# Uniform large deviations and metastability of random dynamical systems

Jifa<EMAIL_ADDRESS>Jian
<EMAIL_ADDRESS>Jianliang
<EMAIL_ADDRESS>Tusheng
<EMAIL_ADDRESS>
1\. Mathematics and Science College, Shanghai Normal University, Shanghai
200234, China
2\. School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
3\. School of Mathematical Sciences, University of Science and Technology of
China,
Hefei 230026, China.
4\. Department of Mathematics, University of Manchester, Oxford Road,
Manchester, M13 9PL, UK

Abstract: In this paper, we first provide a criterion on uniform large
deviation principles (ULDP) of stochastic differential equations under
Lyapunov conditions on the coefficients, which can be applied to stochastic
systems with coefficients of polynomial growth and possible degenerate driving
noises. In the second part, using the ULDP criterion we preclude the
concentration of limiting measures of invariant measures of stochastic
dynamical systems on repellers and acyclic saddle chains and extend Freidlin
and Wentzell’s asymptotics theorem to stochastic systems with unbounded
coefficients. Of particular interest, we determine the limiting measures of
the invariant measures of the famous stochastic van der Pol equation and van
der Pol Duffing equation whose noises are naturally degenerate. We also
construct two examples to match the global phase portraits of Freidlin and
Wentzell’s unperturbed systems and to explicitly compute their transition
difficulty matrices. Other applications include stochastic May-Leonard system
and random systems with infinitely many equivalent classes.

Keywords: Uniform large deviations; Lyapunov conditions; invariant measure;
asymptotic measure; stochastic van der Pol equation

AMS Subject Classification: 60B10; 60F10; 60H10; 37A50, 37C70.

##  1 Introduction

Let $b$ be a measurable vector field on $\mathbb{R}^{d}$. Consider the
dynamical system

$dX(t)=b(X(t))\,dt,\quad t\geq 0,\quad X(0)=x\in\mathbb{R}^{d},$ (1.1)

and its random perturbation

$dX^{\varepsilon}(t)=b(X^{\varepsilon}(t))\,dt+\sqrt{{\varepsilon}}\sigma(X^{\varepsilon}(t))dB(t),\quad
t\geq 0,\quad X^{\varepsilon}(0)=x\in\mathbb{R}^{d},$ (1.2)

where $\\{B(t)\\}_{t\geq 0}$ is an _m_ -dimensional Brownian motion on a
complete probability space $(\Omega,\mathcal{F},\\{{\mathcal{F}}_{t}\\}_{t\geq
0},\mathbb{P})$ with a filtration $\\{{\mathcal{F}}_{t}\\}_{t\geq 0}$
satisfying the usual conditions,
$\sigma(\cdot):\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}\otimes\mathbb{R}^{m}$
is a measurable mapping, and ${\varepsilon}$ is a strictly positive constant.

The deterministic system (1.1) usually has several invariant measures. It is
well-known that the noise often stabilizes the deterministic system, which
means that the perturbed system (1.2) has a unique invariant measure, denoted
by $\mu^{\varepsilon}$. In order to understand how the random perturbations
influence the behavior of the deterministic dynamical system, one of the
fundamental problems is to identify which invariant measure of (1.1)
$\mu^{\varepsilon}$ converges to as $\varepsilon$ tends to 0, which is the
famous asymptotic measure problem proposed by Kolmogorov, see [28].

Except for potential systems (e.g., [14]) and monostable systems (e.g., [16,
9, 36]), it is extremely hard (if not impossible) to describe precisely the
limit measure $\mu$ of the family $\mu^{\varepsilon}$. Instead, people try to
identify where the support of the limiting measure $\mu$ is. This is equally
non-trivial. The existing results [34, 9, 14, 36] state that the limiting
measures are concentrated on stable limit sets of (1.1) under the assumptions
that coefficients are bounded, the driving noise is non-degenerate (see, e.g.,
(${\bf A}_{1}$) or (${\bf A}_{2}$), and (${\bf A}_{4}$) below). However, many
important models from physics are nonlinear systems with polynomial
coefficients, e.g., the Lorentz system [20]; the Rössler system [25]; the Chua
circuit [7] and the FitzHugh-Nagumo system [18] et al, and also all randomly
forced damping Hamiltonian systems originating from Langevin dynamics
naturally have degenerate noise, which include the well-known stochastic van
der Pol equation and van der Pol Duffing equation. The purpose of this paper
is to provide a framework and general results on the support of limiting
measures under mild conditions, which in particular can be applied to the
above interesting models.

