R centered at $u\in V$, respectively. The notation ${{\mathcal {L}}}_2(H_1,H_2)$ will be used for the space of Hilbert–Schmidt operators from a separable Hilbert space $H_1$ to a separable Hilbert space $H_2$ with norm $\Vert \cdot \Vert _{{{\mathcal {L}}}_2(H_1,H_2)}$.

Throughout this paper, we assume that $(\Omega , \mathcal {F}, \{ { \mathcal {F}} _t\} _{t\in \mathbb {R}}, P )$ is a complete filtered probability space satisfying the usual conditions, and W is a cylindrical Wiener process in H with identity covariance operator. Then there exists a separable Hilbert space U such that the embedding $H \hookrightarrow U$ is Hilbert–Schmidt and W is a U-valued Wiener process.

For the nonlinear drift F in (1.1), we assume that $F: \mathbb {R}$ $\rightarrow \mathbb {R}$ is differentiable such that $F(0)=0$ and for all $u, u_1, u_2 \in \mathbb {R}$,

$$\begin{aligned} F (u) u \ge \lambda _1 |u|^p, \end{aligned}$$

(2.1)

$$\begin{aligned} |F(u_1)- F(u_2) | \le \lambda _2 \left( |u_1|^{p-2} + |u_2|^{p-2} \right) | u_1-u_2|, \end{aligned}$$

(2.2)

$$\begin{aligned} {\frac{\partial F}{\partial u}} (u) \ge 0, \end{aligned}$$

(2.3)

where $\lambda _1>0$, $\lambda _2>0$ and $p > 2$ are constants

Note that if $F(u) =|u|^{p-2} u$ for $u\in \mathbb {R}$, then F satisfies all conditions (2.1)–(2.3).

For the linear operator Q in (1.1), we generally assume that

$$\begin{aligned} Q\in {{\mathcal {L}}}_2(H, V ). \end{aligned}$$

(2.4)

As an example, we consider a linear operator Q as defined below. Let $\{a_j\}_{j=1}^\infty$ be a sequence of real numbers such that $\sum \limits _{j=1}^\infty a_j^2<\infty$, and $\{e_j\}_{j=1}^\infty$ be an orthonormal basis of H. Then define an operator $Q: H \rightarrow V$ by, for every $u=\sum \limits _{j=1}^\infty c_j e_j \in H$,

$$\begin{aligned} Qu= \sum _{j=1}^\infty c_ja_j (I-\Delta )^{-\frac{1}{2}} e_j, \end{aligned}$$

(2.5)

where I is the identity operator. One can verify that $Q: H \rightarrow V$ is a bounded linear operator such that

$$ \Vert Q u \Vert _V^2 \le \Vert u \Vert ^2 \sum \limits _{j=1}^\infty a_j^2, \quad \forall \ u \in H. $$

Furthermore, $Q: H \rightarrow V$ is a Hilbert–Schmidt operator with

$$ \Vert Q \Vert _{{{\mathcal {L}}}_2 (H,V)}^2 = \sum \limits _{j=1}^\infty a_j^2 <\infty , $$

and hence this Q satisfies (2.4).

To deal with the non-compactness of Sobolev embeddings on unbounded domains, we need to consider a weighted space. Let ${\kappa }: \mathbb {R}^n \rightarrow \mathbb {R}$ be a weight function given by

$$\begin{aligned} {\kappa } (x) = \left( 1 + |x|^2 \right) ^{\frac{1}{2}}, \quad \forall \ x\in \mathbb {R}^n. \end{aligned}$$

(2.6)

Let $L^2_{\kappa } (\mathbb {R}^n)$ be the weighted space given by

$$ L^2_{\kappa } (\mathbb {R}^n) =\left\{ u\in H: \ \Vert u \Vert _{L^2_{\kappa } (\mathbb {R}^n)} = \left( \int _{\mathbb {R}^n} {\kappa }^2 (x) |u(x)|^2 dx \right) ^{\frac{1}{2}} <\infty \right\} . $$

When proving the exponential tightness of invariant measures, we further assume that

$$\begin{aligned} Q\in {{\mathcal {L}}}_2(H, V ) \cap {{\mathcal {L}}}_2\Big (H, L^2_{\kappa } (\mathbb {R}^n) \Big ). \end{aligned}$$

(2.7)

Based on (2.5), one can construct a linear operator satisfying (2.7). Indeed, for every $u=\sum \limits _{j=1}^\infty c_j e_j \in H$, denote by

$$\begin{aligned} \widetilde{Q}u = {\frac{1}{\kappa }} \sum _{j=1}^\infty c_ja_j (I-\Delta )^{-\frac{1}{2}} e_j, \end{aligned}$$

(2.8)

where $\kappa$ is the weight function as defined by (2.6), $\{a_j\}_{j=1}^\infty$ and $\{e_j\}_{j=1}^\infty$ are the same sequences as in (2.5). Then we have

$$ \big \Vert \widetilde{Q}\big \Vert ^2 _{{{\mathcal {L}}}_2(H,V)} = \sum _{j=1}^\infty \big \Vert \widetilde{Q} e_j \big \Vert ^2_V = \sum _{j=1}^\infty \Big \Vert a_j \kappa ^{-1} (I-\Delta )^{-\frac{1}{2}} e_j \Big \Vert ^2 + \sum _{j=1}^\infty \Big \Vert \nabla \Big ( a_j \kappa ^{-1} (I-\Delta )^{-\frac{1}{2}} e_j\Big ) \Big \Vert ^2 $$

$$ \le \sum _{j=1}^\infty \Big \Vert a_j (I-\Delta )^{-\frac{1}{2}} e_j \Big \Vert ^2 + 2 \sum _{j=1}^\infty \Big \Vert a_j \nabla ( \kappa ^{-1} ) (I-\Delta )^{-\frac{1}{2}} e_j) \Big \Vert ^2 + 2 \sum _{j=1}^\infty \Big \Vert a_j \kappa ^{-1} \nabla (I-\Delta )^{-\frac{1}{2}} e_j) \Big \Vert ^2 $$

$$ \le 3 \sum _{j=1}^\infty \Big \Vert a_j (I-\Delta )^{-\frac{1}{2}} e_j \Big \Vert ^2 + 2 \sum _{j=1}^\infty \Big \Vert a_j \nabla (I-\Delta )^{-\frac{1}{2}} e_j) \Big \Vert ^2 $$

$$\begin{aligned} \le 3 \sum _{j=1}^\infty \Big \Vert a_j (I-\Delta )^{-\frac{1}{2}} e_j \Big \Vert ^2_V = 3 \sum _{j=1}^\infty \big \Vert a_j e_j \big \Vert ^2 = 3 \sum _{j=1}^\infty a_j ^2<\infty . \end{aligned}$$

(2.9)

On the other hand, we have

$$\begin{aligned} \big \Vert \widetilde{Q}\big \Vert ^2 _{{{\mathcal {L}}}_2(H,L^2_\kappa (\mathbb {R}^n) )}{} & {} = \sum _{j=1}^\infty \big \Vert \widetilde{Q} e_j \big \Vert ^2_{L^2_\kappa (\mathbb {R}^n)} = \sum _{j=1}^\infty \Big \Vert a_j (I-\Delta )^{-\frac{1}{2}} e_j \Big \Vert ^2\\{} & {} \quad \le \sum _{j=1}^\infty \Big \Vert a_j e_j \Big \Vert ^2 = \sum _{j=1}^\infty \big \Vert a_j \big \Vert ^2 <\infty , \end{aligned}$$

which along with (2.9) shows that $\widetilde{Q}$ as defined by (2.8) satisfies (2.7).

We remark that condition (2.7) will be used when derive the exponential estimates of solutions in $L^2_{\kappa } (\mathbb {R}^n)$ in Lemma 3.14, which is a key for proving the exponential tightness of invariant measures in Lemma 3.15. Moreover, in order to overcome the non-compactness of Sobolev embeddings on $\mathbb {R}^n$, we have to establish the uniform tail-ends estimates of solutions in Lemma 3.4 for which the condition $Q\in {{\mathcal {L}}}_2(H, H )$ is needed. Note that the assumption (2.7) excludes the space-time white noise and spatial homogeneous noise. For these types of noise, the exponential estimates of solutions in $L^2_{\kappa } (\mathbb {R}^n)$ and the uniform tail-ends estimates are unavailable, and hence the arguments of this paper do not apply to the case of either space-time white noise or spatial homogeneous noise.

Definition 2.1

Suppose $u_0 \in L^2(\Omega ,{{\mathcal {F}}}; H)$. A continuous H-valued ${{\mathcal {F}}}_t$-adapted stochastic process u is called a solution of (1.1) and (1.2) if

$$\begin{aligned} u\in L^2(\Omega , C([0, T], H)) \cap L^2(\Omega , L^2(0, T; V)) \cap L^p(\Omega , L^p(0, T; L^p(\mathbb {R}^n))), \end{aligned}$$

(2.10)

such that for all $t\ge 0$ and $\xi \in V\cap L^p(\mathbb {R}^n)$,

$$\begin{aligned}{} & {} ( u(t), \xi ) +\lambda \int _0^t ( u(s), \xi ) ds + \int _0^t ( \nabla u(s), \nabla \xi ) ds + \int _0^t\int _{\mathbb {R}^n} F(x, u(s,x )) \xi (x) dx ds\nonumber \\{} & {} \quad =(u_0, \xi ) + \sqrt{{\varepsilon }} ( QW(t), \xi ), \end{aligned}$$

(2.11)

P-almost surely.

Under conditions (2.1)–(2.3) and (2.7), one can show that for every $u_0 \in L^2(\Omega , {{\mathcal {F}}}_0; H)$, problem (1.1) and (1.2) has a unique solution u in the sense of Definition 2.1 (see, e.g., [22]). Given ${\varepsilon }>0$, we will write the solution of (1.1) and (1.2) as $u^{\varepsilon }(\cdot , u_0)$, $u^{\varepsilon }(u_0)$ or simply as $u^{\varepsilon }$ if no confusion occurs.

Note that in [9], the authors deal with the fractional version of (1.1) and (1.2) with the fractional Laplace operator $(-\Delta )^\alpha$ for $\alpha \in (0,1)$. But all the results of [9] are valid for the standard Laplace operator when $\alpha =1$, and the proof in the case is simpler. Then we find that for every ${\varepsilon }>0$, system (1.1) and (1.2) has a unique ergodic invariant measure $\mu ^{\varepsilon }$ in H which is actually supported on V. Moreover, $\mu ^{\varepsilon }\rightarrow \delta _0$ weakly as ${\varepsilon }\rightarrow 0$, where $\delta _0$ is the probability measure concentrated at zero.

In this paper, we will investigate the LDP of the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$ of the invariant measures of (1.1) and (1.2) as ${\varepsilon }\rightarrow 0$.

3 Deterministic Controlled Equations

Given $T>0$ and $h\in L^2(0,T; H)$, consider the deterministic controlled equation on (0, T):

$$\begin{aligned} {\frac{d u_h}{dt}} (t) +\lambda u_h(t) -\Delta u_h (t) + F( u_h(t)) = Qh(t) \quad t\in (0, T), \end{aligned}$$

(3.1)

with initial condition

$$\begin{aligned} u_h( 0, x ) = u_0 (x), \quad x\in {\mathbb {R}}^n . \end{aligned}$$

(3.2)

Note that if $h\equiv 0$, then Eq. (3.1) reduces to the limiting equation (1.3). As before, a solution of (3.1) and (3.2) is understood in the following sense.

Definition 3.1

Given $u_0 \in H$ and $h\in L^2(0,T; H)$, a function $u_h$ is called a solution of (3.1) and (3.2) on [0, T] if $u\in C([0,T], H) \cap L^2(0,T; V) \cap L^p(0,T;L^p(\mathbb {R}^n))$ and for all $t\in [0,T]$ and $\xi \in V \cap L^p(\mathbb {R}^n)$,

$$ ( u_h(t), \xi ) +\lambda \int _{0}^t (u_h(s), \xi ) ds + \int _ {0}^t ( \nabla u_h(s), \nabla \xi ) ds + \int _ {0}^t\int _{\mathbb {R}^n} F(u_h(s)) \xi (x) dx ds $$

$$ =(u_0, \xi ) +\int _ {0}^t (Qh(s), \xi ) ds. $$

Similar to problem (1.1) and (1.2), if (2.1)–(2.3) are fulfilled, then for every $u_0 \in H$, there exists a unique solution u to (3.1) and (3.2) in the sense of Definition 3.1. Next, we establish the uniform estimates of solutions of (3.1) and (3.2) in H.