We now give some details about the existing results which are relevant to the
current paper. Freidlin and Wentzell [34, 9] introduced the notion of a
uniform large deviation principle (ULDP) and revealed the close connections to
the long-time influence of small random perturbations of deterministic
dynamical systems, especially to the support of the limiting measures $\mu$.
Let us recall the conditions introduced in [9].
(${\bf A}_{1}$). The coefficients $b$ and $\sigma$ are bounded, locally
Lipschitz continuous and uniformly continuous on $\mathbb{R}^{d}$ and there
exists a positive constant $\lambda>0$ such that

$\beta^{*}\sigma(x)\sigma^{*}(x)\beta\geq\lambda|\beta|^{2}\ {\rm for\ all}\
\beta,x\in\mathbb{R}^{d}.$

(${\bf A}_{2}$). $d=m$, $\sigma$ is an identity matrix and the drift $b$ is
globally Lipschitz continuous. Under the assumption (${\bf A}_{1}$) or (${\bf
A}_{2}$), Freidlin and Wentzell proved that the family of the laws of the
solution processes $\\{X^{\varepsilon,x}\\}$ of stochastic differential
equations (SDEs) (1.2) satisfies a ULDP with respect to the initial value
$x\in\mathbb{R}^{d}$; see [9, Theorem 3.1, p.135] etc.. Since then, when
people apply ULDP to study the metastable behavior of stochastic dynamical
systems, they usually impose the condition (${\bf A}_{1}$) or (${\bf A}_{2}$)
(see [12, 8, 32, 10, 11, 24]). To describe the previous results on the support
of the limiting measures of the family of the invariant measures of SDEs
(1.2), we introduce two further conditions.
(${\bf A}_{3}$). There are ${\varepsilon}_{0},\gamma>0$ and a nonnegative
$C^{2}$ Lyapunov function $V$ with $\lim_{|x|\rightarrow\infty}V(x)=\infty$
such that

$\frac{{\varepsilon}}{2}{\rm
Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)+\langle b(x),\nabla
V(x)\rangle\leq-\gamma\ {\rm for}\ \ |x|\gg 1,\
0<{\varepsilon}\leq{\varepsilon}_{0}.$

(${\bf A}_{4}$). The deterministic system (1.1) admits a finite number of
compact maximal equivalent classes (defined in terms of the quasi-potential
associated with the rate function of ULDP, see Section 3)
$K_{1},K_{2},\cdots,K_{l}$ such that the set of limit points of any trajectory
of the dynamical system (1.1) (the so called $\omega-$limit set) is contained
in one of the $K_{i}^{\prime}$s. Under the assumptions (${\bf A}_{1}$), (${\bf
A}_{3}$), (${\bf A}_{4}$), Freidlin and Wentzell set up a procedure to
determine on which stable equivalent classes the limiting measures of the
invariant measures $\mu^{\varepsilon}$ of SDEs (1.2) will be supported by
estimating the escape time of the solutions $X^{\varepsilon}$ from a domain
and the so called transitive difficulty matrix
$(\mathbb{V}(K_{i},K_{j})_{l\times l}$ (see the definition in Section 3, and
also [9, Theorems 4.1 and 4.2] of Chapter 6). These results are now called the
Freidlin and Wentzell asymptotics theorem. They imply that the support of the
limiting measures of the invariant measures $\mu^{\varepsilon}$ stays away
from the so called repelling equivalent classes $K_{i}^{\prime}$s and acyclic
saddle or trap chains $K_{i}^{\prime}$s. Xu et al. [36] proved that the above
implied results still hold under (${\bf A}_{1}$) and (${\bf A}_{3}$) but
without the finiteness of the number of compact equivalent classes. Assuming
that the diffusion matrix is uniformly elliptic, imposing the conditions
(${\bf A}_{3}$) and (${\bf A}_{4}$) and using Freidlin and Wentzell’s ULDP
technique, Hwang and Sheu [15] studied the long time behavior and the
exponential rate of convergence to the invariant measures of the system (1.2).

Now we highlight the main contributions of the current work.

$\bullet$ ULDP under mild conditions and improvement of the Freidlin and
Wentzell asymptotics theorem. The ULDPs are essential to the study of the
metastable problems of small random perturbation of the dynamical system
(1.1). The current conditions (${\bf A}_{1}$) or (${\bf A}_{2}$) for ULDP to
hold are rather restrictive, which excludes many interesting models where the
coefficients of the systems are unbounded and the diffusion coefficients could
be degenerate. In the first part of the paper, we obtain a general result on
ULDPs of SDEs under Lyapunov conditions on the coefficients, which can be
applied to stochastic systems with unbounded coefficients and degenerate
diffusions, and significantly extends results in literature. This part of the
work is of independent interest and is also essential to the study of the
support of the limiting measures. As a result, we improve the Freidlin and
Wentzell asymptotics theorem et al. so that they apply to systems with
unbounded coefficients.