Lemma 3.2

Let (2.1)–(2.3) and (2.7) hold, $u_0\in H$ and $h\in L^2 (0,T; H)$. If $u_h$ is the solution of (3.1) and (3.2) on [0, T], then for all $t\in [ 0,T]$,

$$ \Vert u_h (t) \Vert ^2 +\int _{0}^t e^{- \lambda (t-s)} \left( \Vert u_h (s)\Vert ^2_V + \Vert u_h(s) \Vert ^p_{L^p(\mathbb {R}^n)} \right) ds $$

$$\begin{aligned} \le L_1 \left( e^{- \lambda t } \Vert u_0 \Vert ^2 + \int _{0}^t \Vert Q h(s) \Vert ^2 ds \right) , \end{aligned}$$

(3.3)

and

$$\begin{aligned} \int _{0}^t \left( \Vert u_h (s) \Vert ^2_V + \Vert u_h(s) \Vert ^p_{L^p(\mathbb {R}^n)} \right) ds \le L_1 \left( \Vert u_0 \Vert ^2 + \int _{0}^t \Vert Q h(s) \Vert ^2 ds \right) , \end{aligned}$$

(3.4)

where $L_1=L_1 (\lambda , \lambda _1)>0$ is a constant depending only on $\lambda$ and $\lambda _1$, but not on $u_0$, h or t.

Proof

By (3.1) and (3.2) we have

$$\begin{aligned} {\frac{1}{2}} {\frac{d}{dt}} \Vert u_h(t) \Vert ^2 + \lambda \Vert u_h (t) \Vert ^2 + \Vert \nabla u_h (t) \Vert ^2 +\int _{\mathbb {R}^n} F(u_h (t)) u_h (t) dx = ( Qh(t), u_h(t)). \end{aligned}$$

(3.5)

By (2.1), (3.5) and Young’s inequality we get

$$\begin{aligned} {\frac{d}{dt}} \Vert u_h(t) \Vert ^2 + {\frac{3}{2}} \lambda \Vert u_h (t) \Vert ^2 + 2 \Vert \nabla u_h (t) \Vert ^2 + 2\lambda _1 \Vert u_h(t) \Vert ^p_{L^p(\mathbb {R}^n)} \le {\frac{2}{\lambda }} \Vert Qh(t) \Vert ^2. \end{aligned}$$

(3.6)

Multiplying (3.6) by $e^{ \lambda t}$ and then integrating on (0, t), we obtain for all $t\in [0, T]$,

$$ \Vert u_h (t) \Vert ^2 +\int _{0}^t e^{- \lambda (t-s)} \left( {\frac{1}{2}} \lambda \Vert u_h (s)\Vert ^2 + 2 \Vert \nabla u_h (s)\Vert ^2 + 2\lambda _1 \Vert u_h(s) \Vert ^p_{L^p(\mathbb {R}^n)} \right) ds $$

$$\begin{aligned} \le e^{- \lambda t } \Vert u_0 \Vert ^2 + {\frac{2}{ \lambda }} \int _{0}^t e^{- \lambda (t-s) }\Vert Q h(s) \Vert ^2 ds. \end{aligned}$$

(3.7)

On the other hand, integrate (3.6) on (0, t) to obtain

$$\begin{aligned} \int _0^t \left( {\frac{3}{2}} \lambda \Vert u_h (s) \Vert ^2 + 2 \Vert \nabla u_h (s) \Vert ^2 + 2\lambda _1 \Vert u_h(s) \Vert ^p_{L^p(\mathbb {R}^n)} \right) ds \le \Vert u_0\Vert ^2 + {\frac{2}{\lambda }} \int _0^t \Vert Qh(s) \Vert ^2ds. \end{aligned}$$

(3.8)

Then (3.3) and (3.4) follow from (3.7) to (3.8) immediately. $\square$

Lemma 3.3

Let (2.1)–(2.3) and (2.7) hold and $h\in L^2 (0, T; H)$. If $u_0\in H$ and $u_h$ is the solution of (3.1) and (3.2) on [0, T] , then for all $t\in (0,T]$,

$$\begin{aligned} \Vert u_h (t)\Vert ^2_V \le L_2 \left( 1+ t^{-1} \right) \left( \Vert u_0\Vert ^2 +\int _0^t \Vert Q h(s) \Vert ^2 ds \right) \end{aligned}$$

(3.9)

In addition, if $u_0\in V$, then for all $t\in [0,T]$,

$$\begin{aligned} \Vert u_h (t)\Vert ^2 _V + \int _0^t e^{- \lambda (t-s)} \Vert u_h (s)\Vert ^2_{H^{2} (\mathbb {R}^n) } ds \le L_2 \left( \Vert u_0\Vert ^2_V + \int _0^T\Vert Q h(s) \Vert ^2 ds \right) , \end{aligned}$$

(3.10)

where $L_2=L_2 (\lambda , \lambda _1)>0$ is a constant independent of $u_0$, h and t.

Proof

By (3.1) we have

$$ {\frac{1}{2}} {\frac{d}{dt}} \Vert \nabla u_h (t)\Vert ^2 + \lambda \Vert \nabla u_h (t)\Vert ^2 + \Vert \Delta u_h (t)\Vert ^2 $$

$$\begin{aligned} = \int _{\mathbb {R}^n} F( u_h (t)) \ \Delta u_h (t) dx - (Q h(t), \Delta u_h (t)). \end{aligned}$$

(3.11)

For the first term on the right-hand side of (3.11) by (2.3) we have for all $t\in [0,T]$,

$$\begin{aligned} \int _{\mathbb {R}^n} F( u_h (t)) \ \Delta u_h (t) dx =- \int _{\mathbb {R}^n} F^\prime ( u_h (t)) |\nabla u_h (t) |^2 dx \le 0. \end{aligned}$$

(3.12)

For the last term on the right-hand side of (3.11), we have

$$\begin{aligned} - (Q h(t), \Delta u_h (t)) \le {\frac{1}{2}} \Vert \Delta u_h (t) \Vert ^2 + {\frac{1}{2}} \Vert Q h(t) \Vert ^2. \end{aligned}$$

(3.13)

It follows from (3.11) to (3.13) that

$$\begin{aligned} {\frac{d}{dt}} \Vert \nabla u_h (t)\Vert ^2 + 2 \lambda \Vert \nabla u_h (t)\Vert ^2 + \Vert \Delta u_h (t)\Vert ^2 \le \Vert Q h(t)\Vert ^2. \end{aligned}$$

(3.14)

By (3.14) we get

$$ {\frac{d}{dt}} \Vert \nabla u_h (t)\Vert ^2 \le \Vert Q h (t)\Vert ^2, $$

and thus we have

$$ t {\frac{d}{dt}} \Vert \nabla u_h (t)\Vert ^2 \le t \Vert Q h (t)\Vert ^2, $$

that is,

$$ {\frac{d}{dt}} \left( t \Vert \nabla u_h (t)\Vert ^2 \right) \le \Vert \nabla u_h (t)\Vert ^2 + t \Vert Q h (t)\Vert ^2. $$

Then for all $t\in [0,T]$,

$$ t \Vert \nabla u_h (t)\Vert ^2 \le \int _{0}^t \Vert \nabla u_h (s)\Vert ^2 ds + t\int _0^t \Vert Q h(s)\Vert ^2 ds, $$

which along with (3.4) implies that for all $t\in (0,T]$,

$$\begin{aligned} \Vert \nabla u_h (t)\Vert ^2 \le t^{-1} L_1 \left( \Vert u_0\Vert ^2 +\int _0^t \Vert Q h(s) \Vert ^2 ds \right) + \int _0^t \Vert Q h(s)\Vert ^2 ds. \end{aligned}$$

(3.15)

Then (3.9) follows from (3.15) and (3.3) immediately.

On the other hand, by (3.14) we obtain, for all $t\in [0,T]$

$$ \Vert \nabla u_h (t)\Vert ^2 + \int _0^t e^{- \lambda (t-s)} \left( \lambda \Vert \nabla u_h (s)\Vert ^2 + \Vert \Delta u_h (s)\Vert ^2 \right) ds $$

$$ \le e^{-\lambda t} \Vert \nabla u(0)\Vert ^2 + \int _0^t e^{- \lambda (t-s)} \Vert Q h(s) \Vert ^2, ds $$

which along with (3.3) implies (3.10), and thus completes the proof. $\square$

The next lemma is concerned with the tail-ends estimates of solutions of (3.1) and (3.2).

Lemma 3.4

Let (2.1)–(2.3) and (2.7) hold. Then for every $R>0$, $\delta >0$ and every compact subset K of H, there exists $m_0=m_0(R, \delta , K)>0$ such that for every $T>0$, $h\in L^2(0,T; H)$ with $\Vert h\Vert _{L^2(0,T; H)} \le R$ and every $u_0\in K$, the solution $u_h$ of (3.1) and (3.2) on [0, T] satisfies the uniform estimates:

$$ \sup _{0\le t\le T} \int _{|x|\ge m_0} |u_h(t, u_0) (x)|^2 dx <\delta . $$

Proof

Let $\theta : \mathbb {R}^n \rightarrow [0,1]$ be a smooth function such that

$$\begin{aligned} \theta (x)=0 \ \ \text {for } |x| \le {\frac{1}{2}}; \ \text {and} \ \theta (x) = 1 \ \ \text {for } \ |x| \ge 1. \end{aligned}$$

(3.16)

Given $m\in \mathbb {N}$, let $\theta _m (x) = \theta \left( {\frac{x}{m}} \right)$. By (3.1) we have

$$\begin{aligned}{} & {} {\frac{d}{dt}} \Vert \theta _m u_h (t) \Vert ^2 + 2 \lambda \Vert \theta _m u_h (t) \Vert ^2 +2 \big ( \nabla u_h (t), \nabla \big (\theta _m^2 u_h (t)\big ) \big )\nonumber \\{} & {} =- 2 \int _{ \mathbb {R}^n}F( u_h (t)) \theta _m^2 (x) u_h (t) dx + 2 \big (\theta _m Qh(t), \theta _m u_h (t) \big ). \end{aligned}$$

(3.17)

Note that there exists $c_1>0$ such that $|\nabla \theta (x)| \le c_1$ for all $x\in \mathbb {R}^n$. Then for the third term on the left-hand side of (3), we have

$$ -2 \left( \nabla u_h (t), \nabla \big (\theta _m^2 u_h (t)\big ) \right) $$

$$\begin{aligned}{} & {} \le - 2\int _{\mathbb {R}^n} \theta _m^2 (x) |\nabla u_h(t,x)|^2 dx +4c_1m^{-1 } \int _{\mathbb {R}^n} \theta _m (x) |u_h (t,x)| \ |\nabla u_h (t,x)| dx\nonumber \\{} & {} \le 2c_1^2\,m^{-2} \Vert u_h(t) \Vert ^2. \end{aligned}$$

(3.18)

For the first term on the right-hand side of (3), by (2.1) we get

$$\begin{aligned} - 2 \int _{ \mathbb {R}^n}F( u_h(t,x)) \theta _m^2 (x) u_h(t,x ) dx \le 0. \end{aligned}$$

(3.19)

For the last term on the right-hand side of (3), we have

$$\begin{aligned} 2 (\theta _m Qh(t), \theta _m u_h (t) ) \le \lambda \Vert \theta _m u_h (t)\Vert ^2 + {\frac{1}{\lambda }} \Vert \theta _m Qh(t)\Vert ^2. \end{aligned}$$

(3.20)

To deal with the right-hand side of (3.20), we take an orthonormal basis of H, $\{e_j\}_{j=1}^\infty$, and write

$$ h(t) = \sum _{j=1}^\infty c_j (t) e_j, \quad c_j(t) =(h(t), e_j). $$

Then we have

$$\begin{aligned} \Vert \theta _m Qh(t)\Vert ^2 = \left\| \sum _{j=1}^\infty c_j (t) \theta _m Qe_j\right\| ^2 \le \sum _{j=1}^\infty c_j^2(t) \sum _{j=1}^\infty \Vert \theta _m Q e_j\Vert ^2 = \Vert h(t) \Vert ^2 \sum _{j=1}^\infty \Vert \theta _m Q e_j\Vert ^2. \end{aligned}$$

(3.21)

Since $Q: H\rightarrow H$ is a Hilbert–Schmidt operator, we know $\sum _{j=1}^\infty \Vert Qe_j \Vert ^2 <\infty$, and thus, for every $\delta >0$, there exists $J=J(\delta )>0$ such that

$$\begin{aligned} \sum _{j=J+1}^\infty \Vert Qe_j \Vert ^2 <{\frac{1}{2}} \delta . \end{aligned}$$

(3.22)

By (3.22) we get

$$\begin{aligned} \sum _{j=1}^\infty \Vert \theta _m Q e_j\Vert ^2 \le \sum _{j=1}^J \Vert \theta _m Q e_j\Vert ^2 +\sum _{j=J+1}^\infty \Vert Q e_j\Vert ^2 \le \sum _{j=1}^J \int _{|x|>{\frac{1}{2}} m} |Q e_j |^2 (x) dx + {\frac{1}{2}} \delta . \end{aligned}$$

(3.23)

For each $j=1,\ldots , J$, since $Qe_j \in H$, we know that there exists $m_j =m_j (j, \delta )>0$ such that

$$\begin{aligned} \int _{|x|>{\frac{1}{2}} m_j} |Q e_j |^2 (x) < {\frac{\delta }{2 J}}. \end{aligned}$$

(3.24)

Let $m_J (\delta )= \max \limits _{1\le j\le J} m_j$. By (3.21) and (3.23) and (3.24) we get for all $m\ge m_J$,

$$\begin{aligned} \Vert \theta _m Qh(t)\Vert ^2 \le \delta \Vert h(t) \Vert ^2. \end{aligned}$$

(3.25)

By (3.20) and (3.25) we get for all $m\ge m_J$,

$$\begin{aligned} 2 (\theta _m Qh(t), \theta _m u_h (t) ) \le \lambda \Vert \theta _m u_h (t)\Vert ^2 + {\frac{\delta }{\lambda }} \Vert h(t)\Vert ^2. \end{aligned}$$

(3.26)

It follows from (3)–(3.19) and (3.26) that for all $m\ge m_J$ and $t>0$,

$$\begin{aligned} {\frac{d}{dt}} \Vert \theta _m u_h (t) \Vert ^2 + \lambda \Vert \theta _m u_h (t) \Vert ^2 \le 2c_1 ^2\,m^{-2} \Vert u_h (t) \Vert ^2 + {\frac{\delta }{\lambda }} \Vert h(t)\Vert ^2, \end{aligned}$$

(3.27)

where $c_1>0$ is independent of m and $\delta$. Solve (3.27) to obtain for all $m\ge m_J$ and $t>0$,

$$ \Vert \theta _m u_h (t) \Vert ^2 \le e^{-\lambda t} \Vert \theta _m u_0 \Vert ^2 + 2c_1^2 m^{-2} \int _0^t e^{- \lambda (t-s )} \Vert u_h (s) \Vert ^2 ds $$

$$ + {\frac{\delta }{\lambda }} \int _0^t e^{-\lambda (t-s)} \Vert h(s)\Vert ^2 ds, $$

which along with Lemma 3.2 shows that for all $m\ge m_J$ and $t\in [0,T]$,

$$ \Vert \theta _m u_h (t) \Vert ^2 \le \Vert \theta _m u_0 \Vert ^2 + c_3 m^{-2} \left( \Vert u_0\Vert ^2 + \int _0^T \Vert Q h(s) \Vert ^2 ds \right) + {\frac{\delta }{\lambda }} \int _0^T \Vert h(s)\Vert ^2 ds $$

$$\begin{aligned} \le \Vert \theta _m u_0 \Vert ^2 + c_3 m^{-2} \left( \Vert u_0\Vert ^2 + R^2 \Vert Q\Vert ^2_{L(H,H)} \right) + {\frac{R \delta }{\lambda }}, \end{aligned}$$

(3.28)

where $c_3>0$ is independent of m and $\delta$, and $\Vert Q\Vert _{L(H,H)} =\sup \limits _{\Vert v\Vert \le 1}\Vert Qv\Vert$. Since $u_0 \in K$ and K is compact in H, by (3.28) we find that there exists $m_0 =m_0 (R, \delta , K) \ge m_J$ such that for all $m\ge m_0$ and $t\in [0,T]$,

$$ \int _{|x| \ge m} | u_h (t) (x) |^2 dx \le \Vert \theta _m u_h (t) \Vert ^2 \le \delta + {\frac{R \delta }{\lambda }}, $$

which completes the proof. $\square$

Next, we establish the continuity of the solutions of the controlled equation (3.1) with respect to initial data as well as controls.

Lemma 3.5

Let (2.1)–(2.3) and (2.7) hold. Then for all $u_{0,1}, u_{0,2} \in H$ and $h_1, h_2\in L^2 (0,T; H)$, the solutions $u_{h_1} (\cdot , u_{0,1})$ and $u_{h_2}(\cdot , u_{0,2})$ of (3.1) and (3.2) with initial data $u_{0,1}$ and $u_{0,2}$, respectively, satisfy for all $t\in [0,T]$,

$$ \Vert u_{h_1} (t, u_{0,1} )-u_{h_2} (t, u_{0,2}) \Vert ^2 \le e^{-\lambda t} \Vert u_{0,1}-u_{0,2} \Vert ^2 +{\frac{2}{ \lambda }} \int _0^t e^{-{ \lambda } (t-s)} \Vert Q(h_1-h_2) (s)\Vert ^2 ds. $$

Proof

Let $v= u_{h_1} -u_{h_2}$. By (3.1) we have

$$ {\frac{1}{2}} {\frac{d}{dt}} \Vert v(t) \Vert ^2 + \lambda \Vert v(t) \Vert ^2 + \Vert \nabla v(t)\Vert ^2 $$

$$\begin{aligned} =- \int _{\mathbb {R}^n} (F(u_{h_1}(t,x))-F( u_{h_2}(t,x) )) v(t,x) dx +(Q(h_1 -h_2 ) (t), v(t) ). \end{aligned}$$

(3.29)

By (2.3) and (3.29) we have

$$\begin{aligned} {\frac{1}{2}} {\frac{d}{dt}} \Vert v(t) \Vert ^2 + \lambda \Vert v(t) \Vert ^2 + \Vert \nabla v(t)\Vert ^2 \le {\frac{1}{4}} \lambda \Vert v(t) \Vert ^2 +{\frac{1}{\lambda }} \Vert Q(h_1-h_2) (t) \Vert ^2. \end{aligned}$$

(3.30)

Multiply (3.30) by $e^{\lambda t}$ to obtain

$$ \Vert v(t) \Vert ^2 + \int _0^t e^{- \lambda (t-s)} \left( {\frac{1}{2}} \lambda \Vert v(s) \Vert ^2 +2 \Vert \nabla v(s) \Vert ^2 \right) ds $$

$$ \le e^{- \lambda t} \Vert v(0)\Vert ^2 +{\frac{2}{\lambda }} \int _0^t e^{- \lambda (t-s)} \Vert Q(h_1-h_2) (s)\Vert ^2 ds, $$

which completes the proof. $\square$

Given $T>0$, $u_0\in H$ and $u\in C([0,T], H)$, define the rate function (or action functional):

$$\begin{aligned} I_{T, u_0} (u) = \inf \left\{ {\frac{1}{2}} \int _{0}^{T} \Vert h (t) \Vert ^2 dt:\ h\in L^2(0,T; H), \ u(0) =u_0,\ \ u =u_h \right\} , \end{aligned}$$

(3.31)

where $u_h$ is the solution of (3.1) and (3.2) with initial condition $u_0$ at initial time 0. Again, $\inf \emptyset$ is set to $+\infty$. Given $s \ge 0$, the s-level set of $I_{T,u_0}$ is defined by

$$\begin{aligned} I_{T,u_0}^s =\left\{ u\in C([0,T], H): I_{T, u_0} (u) \le s \right\} . \end{aligned}$$

(3.32)

For convenience, given ${\varepsilon }\in (0,1)$, $T>0$ and $u_0\in H$, let $\nu ^{\varepsilon }_{u_0}$ be the distribution of the solution $u^{\varepsilon }(\cdot , u_0)$ of (1.1) and (1.2) in C([0, T], H). It follows from [23] that if $Q\in {{\mathcal {L}}}_2 (H, H^k(\mathbb {R}^n))$ for some $k\ge {\frac{1}{2}} n(p-2)p^{-1}$, then the family $\{\nu ^{\varepsilon }_{u_0}\} _{{\varepsilon }\in (0,1)}$ satisfies the Freidlin–Wentzell uniform LDP on C([0, T], H) with rate function $I_{T, u_0}$ uniformly with respect to $u_0$ bounded in H. This result is also valid for $Q\in {{\mathcal {L}}}_2 (H, H))$, which can be proved by the argument of [24] with minor modifications as stated below.

Proposition 3.6

If (2.1)–(2.3) and (2.7) hold, then for every $T>0$, the family $\{u^{\varepsilon }(u_0) \}_{0<{\varepsilon }\le 1}$ of the solutions of (1.1) and (1.2) satisfies the Freidlin–Wentzell uniform LDP on C([0, T], H) with rate function $I_{T,u_0}$, uniformly with respect to $u_0$ in every bounded subset of H; more precisely:

• 1. For every $R>0$, $s\ge 0$, $\delta _1>0$ and $\delta _2>0$, there exists ${\varepsilon }_0>0$ such that

  $$\begin{aligned} \inf _{\Vert u_0\Vert \le R} \ \inf _{ \xi \in I_{T, u_0}^s } \left( \nu ^{{\varepsilon }}_{u_0} ( B_{C([0,T], H)}(\xi , \delta _1) ) -e^{- {\frac{I_{T,u_0} (\xi ) +\delta _2}{{\varepsilon }}} } \right) \ge 0, \quad \forall \ {\varepsilon }\le {\varepsilon }_0, \end{aligned}$$

  (3.33)

  where $B_{C([0,T], H)}(\xi , \delta _1) =\left\{ u\in C([0,T], H): \Vert u-\xi \Vert _{C([0,T], H)} <\delta _1 \right\}$.

• 2. For every $R>0$, $s_0\ge 0$, $\delta _1>0$ and $\delta _2>0$, there exists ${\varepsilon }_0>0$ such that

  $$ \sup _{\Vert u_0\Vert \le R} \nu ^{{\varepsilon }}_{u_0} \left( {C([0,T], H)} {\setminus } {{\mathcal {N}}}\big (I _{T, u_0} ^s, \delta _1\big ) \right) \le e^{-{\frac{s-\delta _2}{{\varepsilon }}} }, \quad \forall \ {\varepsilon }\le {\varepsilon }_0, \ \forall \ s\le s_0, $$

  where ${{\mathcal {N}}}(I_{T,u_0}^s,\delta _1) =\left\{ u\in {C([0,T], H)}: \mathrm{{dist}} \big (u, I_{T,u_0} ^s\big ) <\delta _1 \right\}$.

In order to establish the LDP of the invariant measures of the stochastic equation (1.1), we also need the Dembo–Zeitouni uniform LDP of solutions, for which the following continuity of level sets of $I_{T,u_0}$ is useful.

Lemma 3.7

Suppose (2.1)–(2.3) hold. If $u_{0,n} \rightarrow u_0$ in H, then for every $T>0$ and $s\ge 0$, the sequence $\{I_{T, u_{0,n}}^s\}_{n=1}^\infty$ converges to $I_{T, u_0}^s$ in the Hausdorff metric in C([0, T], H) as $n \rightarrow \infty$; more precisely,

$$ \max \left\{ \sup _{v\in I^s_{T,u_0}} \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0,n}}\big ), \ \ \sup _{v\in I^s_{T,u_{0,n}}} \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0}}\big ) \right\} \rightarrow 0, \ \ \text {as } \ n\rightarrow \infty . $$

Proof

Let $v\in I^s_{T,u_0}$. Then there exists $h\in L^2(0,T; H)$ such that ${\frac{1}{2}}\int _0^T \Vert h(s) \Vert ^2 ds \le s$ and $v= u_h (\cdot , u_0)$, where $u_h (\cdot , u_0)$ is the solution of (3.1) and (3.2) with initial value $u_0$. Let $u_h (\cdot , u_{0,n})$ be the solution of (3.1) and (3.2) with initial data $u_{0,n}$. Then we have $u_h (\cdot , u_{0,n}) \in I^s_{T, u_{0,n}}$ and hence by Lemma 3.5 we obtain

$$ \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0,n}}\big ) \le \textrm{dist}_{C([0,T],H)} \big (v, u_h(\cdot , u_{0,n} )\big ) $$

$$ = \textrm{dist}_{C([0,T],H)} ( u_h(\cdot , u_{0} ), \ u_h(\cdot , u_{0,n} )) \le \Vert u_{0,n} -u_0\Vert , $$

which implies that

$$\begin{aligned} \sup _{v\in I^s_{T,u_0}} \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0,n}}\big ) \le \Vert u_{0,n} -u_0\Vert . \end{aligned}$$

(3.34)

Similarly, if $v\in I^s_{T,u_{0,n}}$, then there exists $h\in L^2(0,T; H)$ such that ${\frac{1}{2}}\int _0^T \Vert h(s) \Vert ^2 ds \le s$ and $v= u_h (\cdot , u_{0,n} )$, where $u_h (\cdot , u_{0,n} )$ is the solution of (3.1) and (3.2) with initial value $u_{0,n}$. Let $u_h (\cdot , u_{0})$ be the solution of (3.1) and (3.2) with initial data $u_{0}$. Then we have $u_h (\cdot , u_{0}) \in I^s_{T, u_{0}}$ and hence by Lemma 3.5 again we obtain

$$ \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0}}\big ) \le \textrm{dist}_{C([0,T],H)} (v, u_h(\cdot , u_{0} )) $$

$$ = \textrm{dist}_{C([0,T],H)} ( u_h(\cdot , u_{0,n} ), \ u_h(\cdot , u_{0} )) \le \Vert u_{0,n} -u_0\Vert , $$

which implies that

$$\begin{aligned} \sup _{v\in I^s_{T,u_{0,n}}} \textrm{dist}_{C([0,T],H)} \big (v, I^s_{T, u_{0}}\big ) \le \Vert u_{0,n} -u_0\Vert . \end{aligned}$$

(3.35)

Since $u_{0,n}\rightarrow u_0$ in H, the lemma follows from (3.34) and (3.35). $\square$

In terms of Proposition 3.6 and Lemma 3.7, by [19, Theorem 2.7] we obtain the following Dembo–Zeitouni uniform LDP of solutions of (1.1) and (1.2).

Proposition 3.8

If (2.1)–(2.3) and (2.7) hold, then for every $T>0$, the family $\{u^{\varepsilon }(u_0) \}_{0<{\varepsilon }\le 1}$ of the solutions of (1.1) and (1.2) satisfies the Dembo–Zeitouni uniform LDP on C([0, T], H) with rate function $I_{T,u_0}$, uniformly with respect to $u_0$ in every compact subset of H; more precisely:

• 1. For every compact subset ${{\mathcal {Z}}}$ of H and every open subset G of C([0, T], H),

  $$ \liminf _{{\varepsilon }\rightarrow 0} \inf _{u_0 \in {{\mathcal {Z}}}} \left( {\varepsilon }\ln P(u^{\varepsilon }( u_0) \in G ) \right) \ge -\sup _{u_0 \in {{\mathcal {Z}}}} \inf _{v\in G} I_{T, u_0} (v). $$

• 2. For every compact subset ${{\mathcal {Z}}}$ of H and every closed subset F of C([0, T], H),

  $$ \limsup _{{\varepsilon }\rightarrow 0} \sup _{u_0 \in {{\mathcal {Z}}}} \left( {\varepsilon }\ln P(u^{\varepsilon }( u_0) \in F ) \right) \le -\inf _{u_0 \in {{\mathcal {Z}}}} \inf _{v\in F} I_{T, u_0} (v). $$

Proof

By Lemma 3.7, we know from [19, Theorem 2.7] that the Freidlin–Wentzell and the Dembo–Zeitouni uniform LDP on C([0, T], H) over a compact subset of H are equivalent, and thus the conclusion follows from Proposition 3.6. $\square$

By (2.1) we find that zero is the unique stationary solution of the deterministic equation and it attracts all bounded subsets in H exponentially. In other words, the deterministic equation has a global attractor ${{\mathcal {A}}}$ which is a singleton, i.e., ${{\mathcal {A}}}=\{0\}$. In terms of ${{\mathcal {A}}}$, by the idea of [18] we define a rate function $J: H\rightarrow [0, +\infty ]$ by: for every $v\in H$,

$$\begin{aligned} J(v) =\lim _{\delta \rightarrow 0} \ \inf \left\{ I_{r,0} (u): \ r>0,\ u\in C([0,r], H), u(0) =0,\ u(r) \in B_H(v, \delta ) \right\} . \end{aligned}$$

(3.36)

The rate function J is also called a quasipotential in the literature. Given $s\ge 0$, the s-level set of J is denoted by:

$$ J^s =\left\{ v \in H: \ J(v) \le s \right\} . $$

We first prove J given by (3.36) is a good rate function in H by the uniform tail-ends estimates.

Lemma 3.9

Let (2.1)–(2.3) and (2.7) hold and J be the function as defined by (3.36). Then for every $s\ge 0$, the s-level set $J^s$ of J is compact in H and bounded in V.

Proof

Given $s\ge 0$, we first prove $J^s$ is bounded in V; that is, there exists $c_1=c_1(s) >0$ such that

$$\begin{aligned} \Vert v\Vert _V \le c_1, \quad \forall \ v\in J^s. \end{aligned}$$

(3.37)

Given $v\in J^s$, by (3.36) we find that for every $m\in \mathbb {N}$, there exist $r_m>0$ and $u_m\in C([0,r_m], H)$ such that

$$ u_m (0) =0, \quad \Vert u_m (r_m) -v \Vert< {\frac{1}{m}}, \quad I_{r_m} (u_m)< s+{\frac{1}{m}}, $$

which further implies that there exists $h_m \in L^2(0, r_m; H)$ such that for all $m\in \mathbb {N}$,

$$\begin{aligned} \int _0^{r_m} \Vert h_m (s) \Vert ^2 ds < 2 \left( s+{\frac{1}{m}} \right) \le 2 (s+1) \end{aligned}$$

(3.38)

and

$$\begin{aligned} \lim _{m\rightarrow \infty } \Vert u_{h_m} (r_m, u_{0,m} ) -v \Vert =0, \end{aligned}$$

(3.39)

where $u_{h_m} (t, u_{0,m})$ is the solution of (3.1) and (3.2) with initial value $u_{0,m} =u_m (0)\in {{\mathcal {A}}}$ and control $h_m$.

On the other hand, by (3.10) and (3.38) we have

$$\begin{aligned} \sup _{t\in [0, r_m]} \Vert u_{h_m} (t, u_{0,m}) \Vert _V \le c_2, \quad \forall \ m\in \mathbb {N}, \end{aligned}$$

(3.40)

where $c_2=c_2(\lambda , \lambda _1, s)>0$ is a constant independent of m. It follows from (3.39) and (3.40) that $\Vert v\Vert _V \le c_2$ for all $v\in J^s$, which gives (3.37).

It follows from (3.38) and Lemma 3.4 that for every $\delta >0$, there exists $k_0=k_0 (\delta , s)>0$ such that

$$\begin{aligned} \sup _{0\le t\le r_m} \int _{|x| \ge k_0} |u_{h_m} (t, u_{0,m}) (x)|^2 dx < {\frac{1}{16}}\delta ^2. \end{aligned}$$

(3.41)

By (3.39) and (3.41) we get

$$\begin{aligned} \int _{|x|\ge k_0} |v(x)|^2 dx \le {\frac{1}{16}}\delta ^2, \quad \forall \ v\in J^s. \end{aligned}$$

(3.42)

By (3.37) we have $J^s \subseteq {\overline{B}} _V(0, c_1)$. Since the embedding $H^1 (|x|<k_0) \hookrightarrow L^2(|x|< k_0)$ is compact, we infer that $J^s$ has a finite open cover of radius ${\frac{1}{4}} \delta$ in $L^2(|x|<k_0)$, which together with (3.42) shows that the set $J^s$ has a finite open cover of radius $\delta$ in H for every $\delta >0$, and hence it is precompact in H.

It remains to show $J^s$ is closed in H, which is similar to that of [18]. Suppose $v_m \in J^s$ and $v\in H$ such that $v_m \rightarrow v$ in H. We need to prove $v\in J^s$. Since $v_m \rightarrow v$ in H, for every $\delta _1>0$, there exists $m_0 =m_0 (\delta _1)>0$ such that

$$\begin{aligned} \Vert v_m -v\Vert <{\frac{1}{2}} \delta _1, \quad \forall \ m\ge m_0. \end{aligned}$$

(3.43)

Since $v_{m_0} \in J^s$, for every $\delta _2>0$, there exist $r>0$ and $u\in C([0,r], H)$ such that $u(0) \in {{\mathcal {A}}}$, $u(r) \in B_H(v_{m_0}, {\frac{1}{2}} \delta _1 )$ and $I_r(u) < s+ \delta _2$. Therefore, there exists $h\in L^2(0, r; H)$ such that

$$\begin{aligned} {\frac{1}{2}} \int _0^r \Vert h(s)\Vert ^2 ds< s + \delta _2, \quad \quad \Vert u_h (r, u_0) - v_{m_0}\Vert < {\frac{1}{2}} \delta _1, \end{aligned}$$

(3.44)

where $u_h(t,u_0)$ is the solution of (3.1) and (3.2) with initial condition $u_0 =u(0) \in {{\mathcal {A}}}$. By (3.43) and (3.44) we have

$$ \Vert u_h (r, u_0) - v \Vert < \delta _1, $$

which along with (3.44) implies that

$$ J(v) \le s + \delta _2, \quad \forall \ \delta _2>0, $$

Then we see $v\in J^s$ and thus $J^s$ is closed in H. $\square$

We remark that the uniform tail-ends argument in the proof of Lemma 3.9 is closely related to the following theorem for compact subsets of $L^2(\mathbb {R}^n)$ (see, e.g., [1, Corollary 4.27]):

Proposition 3.10

(Kolmogorov–Riesz–Fréchet) Let $\Gamma$ be a bounded subset of $L^2(\mathbb {R}^n)$. Assume that

$$ \lim _{h\rightarrow 0}\sup _{f\in \Gamma } \Vert f(\cdot +h) -f(\cdot )\Vert =0, $$

and for every $\delta >0$, there exists a bounded measurable subset $\mathcal {O} \subseteq \mathbb {R}^n$ such that

$$ \sup _{f\in \Gamma } \Vert f \Vert _{L^2(\mathbb {R}^n {\setminus } \mathcal {O})} <\delta . $$

Then $\Gamma$ is precompact in $L^2(\mathbb {R}^n)$.

Lemma 3.11

Let (2.1)–(2.3) and (2.7) hold. Then for every positive $\delta _1, \delta _2$ and l, there exists $\delta >0$ such that for all $t>0$,

$$ \left\{ u(t): \ u\in C([0,t],H), \ u(0) \in B_H(0,\delta ), \ I_{t, u(0)} (u) \le l-\delta _1 \right\} \subseteq J^l_{\delta _2}, $$

where $J^l_{\delta _2}$ is the open $\delta _2$-neighborhood of the l-level set of J.

Proof

We argue by contradiction. Suppose it is false. Then there exist $\delta _1>0, \delta _2>0$ and $l>0$ such that for every $m>0$, there exist $t_m>0$, $u_{0,m}\in B_H(0, {\frac{1}{m}} )$ and $h_m\in L^2(0,t_m; H)$ with ${\frac{1}{2}} \Vert h_m \Vert ^2 _{L^2(0,t_m; H)} <l-{\frac{1}{2}} \delta _1$ such that the solution $u_{h_m}$ of (3.1) and (3.2) with initial condition $u_{0,m}$ and control $h_m$ on $[0,t_m]$ satisfies:

$$\begin{aligned} u_{h_m} (t_m) \notin J^l_{\delta _2}. \end{aligned}$$

(3.45)

Note that

$$\begin{aligned} \Vert u_{0,m} \Vert <{\frac{1}{m}}. \end{aligned}$$

(3.46)

Let $v_{h_m}$ be the solution of (3.1) and (3.2) with initial condition zero and control $h_m$ on $[0, t_m]$. Then by Lemma 3.5 and (3.46) we get for all $t\in [0,t_m]$,

$$\begin{aligned} \Vert v_{h_m} (t) - u_{h_m} (t) \Vert <{\frac{1}{m}}. \end{aligned}$$

(3.47)

By (3.45) and (3.47) we infer that there exists $m_0=m_0(\delta _2)>0$ such that

$$\begin{aligned} v_{h_m} (t_m) \notin J^l_{{\frac{1}{2}}\delta _2}, \quad \forall \ m \ge m_0. \end{aligned}$$

(3.48)

On the other hand, since $v_{h_m} (0) =0$, by the definition of J, we find that

$$ J(v_{h_m} (t_m)) \le I_{t_m, 0} (v_{h_m}) \le {\frac{1}{2}} \Vert h_m\Vert ^2_{L^2(0,t_m;H)} <l-{\frac{1}{2}}\delta _1 \le l, $$

and thus $v_{h_m} (t_m) \in J^l$. This is in contradiction with (3.48) and hence completes the proof.

$\square$

Lemma 3.12

Let (2.1)–(2.3) and (2.7) hold. Then for every $R>0$ and $\delta >0$, there exists $T>0$ such that

$$ \inf \left\{ I_{T,u_0} (u): \ \Vert u_0\Vert \le R, u\in C([0,T], H), u(0)=u_0, \ u(T)\notin B_H(0,\delta ) \right\} >0. $$

Proof

Suppose the statement is false. Then there exist $R>0$ and $\delta >0$ such that for every $m\in \mathbb {N}$, there exist $u_{0,m} \in H$ with $\Vert u_{0,m}\Vert \le R$ and $h_m\in L^2 (0, m; H)$ with $\Vert h_m \Vert ^2 _{L^2(0,m; H)} \le {\frac{1}{\,}m}$ such that the solution $u_{h_m}$ of (3.1) and (3.2) on [0, m] with initial value $u_{0,m}$ and control $h_m$ satisfies:

$$\begin{aligned} u_{h_m} (m) \notin B_H(0,\delta ), \quad \forall \ m\in \mathbb {N}. \end{aligned}$$

(3.49)

Let $v_m$ be the solution of the deterministic equation (1.3) on [0, m] with initial condition $u_{0,m}$, which is also the solution of the controlled equation (3.1) with $h=0$. By Lemma 3.5 we have for all $t\in [0, m]$,

$$ \Vert v_m (t) - u_{h_m} (t)\Vert ^2 \le {\frac{2}{\lambda }} \int _0^m \Vert Q h_m (s)\Vert ^2 ds $$

$$ \le {\frac{2}{\lambda }} \Vert Q\Vert ^2_{L(H,H)}\Vert h_m\Vert ^2_{L^2(0,m;H)} \le 2\,m^{-1} \lambda ^{-1} \Vert Q\Vert ^2_{L(H,H)}. $$

Therefore, by (3.49) we see that there exists $m_0>0$ such that

$$\begin{aligned} v_m (m) \notin B_H\Big (0,{\frac{1}{2}} \delta \Big ), \quad \forall \ m\ge m_0. \end{aligned}$$

(3.50)

On the other hand, since ${{\mathcal {A}}}=\{0\}$ is the global attractor of the deterministic equation and $\Vert u_{0,m}\Vert \le R$ for all $m\in \mathbb {N}$, we must have $v_m (m) \in B_H(0,{\frac{1}{2}} \delta )$ when m is sufficiently large, which is in contradiction with (3.50) and thus completes the proof. $\square$

Lemma 3.13

If (2.1)–(2.3) and (2.7) are valid, then we have

$$\begin{aligned} \lim _{R\rightarrow \infty } \limsup _{{\varepsilon }\rightarrow 0} {\varepsilon }\ln \mu ^{\varepsilon }\big (\overline{B}_H^c (0, R) \big ) =-\infty , \end{aligned}$$

(3.51)

and

$$\begin{aligned} \lim _{R\rightarrow \infty } \limsup _{{\varepsilon }\rightarrow 0} {\varepsilon }\ln \mu ^{\varepsilon }\big (\overline{B}_V^c (0, R) \big ) =-\infty , \end{aligned}$$

(3.52)

where $\overline{B}_H^c (0, R)=H{\setminus } \overline{B}_H(0,R)$ and $\overline{B}_V^c (0, R)=V{\setminus } \overline{B}_V(0,R)$.

Proof

By Itô’s formula we get from (1.1) that

$$ d\Vert u^{\varepsilon }(t)\Vert ^2 + 2\big ( \lambda \Vert u^{\varepsilon }(t) \Vert ^2 + \Vert \nabla u^{\varepsilon }(t) \Vert ^2\big )dt $$

$$\begin{aligned} =-2 \int _{\mathbb {R}^n} F(u^{\varepsilon }(t,x)) u^{\varepsilon }(t,x) dx dt + {\varepsilon }\Vert Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)}dt + 2\sqrt{{\varepsilon }} (u^{\varepsilon }, Q dW). \end{aligned}$$

(3.53)

For every $\delta _1>0$, $\gamma >0$ and ${\varepsilon }>0$, let $\xi (t,s) = e^{\delta _1 t} e^{{\frac{\gamma }{{\varepsilon }}} s}$. Then by (3.53) we obtain

$$\begin{aligned}{} & {} d\xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \le \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \left( \delta _1 +{\frac{2\gamma ^2}{{\varepsilon }}}\Vert Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)} \Vert u^{\varepsilon }(t) \Vert ^2 \right. \nonumber \\{} & {} \quad \left. +\gamma \Vert Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt -{\frac{2\gamma }{{\varepsilon }}} \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \left( \lambda \Vert u^{\varepsilon }(t) \Vert ^2 +\Vert \nabla u^{\varepsilon }(t) \Vert ^2 \right. \nonumber \\{} & {} \quad \left. + \int _{\mathbb {R}^n} F(u^{\varepsilon }(t,x)) u^{\varepsilon }(t,x) dx \right) dt +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) (u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.54)

By (2.1) and (3.54) we get

$$ d\xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \le \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \left( \delta _1 +\gamma \Vert Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt $$

$$\begin{aligned} -{\frac{2\gamma }{{\varepsilon }}} \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \big (\lambda - \gamma \Vert Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \big ) \Vert u^{\varepsilon }(t) \Vert ^2 dt +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) (u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.55)

Let $\delta _1={\frac{1}{2}}\lambda$, and choose $\gamma$ small enough such that

$$ \gamma \Vert Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \le {\frac{1}{2}} \lambda . $$

Then by (3.55) we get

$$\begin{aligned} d\xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \le \lambda \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t) \Vert ^2 \right) dt +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert u^{\varepsilon }(t)\Vert ^2\big ) (u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.56)

It follows from (3.56) that

$$ \mathbb {E}\left( e^{{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(0)\Vert ^2} \right) + \lambda \mathbb {E}\left( \int _0^t e^{{\frac{1}{2}} \lambda s} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(s)\Vert ^2} \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(s) \Vert ^2 \right) ds \right) . $$

Note that $e^r (1-r) \le 1$ for all $r\ge 0$. We get that

$$\mathbb {E}\left( e^{{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(0)\Vert ^2} \right) + \lambda \int _0^t e^{{\frac{1}{2}} \lambda s} ds \le e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(0)\Vert ^2} + 2 e^{{\frac{1}{2}} \lambda t}, $$

which yields that

$$\begin{aligned} \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t)\Vert ^2} \right) \le e^{-{\frac{1}{2}} \lambda t}\mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(0)\Vert ^2}\right) + 2. \end{aligned}$$

(3.57)

Then for every fixed $u_0 \in H$, $R>0$ and $t>0$, we get

$$ P \left( u^{\varepsilon }(t, u_0 ) \notin \overline{B} _H(0, R) \right) = P \left( \Vert u^{\varepsilon }(t, u_0) \Vert>R \right) = P \left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t, u_0) \Vert ^2} > e^{ {\frac{\gamma }{{\varepsilon }}}R ^2 } \right) $$

$$\begin{aligned} \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 } \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t, u_0) \Vert ^2} \right) \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 } \left( e^{-{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2} + 2 \right) . \end{aligned}$$

(3.58)

By the invariance of $\mu ^{\varepsilon }$ we have for all $t\ge 0$,

$$ \mu ^{\varepsilon }\big ( \overline{B}^c_H(0, R) \big ) = \int _H P \left( u^{\varepsilon }(t, u_0) \notin \overline{B} _H(0, R) \right) d\mu ^{\varepsilon }(u_0) $$

which along with the Fatou’s lemma and (3.58) implies that

$$ \mu ^{\varepsilon }\big ( \overline{B} ^c_H(0, R) \big ) \le \int _H \limsup _{t\rightarrow \infty } P \left( u^{\varepsilon }(t, u_0) \notin \overline{B} _H(0, R) \right) d\mu ^{\varepsilon }(u_0) $$

$$ \le \int _H e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2} + 2 \right) d\mu ^{\varepsilon }(u_0) = 2 e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 }, $$

which yields (3.51).

Next, we prove (3.52). By Itô’s formula we get from (1.1) that

$$ d\Vert \nabla u^{\varepsilon }(t)\Vert ^2 + 2( \lambda \Vert \nabla u^{\varepsilon }(t) \Vert ^2 + \Vert \Delta u^{\varepsilon }(t) \Vert ^2)dt $$

$$\begin{aligned} =2 \int _{\mathbb {R}^n} F( u^{\varepsilon }(t,x)) \Delta u^{\varepsilon }(t,x) dx dt + {\varepsilon }\Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)}dt - 2\sqrt{{\varepsilon }} (\Delta u^{\varepsilon }, Q dW). \end{aligned}$$

(3.59)

Then by (3.59) we obtain

$$ d\xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) $$

$$ \le \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \left( \delta _1 +{\frac{2\gamma ^2}{{\varepsilon }}}\Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)} \Vert \nabla u^{\varepsilon }(t) \Vert ^2 +\gamma \Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt $$

$$ -{\frac{2\gamma }{{\varepsilon }}} \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \left( \lambda \Vert \nabla u^{\varepsilon }(t) \Vert ^2 +\Vert \Delta u^{\varepsilon }(t) \Vert ^2 - \int _{\mathbb {R}^n} ( F( u^{\varepsilon }(t,x) ) \Delta u^{\varepsilon }(t,x) dx \right) dt $$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) (-\Delta u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.60)

By (3.12) and (3.60) we get

$$ d\xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \le \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \left( \delta _1 +\gamma \Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt $$

$$ -{\frac{2\gamma }{{\varepsilon }}} \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \left( \lambda - \gamma \Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) \Vert \nabla u^{\varepsilon }(t) \Vert ^2 dt $$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) (-\Delta u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.61)

Let $\delta _1={\frac{1}{2}}\lambda$, and choose $\gamma$ small enough such that

$$ \gamma \Vert \nabla Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \le {\frac{1}{2}}\lambda . $$

Then by (3.61) we get

$$ d\xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \le \lambda \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t) \Vert ^2 \right) dt $$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi \big (t, \Vert \nabla u^{\varepsilon }(t)\Vert ^2\big ) (-\Delta u^{\varepsilon }(t), QdW). \end{aligned}$$

(3.62)

It follows that

$$ \mathbb {E}\left( e^{{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(0)\Vert ^2} \right) $$

$$ + \lambda \mathbb {E}\left( \int _0^t e^{{\frac{1}{2}} \lambda s} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(s)\Vert ^2} \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(s) \Vert ^2 \right) ds \right) . $$

By $e^r (1-r) \le 1$ for all $r\ge 0$, we get that

$$ \mathbb {E}\left( e^{{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(0)\Vert ^2} \right) + 2 e^{{\frac{1}{2}}\lambda t}, $$

which yields that

$$\begin{aligned} \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t)\Vert ^2} \right) \le e^{-{\frac{1}{2}} \lambda t}\mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(0)\Vert ^2}\right) + 2. \end{aligned}$$

(3.63)

Then for every fixed $u_0 \in V$, $R>0$ and $t>0$, we get from (3.63) that

$$ P \left( \nabla u^{\varepsilon }(t, u_0) \notin \overline{B} _H(0, R) \right) = P \left( \Vert \nabla u^{\varepsilon }(t,u_0) \Vert>R \right) = P \left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t,u_0 ) \Vert ^2} > e^{ {\frac{\gamma }{{\varepsilon }}}R ^2 } \right) $$

$$\begin{aligned} \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 } \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t,u_0) \Vert ^2} \right) \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 } \left( e^{-{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u_0 \Vert ^2} + 2 \right) . \end{aligned}$$

(3.64)

By the invariance of $\mu ^{\varepsilon }$ we have for all $t\ge 0$,

$$ \mu ^{\varepsilon }\big (\overline{B} ^c_V(0, R) \big ) = \int _H P \left( u^{\varepsilon }(t, u_0) \notin \overline{B}_V(0, R) \right) d\mu ^{\varepsilon }(u_0) $$

$$ = \int _H P \left( \Vert u^{\varepsilon }(t, u_0)\Vert ^2_V >R^2 \right) d\mu ^{\varepsilon }(u_0) $$

$$ \le \int _H P \left( \Vert u^{\varepsilon }(t, u_0)\Vert ^2> {\frac{1}{2}} R^2 \right) d\mu ^{\varepsilon }(u_0) + \int _H P \left( \Vert \nabla u^{\varepsilon }(t, u_0)\Vert ^2 >{\frac{1}{2}} R^2 \right) d\mu ^{\varepsilon }(u_0) $$

$$ = \int _H P \left( \Vert u^{\varepsilon }(t, u_0)\Vert \notin \overline{B}_H\Bigg (0, {\frac{1}{\sqrt{2}}} R\Bigg ) \right) d\mu ^{\varepsilon }(u_0) $$

$$ + \int _H P \left( \Vert \nabla u^{\varepsilon }(t, u_0)\Vert \notin \overline{B}_H\Bigg (0, {\frac{1}{\sqrt{2}}} R \Bigg ) \right) d\mu ^{\varepsilon }(u_0) $$

which along with the Fatou’s lemma, (3.58) and (3.64) implies that

$$ \mu ^{\varepsilon }\big ( \overline{B}^c_V(0, R) \big ) \le \int _H \limsup _{t\rightarrow \infty } P \left( \Vert u^{\varepsilon }(t, u_0)\Vert \notin \overline{B}_H\Bigg (0, {\frac{1}{\sqrt{2}}} R\Bigg ) \right) d\mu ^{\varepsilon }(u_0) $$

$$ + \int _H \limsup _{t\rightarrow \infty } P \left( \Vert \nabla u^{\varepsilon }(t, u_0)\Vert \notin \overline{B}_H\Bigg (0, {\frac{1}{\sqrt{2}}} R \Bigg ) \right) d\mu ^{\varepsilon }(u_0) $$

$$ \le \int _H e^{- {\frac{\gamma }{2{\varepsilon }}}R ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2} + 2 \right) d\mu ^{\varepsilon }(u_0) $$

$$ + \int _V e^{- {\frac{\gamma }{2{\varepsilon }}}R ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} \lambda t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u_0 \Vert ^2} + 2 \right) d\mu ^{\varepsilon }(u_0) = 2 e^{- {\frac{\gamma }{2 {\varepsilon }}}R ^2 }, $$

where we have used $\mu ^{\varepsilon }(V)=1$. This completes the proof. $\square$

For convenience, given $\delta >0$, denote by

$$ {\kappa _\delta } (x) = \left( 1 + |\delta x|^2 \right) ^{\frac{1}{2}}, \quad \forall \ x\in \mathbb {R}^n. $$

The next lemma is concerned with the exponential probability estimates in $L^2_{\kappa } (\mathbb {R}^n)$.

Lemma 3.14

Suppose (2.1)–(2.3) and (2.7) hold. Let $\gamma >0$ be a small number such that

$$\begin{aligned} \gamma \Vert {\kappa } Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} < {\frac{1}{2}} {\lambda }. \end{aligned}$$

(3.65)

Then there exists $\delta \in (0,1)$ such that for every $u_0 \in L^2_{\kappa } (\mathbb {R}^n)$, the solution $u^{\varepsilon }(\cdot , u_0)$ of (1.1) and (1.2) satisfies for all $t\ge 0$,

$$\begin{aligned} \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(t, u_0 )\Vert ^2} \right) \le e^{-{\frac{1}{2}} {\lambda } t}\mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa } u_0 \Vert ^2}\right) + 2. \end{aligned}$$

(3.66)

Proof

By (3.65) we find that there exists $\delta \in (0,1)$ such that

$$\begin{aligned} \delta ^2 + \gamma \Vert {\kappa } Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} < {\frac{1}{2}} {\lambda }. \end{aligned}$$

(3.67)

By Itô’s formula we get from (1.1) that

$$ d\Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2 + 2 \left( \lambda \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2 - ( \kappa _\delta ^2 u^{\varepsilon }(t), \Delta u^{\varepsilon }(t) \right) dt $$

$$\begin{aligned} =-2 \int _{\mathbb {R}^n} \kappa _\delta ^2 (x) F( u^{\varepsilon }(t,x)) u^{\varepsilon }(t,x) dx dt + {\varepsilon }\Vert {\kappa _\delta } Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)}dt + 2\sqrt{{\varepsilon }} ( \kappa _\delta ^2 u^{\varepsilon }, Q dW). \end{aligned}$$

(3.68)

Let $\delta _1={\frac{1}{2}} \lambda$, and $\xi (t,s) = e^{\delta _1 t} e^{{\frac{\gamma }{{\varepsilon }}} s}$. Then by (3.68) we obtain

$$\begin{aligned} d\xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \le \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \left( \delta _1 +{\frac{2\gamma ^2}{{\varepsilon }}}\Vert {\kappa _\delta } Q\Vert ^2_{{{\mathcal {L}}}_2 (H,H)} \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2 +\gamma \Vert {\kappa _\delta } Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt \end{aligned}$$

$$\begin{aligned} -{\frac{2\gamma }{{\varepsilon }}} \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \left( \lambda \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2 - ( \kappa _\delta ^2 u^{\varepsilon }(t), \Delta u^{\varepsilon }(t) ) + \int _{\mathbb {R}^n} \kappa _\delta ^2 (x) F(u^{\varepsilon }(t,x)) u^{\varepsilon }(t,x) dx \right) dt \end{aligned}$$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) ({\kappa _\delta } u^{\varepsilon }(t), {\kappa _\delta } QdW). \end{aligned}$$

(3.69)

By (2.1) we have

$$\begin{aligned} -\int _{\mathbb {R}^n}\kappa _\delta ^2 F(u^{\varepsilon }(t,x)) u^{\varepsilon }(t,x) dx \le 0. \end{aligned}$$

(3.70)

Note that

$$ ( \kappa _\delta ^2 u^{\varepsilon }(t), \Delta u^{\varepsilon }(t) ) =-\int _{\mathbb {R}^n} \kappa _\delta ^2(x) |\nabla u^{\varepsilon }(t, x)|^2dx -2\delta ^2 \int _{\mathbb {R}^n} u^{\varepsilon }(t, x) \ x\cdot \nabla u^{\varepsilon }(t, x) dx $$

$$ \le -\int _{\mathbb {R}^n} \kappa _\delta ^2(x) |\nabla u^{\varepsilon }(x)|^2dx +\delta ^2 \Vert u^{\varepsilon }(t) \Vert ^2 + \int _{\mathbb {R}^n} | \delta x|^2 | \nabla u^{\varepsilon }(x) |^2 dx $$

$$\begin{aligned} = -\int _{\mathbb {R}^n} |\nabla u^{\varepsilon }(x)|^2dx +\delta ^2 \Vert u^{\varepsilon }(t) \Vert ^2 \le \delta ^2 \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2. \end{aligned}$$

(3.71)

It follows from (3.69)–(3.71) that

$$ d\xi (t,{\kappa _\delta } \Vert u^{\varepsilon }(t)\Vert ^2) \le \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \left( \delta _1 +\gamma \Vert {\kappa _\delta } Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) dt $$

$$ -{\frac{2\gamma }{{\varepsilon }}} \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \left( \Big (\lambda -\delta ^2 - \gamma \Vert {\kappa _\delta } Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \Big ) \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2 \right) dt $$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) ({\kappa _\delta } u^{\varepsilon }(t), {\kappa _\delta } QdW). \end{aligned}$$

(3.72)

By (3.67) and (3.72) we get

$$ d\xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \le {\lambda } \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(t) \Vert ^2 \right) dt $$

$$\begin{aligned} +{\frac{2\gamma }{\sqrt{{\varepsilon }}}} \xi (t, \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2) ({\kappa _\delta } u^{\varepsilon }(t), {\kappa _\delta } QdW). \end{aligned}$$

(3.73)

It follows from (3.73) that

$$\begin{aligned} \mathbb {E}\left( e^{{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(0)\Vert ^2} \right) + {\lambda } \mathbb {E}\left( \int _0^t e^{{\frac{1}{2}} {\lambda } s} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(s)\Vert ^2} \left( 1- {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(s) \Vert ^2 \right) ds \right) . \end{aligned}$$

Note that $e^r (1-r) \le 1$ for all $r\ge 0$. We get that

$$\mathbb {E}\left( e^{{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2} \right) \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(0)\Vert ^2} \right) + {\lambda } \int _0^t e^{{\frac{1}{2}} {\lambda }s} ds \le \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(0)\Vert ^2} \right) + 2 e^{{\frac{1}{2}} {\lambda } t}, $$

which yields that

$$ \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(t)\Vert ^2} \right) \le e^{-{\frac{1}{2}} {\lambda } t}\mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa _\delta } u^{\varepsilon }(0)\Vert ^2}\right) + 2, $$

as desired. $\square$

By the uniform estimates given by Lemma 3.14, one can check that the invariant measures of (1.1) and (1.2) are supported on $L^2_\kappa (\mathbb {R}^n)$.

Lemma 3.15

Suppose (2.1)–(2.3) and (2.7) hold. Given $R>0$, let

$$\begin{aligned} {{\mathcal {Z}}}_R= \overline{B}_{L^2_{\kappa } (\mathbb {R}^n)} (0,R) \cap \overline{B} _V(0,R). \end{aligned}$$

(3.74)

Then we have

$$ \lim _{R\rightarrow \infty } \limsup _{{\varepsilon }\rightarrow 0} {\varepsilon }\ln \mu ^{\varepsilon }(H{\setminus } {{\mathcal {Z}}}_R) =-\infty . $$

Proof

Let $\gamma >0$ be a small number such that

$$ \gamma \left( \Vert Q\Vert ^2_{{{\mathcal {L}}}_2(H, V)} + \Vert \kappa Q\Vert ^2_{{{\mathcal {L}}}_2(H,H)} \right) < {\frac{1}{2}} \lambda . $$

Then by Lemma 3.14 we see that there exists $\delta \in (0,1)$ such that (3.66) holds.

For every fixed $u_0 \in {{\mathcal {Z}}}_R$ and $t>0$, we get

$$ P \left( u^{\varepsilon }(t,u_0) \in H{\setminus } {{\mathcal {Z}}}_R \right) \le P \left( u^{\varepsilon }(t,u_0) \in H{\setminus } \overline{B}_{L^2_{\kappa } (\mathbb {R}^n)} (0,R) \right) + P \left( u^{\varepsilon }(t,u_0) \in H{\setminus } \overline{B} _V(0,R) \right) $$

$$ = P \left( \Vert u^{\varepsilon }(t,u_0) \Vert ^2_{L^2_{\kappa } (\mathbb {R}^n)}> R^2 \right) + P \left( \Vert u^{\varepsilon }(t,u_0)\Vert _V^2 > R ^2 \right) $$

$$ \le P \left( \Vert \kappa _\delta u^{\varepsilon }(t,u_0) \Vert ^2> R^2\delta ^2 \right) + P \left( \Vert u^{\varepsilon }(t,u_0)\Vert ^2>{\frac{1}{2}} R ^2 \right) + P \left( \Vert \nabla u^{\varepsilon }(t,u_0)\Vert ^2 >{\frac{1}{2}} R ^2 \right) $$

$$ = P \left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \kappa _\delta u^{\varepsilon }(t, u_0 ) \Vert ^2 }> e^{ {\frac{\gamma }{{\varepsilon }}}R ^2 \delta ^2 } \right) + P \left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u^{\varepsilon }(t, u_0) \Vert ^2 }> e^{ {\frac{\gamma }{2 {\varepsilon }}}R ^2 } \right) + P \left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u^{\varepsilon }(t, u_0 ) \Vert ^2 } > e^{ {\frac{\gamma }{2 {\varepsilon }}}R ^2 } \right) $$

$$ \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 \delta ^2 } \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \kappa _\delta u^{\varepsilon }(t, u_0 ) \Vert ^2 } \right) + e^{- {\frac{\gamma }{2 {\varepsilon }}}R ^2 } \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }} } \Vert u^{\varepsilon }(t, u_0) \Vert ^2 } \right) + e^{- {\frac{\gamma }{2{\varepsilon }}}R ^2 } \mathbb {E}\left( e^{ {\frac{\gamma }{{\varepsilon }} } \Vert \nabla u^{\varepsilon }(t, u_0) \Vert ^2 } \right) . $$

Then by (3.57), (3.63) and (3.66) we get

$$ P \left( u^{\varepsilon }(t,u_0) \in H{\setminus } {{\mathcal {Z}}}_R \right) \le e^{- {\frac{\gamma }{{\varepsilon }}}R ^2 \delta ^2 } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa } u_0 \Vert ^2} + 2 \right) $$

$$\begin{aligned} + e^{- {\frac{\gamma }{2{\varepsilon }}}R ^2 } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2} + 2 \right) + e^{- {\frac{\gamma }{2{\varepsilon }}}R ^2 } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert \nabla u_0 \Vert ^2 } + 2 \right) . \end{aligned}$$

(3.75)

By the invariance of $\mu ^{\varepsilon }$ we have for all $t\ge 0$,

$$ \mu ^{\varepsilon }(H{\setminus } {{\mathcal {Z}}}_R ) = \int _H P \left( u^{\varepsilon }(t, u_0) \in H{\setminus } {{\mathcal {Z}}}_R \right) d\mu ^{\varepsilon }(u_0) $$

which along with the Fatou theorem and (3.75) implies that

$$ \mu ^{\varepsilon }(H{\setminus } {{\mathcal {Z}}}_R ) \le \int _H \limsup _{t\rightarrow \infty } P \left( u^{\varepsilon }(t, u_0) \in H{\setminus } {{\mathcal {Z}}}_R \right) d\mu ^{\varepsilon }(u_0) $$

$$ \le \int _H e^{- {\frac{\gamma }{2 {\varepsilon }} }R ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2} + 2 \right) d\mu ^{\varepsilon }(u_0) $$

$$ + \int _{L^2_{\kappa } (\mathbb {R}^n)} e^{- {\frac{\gamma }{ {\varepsilon }} }R ^2 \delta ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert {\kappa } u_0 \Vert ^2} + 2 \right) d\mu ^{\varepsilon }(u_0) $$

$$ + \int _V e^{- {\frac{\gamma }{2 {\varepsilon }} }R ^2 }\limsup _{t\rightarrow \infty } \left( e^{-{\frac{1}{2}} {\lambda } t} e^{ {\frac{\gamma }{{\varepsilon }}} \Vert u_0 \Vert ^2_V} + 2 \right) d\mu ^{\varepsilon }(u_0) = 2 e^{- {\frac{\gamma }{{\varepsilon }}}R ^2\delta ^2 } + 4 e^{- {\frac{\gamma }{2 {\varepsilon }}}R ^2 } \le 6 e^{- {\frac{\gamma }{2 {\varepsilon }}}R ^2 \delta ^2}, $$

where we have used the fact that $\mu ^{\varepsilon }$ is supported on V and $L^2_{\kappa }(\mathbb {R}^n)$. This completes the proof. $\square$

4 LDP Lower Bounds

We now start to prove Theorem 1.1 by first showing the LDP lower bound of the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$ of invariant measures of (1.1) and (1.2) as ${\varepsilon }\rightarrow 0$. More precisely, in what follows we prove: for every $u\in H$, $\delta _1>0$ and $\delta _2>0$, there exists ${\varepsilon }_0$ such that for all ${\varepsilon }\le {\varepsilon }_0$,

$$\begin{aligned} \mu ^{\varepsilon }(B_H (u,\delta _1)) \ge e^{ - {\frac{J (u) +\delta _2}{{\varepsilon }} } }. \end{aligned}$$

(4.1)

Note that (4.1) is evident if $J(u)=\infty$. If $J(u) <\infty$, then there exist $r>0$ and $h\in L^2(0,r; H)$ such that

$$\begin{aligned} {\frac{1}{2}} \int _0^r\Vert h(t) \Vert ^2 dt< J(u) + {\frac{1}{3}} \delta _2, \quad \text {and} \quad \Vert u_h(r, 0)- u\Vert <{\frac{1}{4}} \delta _1, \end{aligned}$$

(4.2)

where $u_h(\cdot , 0)$ is the solution of (3.1) and (3.2) with $u_0=0$. By Lemma 3.5 we see that $\Vert u_h(r, v) -u_h(r, 0)\Vert < {\frac{1}{4}} \delta _1$ for all $\Vert v\Vert <{\frac{1}{4}} \delta _1$, which along with (4.2) shows that

$$\begin{aligned} \Vert u_h (r, v) -u\Vert<{\frac{1}{2}} \delta _1, \quad \text {for all } \ v\in H \ \text {with} \ \Vert v\Vert <{\frac{1}{4}} \delta _1. \end{aligned}$$

(4.3)

Note that if $\Vert v\Vert <{\frac{1}{4}} \delta _1$, then by (4.3) we have

$$ \Vert u^{\varepsilon }(r, v)- u\Vert \le \Vert u^{\varepsilon }(r, v) -u_h(r,v)\Vert +\Vert u_h(r,v)- u\Vert < \Vert u^{\varepsilon }(r, v) -u_h(r,v)\Vert +{\frac{1}{2}} \delta _1, $$

and hence

$$\begin{aligned} P \left( \Vert u^{\varepsilon }(r, v)- u\Vert<\delta _1 \right) \ge P \left( \Vert u^{\varepsilon }(r, v)- u_h(r,v) \Vert < {\frac{1}{2}} \delta _1 \right) . \end{aligned}$$

(4.4)

By (4.2) we see that for all $v\in H$,

$$ I_{r,v} (u_h (\cdot , v)) < J(u) + {\frac{1}{3}}\delta _2, $$

which along with (3.33) for $s=J(u) + {\frac{1}{3}}\delta _2$, $T=r$, $u_0=v$ and $R= {\frac{1}{4}} \delta _1$ implies that for all $v\in H$ with $\Vert v \Vert <{\frac{1}{4}} \delta _1$,

$$\begin{aligned} P \left( \Vert u^{\varepsilon }(\cdot , v)- u_h(\cdot ,v) \Vert _{C([0,r], H)}< {\frac{1}{2}} \delta _1 \right) \ge e^{ -{\frac{I_{r,v} (u_h(\cdot , v)) +{\frac{1}{3}} \delta _2}{{\varepsilon }} } } \ge e^{ -{\frac{ J(u) +{\frac{2}{3}} \delta _2}{{\varepsilon }} } }. \end{aligned}$$

(4.5)

By (4.4) and (4.5) we get for all $v\in B_H(0, {\frac{1}{4}} \delta _1 )$,

$$\begin{aligned} P \left( u^{\varepsilon }(r, v) \in B_H(u, \delta _1) \right) \ge P \left( \Vert u^{\varepsilon }(\cdot , v)- u_h(\cdot ,v) \Vert _{C([0,r], H)}< {\frac{1}{2}} \delta _1 \right) \ge e^{ -{\frac{ J(u) +{\frac{2}{3}} \delta _2}{{\varepsilon }} } }. \end{aligned}$$

(4.6)

Since $\mu ^{\varepsilon }$ is an invariant measure, by (4.6) we obtain

$$ \mu ^{\varepsilon }(B_H(u, \delta _1)) = \int _H P (u^{\varepsilon }(r, v) \in B_H(u, \delta _1)) \mu ^{\varepsilon }(d v) $$

$$\begin{aligned} \ge \int _{B_H (0, {\frac{1}{4}} \delta _1)} P (u^{\varepsilon }(r, v) \in B_H(u, \delta _1)) \mu ^{\varepsilon }(d v) \ge \mu ^{\varepsilon }\Big (B_H\Big (0, {\frac{1}{4}} \delta _1\Big )\Big ) e^{ -{\frac{ J(u) +{\frac{2}{3}} \delta _2}{{\varepsilon }} } }. \end{aligned}$$

(4.7)

Since $\mu ^{\varepsilon }\rightarrow \delta _0$ weakly, we find that

$$ \liminf _{{\varepsilon }\rightarrow 0} \mu ^{\varepsilon }\Big (B_H\Big (0, {\frac{1}{4}} \delta _1\Big )\Big ) \ge 1 $$

and hence there exists ${\varepsilon }_0={\varepsilon }_0 (\delta _1)>0$ such that for all ${\varepsilon }\in (0, {\varepsilon }_0)$,

$$\begin{aligned} \mu ^{\varepsilon }\Big (B_H\Big (0, {\frac{1}{4}} \delta _1\Big )\Big ) \ge {\frac{1}{2}}. \end{aligned}$$

(4.8)

It follows from (4.7) and (4.8) that for all ${\varepsilon }\in (0, {\varepsilon }_0)$,

$$ \mu ^{\varepsilon }(B_H(u, \delta _1)) \ge {\frac{1}{2}} e^{ \frac{ \delta _2}{3 {\varepsilon }} } e^{ -{\frac{ J(u) + \delta _2}{{\varepsilon }} } }, $$

which implies (4.1) when ${\varepsilon }$ is sufficiently small, and thus completes the proof of the LDP lower bound for $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$.

5 LDP Upper Bounds

In this section, we prove the LDP upper bound for the family $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$; that is, we will show that for any positive numbers $\delta _1$, $\delta _2$ and l, there exists ${\varepsilon }_0>0$ such that for all ${\varepsilon }\le {\varepsilon }_0$,

$$\begin{aligned} \mu ^{\varepsilon }\big (H{\setminus } J^l_{\delta _1}\big ) \le e^{ -{\frac{l-\delta _2}{{\varepsilon }} } }, \end{aligned}$$

(5.1)

where $J^l_{\delta _1}$ is the open $\delta _1$-neighborhood of $J^l$.

By Lemma 3.11, there exists $\delta >0$ such that for all $t>0$,

$$\begin{aligned} \left\{ u(t):\ u\in C([0,t], H), \ u(0)\in B_H(0, \delta ), \ I_t(u) \le l- {\frac{1}{4}} \delta _2 \right\} \subseteq J^l_{ {\frac{1}{2}} \delta _1 }. \end{aligned}$$

(5.2)

By Lemma 3.15, there exist $R >0$ and ${\varepsilon }_1 >0$ such that for all ${\varepsilon }<{\varepsilon }_1$,

$$\begin{aligned} \mu ^{\varepsilon }(H{\setminus } {{\mathcal {Z}}}_R ) \le e^{-{\frac{l}{{\varepsilon }}}}, \end{aligned}$$

(5.3)

where ${{\mathcal {Z}}}_R$ is given by (3.74). By Proposition 3.10 we infer that ${{\mathcal {Z}}}_R$ is a compact subset of H.

On the other hand, for the chosen $R>0$ and $\delta >0$, by Lemma 3.12, we see that there exists $t_0>0$ such that

$$\begin{aligned} \widetilde{\delta } = \inf \left\{ I_{t_0, u_0} (u):\ u_0 \in \overline{B} _H(0,R) \ u\in C([0,t_0], H), \ u(0) =u_0, \ u(t_0) \notin B_H(0,\delta ) \right\} >0. \end{aligned}$$

(5.4)

Given $n\in \mathbb {N}$, define

$$\begin{aligned} {{\mathcal {Y}}}_n =\left\{ u\in C([0, n t_0], H): \ u(0) \in \overline{B} _H(0, R), \ u(kt_0)\in B_H^c (0,\delta )\cap \overline{B} _H (0, R), \ \forall \ k=1,\ldots , n \right\} , \end{aligned}$$

(5.5)

where $B^c_H(0,\delta )=H{\setminus } B_H(0,\delta )$. By (5.4) and (5.5) we infer that for all $n> l\widetilde{\delta }^{-1}$,

$$\begin{aligned} \inf \left\{ I_{nt_0, u_0} (u): \ u_0 \in \overline{B} _H(0, R), \ u\in {{\mathcal {Y}}}_n, \ u(0)=u_0 \right\} >l. \end{aligned}$$

(5.6)

Note that if $I_{nt_0, u_0} (u) =+\infty$ for all $u_0 \in \overline{B} _H(0, R)$ and $u\in {{\mathcal {Y}}}_n$, then the left-hand side of (5.6) is $+\infty$, and hence (5.6) is true in this case.

If there exist $u_0 \in \overline{B} _H(0, R)$ and $u\in {{\mathcal {Y}}}_n$ such that $I_{nt_0, u_0} (u) <+\infty$, then there exists $h\in L^2(0, nt_0; H)$ such that $u=u_h(\cdot , u_0)$ where $u_h$ is the solution of (3.1) and (3.2) with initial condition $u_0=u(0)$. In this case we have

$$\begin{aligned} {\frac{1}{2}} \int _0^{nt_0} \Vert h(s)\Vert ^2 ds ={\frac{1}{2}} \sum _{k=1}^n \int _{(k-1) t_0}^{kt_0} \Vert h(s) \Vert ^2 ds ={\frac{1}{2}} \sum _{k=1}^n \int _{ 0}^{ t_0} \Vert h_k(s) \Vert ^2 ds \end{aligned}$$

(5.7)

where $h_k( s) =h(s+ (k-1) t_0)$ for $s\in [0, t_0]$. Let $v_k (t) =u(t + (k-1) t_0)$ for $t\in [0, t_0]$. Then $v_k$ is the solution of (3.1) and (3.2) on $[0, t_0]$ with initial condition $v_k (0) =u((k-1) t_0)$ when h replaced by $h_k$. Since $u\in {{\mathcal {Y}}}_n$, by (5.5) we see that $v_k (0) =u((k-1) t_0)\in \overline{B} _H (0, R)$ and $v_k (t_0) =u(k t_0) \notin B_H(0,\delta )$ for all $k=1,\ldots , n$. Then by (5.4) we have

$$ I_{t_0, v_k (0)} (v_k) \ge \widetilde{\delta }, \quad \forall \ k=1,\ldots , n, $$

and hence

$$ {\frac{1}{2}}\int _0^{t_0} \Vert h_k (s) \Vert ^2 ds \ge I_{t_0, v_k (0)} (v_k) \ge \widetilde{\delta }, \quad \forall \ k=1,\ldots , n, $$

which along with (5.7) implies that

$$\begin{aligned} {\frac{1}{2}} \int _0^{nt_0} \Vert h(s)\Vert ^2 ds \ge n \widetilde{\delta }. \end{aligned}$$

(5.8)

Note that (5.8) is valid for all $h\in L^2(0,nt_0; H)$ if $u=u_h(\cdot , u_0)$, and thus

$$\begin{aligned} I_{nt_0, u_0} (u) \ge n \widetilde{\delta }, \quad \forall \ u_0\in \overline{B} _H(0, R), \ \ u\in {{\mathcal {Y}}}_n. \end{aligned}$$

(5.9)

Then (5.6) follows from (5.9) for all $n> l\widetilde{\delta }^{-1}$,

We now fix $n_0 \in \mathbb {N}$ such that $n_0 > l\widetilde{\delta }^{-1}$. Then by (5.6) we have

$$\begin{aligned} \inf \left\{ I_{n_0 t_0, u_0} (u): \ u_0 \in \overline{B} _H(0, R), \ u\in {{\mathcal {Y}}}_{n_0}, \ u(0)=u_0 \right\} >l. \end{aligned}$$

(5.10)

Since ${{\mathcal {Y}}}_{n_0}$ is a closed subset of $C([0, n_0 t_0], H)$ and ${{\mathcal {Z}}}_R$ is a compact subset of H, by (5.10) and Proposition 3.8 we find that there exists ${\varepsilon }_2 \in (0, {\varepsilon }_1)$ such that for all ${\varepsilon }\le {\varepsilon }_2$,

$$\begin{aligned} \sup _{v\in {{\mathcal {Z}}}_R } P \left( u^{\varepsilon }(\cdot , v)\in {{\mathcal {Y}}}_{n_0} \right) \le e^{- {\frac{l}{{\varepsilon }} }}. \end{aligned}$$

(5.11)

Let $\widetilde{t} =n_0t_0 + t_0$. By the invariance of $\mu ^{\varepsilon }$ we get

$$ \mu ^{\varepsilon }\big (H{\setminus } J^l_{\delta _1}\big ) =\int _{H} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1} \big ) \mu ^{\varepsilon }(d v) $$

$$ =\int _{H{\setminus } {{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1} \big ) \mu ^{\varepsilon }(d v) + \int _{{{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1} \big ) \mu ^{\varepsilon }(d v) $$

$$ =\int _{H{\setminus } {{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1} \big ) \mu ^{\varepsilon }(d v) $$

$$ + \int _{{{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \ u^{\varepsilon }(\cdot , v) \in {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v) $$

$$ + \int _{{{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v) $$

$$ \le \mu ^{\varepsilon }( {H{\setminus } {{\mathcal {Z}}}_R}) + \int _{{{\mathcal {Z}}}_R} P \big ( \ u^{\varepsilon }(\cdot , v) \in {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v) $$

$$ + \int _{{{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v) $$

$$ \le \mu ^{\varepsilon }( {H{\setminus } {{\mathcal {Z}}}_R}) + \sup _{v\in {{\mathcal {Z}}}_R } P \big ( u^{\varepsilon }(\cdot , v) \in {{\mathcal {Y}}}_{n_0} \big ) $$

$$ + \int _{ {{\mathcal {Z}}}_R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v) $$

which along with (5.3) and (5.11) shows that for all ${\varepsilon }\le {\varepsilon }_2$,

$$\begin{aligned} \mu ^{\varepsilon }\big (H{\setminus } J^l_{\delta _1}\big ) \le 2 e^{- {\frac{l}{{\varepsilon }}}} + \int _{\Vert v\Vert \le R} P \big (u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \big ) \mu ^{\varepsilon }(d v). \end{aligned}$$

(5.12)

For the last term on the right-hand side of(5.12), by (5.5) we have

$$ \int _{\Vert v\Vert \le R} P\left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \right) \ \mu ^{\varepsilon }(d v) $$

$$ \le \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(kt_0, v) \in B_H(0,\delta ) \right) \ \mu ^{\varepsilon }(d v) $$

$$\begin{aligned} + \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \Vert u^{\varepsilon }(kt_0, v) \Vert >R \right) \ \mu ^{\varepsilon }(d v). \end{aligned}$$

(5.13)

For the second term on the right-hand side of (5.13), by (5.3) and the invariance of $\mu ^{\varepsilon }$, we get

$$ \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ \Vert u^{\varepsilon }(kt_0, v) \Vert >R \right) \ \mu ^{\varepsilon }(d v) $$

$$ \le \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( \Vert u^{\varepsilon }(kt_0, v) \Vert>R \right) \ \mu ^{\varepsilon }(d v) \le \sum _{k=1}^{n_0} \int _{H} P \left( \Vert u^{\varepsilon }(kt_0, v) \Vert >R \right) \ \mu ^{\varepsilon }(d v) $$

$$\begin{aligned} =n_0 \mu ^{\varepsilon }\left( \{ u\in H: \ \Vert u\Vert >R\} \right) \le n_0 \mu ^{\varepsilon }(H{\setminus } {{\mathcal {Z}}}_R) \le n_0 e^{ -{\frac{l}{{\varepsilon }}}}. \end{aligned}$$

(5.14)

Let $P(0, u_0; t, \cdot )$ be the transition probability function of $u^{\varepsilon }(t, u_0)$ with initial value $u_0$ at initial time 0. Then for the first term on the right-hand side of (5.13), by the Markov property, we get

$$ \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(kt_0, v) \in B_H(0,\delta ) \right) \ \mu ^{\varepsilon }(d v) $$

$$ = \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} \left( \int _{z\in B_H(0,\delta )} P \left( u^{\varepsilon }(\widetilde{t} -kt_0, z ) \notin J^l_{\delta _1} \right) P(0,v; kt_0, dz) \right) \ \mu ^{\varepsilon }(dv) $$

$$\begin{aligned} \le \sum _{k=1}^{n_0} \left( \sup _{z\in B_H(0,\delta )}P \left( u^{\varepsilon }(\widetilde{t} -kt_0, z ) \notin J^l_{\delta _1} \right) \right) . \end{aligned}$$

(5.15)

By (5.2) we see that

$$\begin{aligned} \left\{ u( \widetilde{t} -kt_0):\ u\in C([0, \widetilde{t} -kt_0], H), \ u(0)\in B_H(0,\delta ), \ I_{\widetilde{t} -kt_0 }(u) \le l- {\frac{1}{4}} \delta _2 \right\} \subseteq J^l_{ {\frac{1}{2}} \delta _1 }. \end{aligned}$$

(5.16)

By (5.16) we find that for every $z\in B_H(0,\delta )$,

$$\begin{aligned} \left\{ \omega \in \Omega : \ u^{\varepsilon }(\cdot , z ) \in {{\mathcal {N}}}\left( I^{l-{\frac{1}{4}} \delta _2}_{\widetilde{t} -kt_0, z },\ {\frac{1}{2}} \delta _1 \right) \right\} \subseteq \left\{ \omega \in \Omega : \ u^{\varepsilon }(\widetilde{t} -kt_0, z ) \in J^l_{\delta _1} \right\} . \end{aligned}$$

(5.17)

Indeed, if $u^{\varepsilon }(\cdot , z ) \in {{\mathcal {N}}}\left( I^{l-{\frac{1}{4}} \delta _2}_{\widetilde{t} -kt_0, z },\ {\frac{1}{2}} \delta _1 \right)$, then there exists $\phi \in I^{l-{\frac{1}{4}} \delta _2}_{\widetilde{t} -kt_0, z }$ such that

$$ \sup _{t\in [0, \widetilde{t} -kt_0] } \Vert u^{\varepsilon }(t, z ) -\phi (t)\Vert <{\frac{1}{2}} \delta _1, $$

and hence by (5.16)

$$ \textrm{dist}_{H} \left( u^{\varepsilon }(\widetilde{t} -kt_0, z), \ J^l \right) \le \Vert u^{\varepsilon }(\widetilde{t} -kt_0, z) -\phi (\widetilde{t} -kt_0)\Vert + \textrm{dist}_{H} \left( \phi (\widetilde{t} -kt_0), \ J^l \right) < \delta _1, $$

which gives (5.17). By (5.17) we see that

$$ P\left( u^{\varepsilon }(\widetilde{t} -kt_0, z) \notin J^l_{\delta _1} \right) \le P \left( u^{\varepsilon }(\cdot , z ) \notin {{\mathcal {N}}}\left( I^{l-{\frac{1}{4}} \delta _2}_{\widetilde{t} -kt_0, z },\ {\frac{1}{2}} \delta _1 \right) \right) $$

which together with Proposition 3.6 implies that there exists ${\varepsilon }_3\in (0, {\varepsilon }_2)$ such that for all ${\varepsilon }\le {\varepsilon }_3$,

$$\begin{aligned} \sup _{z\in {{\mathcal {A}}}_\delta } P\left( u^{\varepsilon }(\widetilde{t} -kt_0, z) \notin J^l_{\delta _1} \right) \le e^{ -{\frac{1}{{\varepsilon }}} (l-{\frac{1}{2}} \delta _2) }. \end{aligned}$$

(5.18)

By (5.15) and (5.18) we get that for all ${\varepsilon }\le {\varepsilon }_3$,

$$\begin{aligned} \sum _{k=1}^{n_0} \int _{\Vert v\Vert \le R} P \left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(kt_0, v) \in B_H(0,\delta ) \right) \ \mu ^{\varepsilon }(d v) \le n_0 e^{ -{\frac{1}{{\varepsilon }}} (l-{\frac{1}{2}} \delta _2) }. \end{aligned}$$

(5.19)

By (5.13), (5.14) and (5.19) we get for all ${\varepsilon }\le {\varepsilon }_3$,

$$\begin{aligned} \int _{\Vert v\Vert \le R} P\left( u^{\varepsilon }(\widetilde{t}, v) \notin J^l_{\delta _1}, \ u^{\varepsilon }(\cdot , v) \notin {{\mathcal {Y}}}_{n_0} \right) \ \mu ^{\varepsilon }(d v) \le 2n_0 e^{- {\frac{l -{\frac{1}{2}}\delta _2 }{{\varepsilon }} }}. \end{aligned}$$

(5.20)

By (5.12) and (5.20) we get for all ${\varepsilon }\le {\varepsilon }_3$,

$$ \mu ^{\varepsilon }\big (H{\setminus } J^l_{\delta _1}\big ) \le 2 e^{- {\frac{l}{{\varepsilon }}}} + 2n_0 e^{- {\frac{l -{\frac{1}{2}}\delta _2 }{{\varepsilon }} }} \le \left( 2 e^{- {\frac{\delta _2}{{\varepsilon }} }} + 2n_0 e^{- {\frac{\delta _2}{2{\varepsilon }} }} \right) e^{- {\frac{l-\delta _2}{{\varepsilon }} }} $$

which implies (5.1) when ${\varepsilon }$ is sufficiently small, and thus the LDP upper bound for $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$ follows.

By the LDP lower and upper bounds of $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$, we immediately obtain the LDP of $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$ in $L^2(\mathbb {R}^n)$, which completes the proof of Theorem 1.1.

Remark 5.1

Recall that the rate function J for $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$ is given by (3.36). On the other hand, following the idea of [5, 20], we can also define a rate function $\widetilde{J}$ by, for every $v\in H$,

$$ \widetilde{J} (v) =\inf \left\{ I_{r,0} (u): r>0, u\in C([0,r], H), u(0)=0, u(r)=v \right\} . $$

One can verify that $\widetilde{J}$ is also a good rate function for $\{\mu ^{\varepsilon }\}_{{\varepsilon }\in (0,1)}$. Then by the uniqueness of rate functions, we have $J=\widetilde{J}$.

Remark 5.2

We have proved Theorem 1.1 on the LDP of invariant measures of (1.1) and (1.2) under the conditions (2.1)–(2.3) and (2.7). Actually, Theorem 1.1 is valid for more general nonlinear function F. Let $F (u) = F_1(u) +F_2(u)$ for all $u\in \mathbb {R}$, where $F_1: \mathbb {R}\rightarrow \mathbb {R}$ satisfies conditions (2.1)–(2.3), and $F_2: \mathbb {R}\rightarrow \mathbb {R}$ is a globally Lipschitz function with Lipschitz constant $L_{F_2}$. If $F_1(0)=0$, $F_2(0)=0$ and $L_{F_2} < \lambda$, then one can verify that Theorem 1.1 is still valid after minor modifications in the proof.