$\bullet$ Computation of the transitive difficulty matrix
$\left(\mathbb{V}(K_{i},K_{j})\right)_{4\times 4}$ of two quasipotential
planar polynomial systems. By the transitive difficulty matrices and our
improved Freidlin and Wentzell’s asymptotics theorem, we prove that the
invariant measure $\mu^{\varepsilon}$ of the random planar dynamical systems
converges weakly to the arc-length measure of stable limit cycles as
$\varepsilon\rightarrow 0$. In the literature, when considering limiting
measures and their concentrations, people always first draw a picture of
global phase portraits and then design a transitive difficulty matrix, instead
of first studying the global phase portraits of the given system (1.1) and
then calculating the transitive difficulty matrix
$\left(\mathbb{V}(K_{i},K_{j})\right)_{l\times l}$ of the given system (1.2),
see e.g., [9, p.151] and [12, p.557]. It seems to us this is the first time to
compute the transitive difficulty matrix explicitly from original given SDEs
(1.2).

$\bullet$ Characterization of the limit measures of the invariant measures of
the van der Pol equation and the van der Pol Duffing equation perturbed by
unbounded random noise. We prove that the invariant measures of the random van
der Pol equation converge weakly to the arc-length measure corresponding to
the unique limit cycle, and that the invariant measures of the random van der
Pol Duffing equation converge weakly to either the convex combination of Delta
measures of the stable equilibria, or the convex combination of Delta measures
of the stable equilibria and the arc-length measure of stable limit cycle. We
stress here that, due to the degenerate driving noise, the Freidlin and
Wentzell asymptotics theorem is not applicable to them.

Both models are damping Hamiltonian systems or Langevin dynamics. The van der
Pol equation describes a self-oscillating triode circuit and the van der Pol
Duffing equation describes single diode circuit (see [23, 30]). The study of
these models goes back as early as 1927s when van der Pol [31] discovered an
“irregular noise” in a diode subject to periodic forcing and also the
coexistence of periodic orbits of different periods, which is the first
experimental observation of deterministic chaos. Through the years, the study
of the dynamical behavior of the van der Pol equation gave rise the birth of
the well-known Smale horseshoe, even differential dynamical systems, see [29,
19, 5, 6]. Because the random van der Pol equation has highly degenerate
driving noise and polynomial coefficients, to characterize the limit measures
of the invariant measures, new difficulties occur and new ideas are needed.

To obtain the existence of invariant measures for the random van der Pol
equations, because the standard energy functionals are not applicable, we use
a carefully designed Lyapunov function (see (4.14)), which not only allows
polynomial driving noise, but also help us obtain the ULDP and prove that both
stochastically and periodically forced van der Pol equation admits a periodic
stationary distribution, see Remark 4.1. The other challenge we encounter is
the continuity of its quasipotential in these systems with degenerate driving
noise. Unlike the non-degenerate case, what we have only been able to show is
that the quasipotential is upper-continuous. Even this is not easy, it
involves carefully constructing sample orbits connecting any pair of states,
which are required to satisfy certain restrictions in order to design good
controls of rate functions of the corresponding random systems; see (3.36),
(3.37), (3.41), (3.42), and (3.51). The upper-continuity of the quasipotential
implies that it is continuous at any equilibrium, see Lemma 3.2. Combined with
the estimate of rare probability of a small neighborhood of the unstable
equilibrium via ULDP, this turns out to be sufficient to characterize the
limiting measures. It is somehow surprising. Due to the degenerate driving
noise and polynomial growth of the coefficients, some other technical
difficulties have also to be overcome, see Lemma 3.2 and Subsection 4.2 for
the details. Finally, we point out that our method used in this part is valid
for any single degree of freedom stochastic damping Hamiltonian systems
because Lemma 3.2 is effective to all second-order stochastic equation.

The paper is organized as follows. Section 2 establishes a general result for
Freidlin and Wentzell’s ULDP, which is essential for later sections. In
Section 3, we present new results on the support of the limiting measures of
the invariant measures of the system (1.2), including the generalization of
the results in [9], [36] and [15] to the case of unbounded coefficients.
Section 4 is devoted to applications. We give several examples of random
perturbations of dynamical systems with unbounded coefficients, including the
realization of Freidlin’s two phase portraits and the computation of their
transitive difficulty matrices; stochastic van der Pol equation and stochastic
van der Pol Duffing equation; stochastic May-Leonard system and an example of
systems with unbounded coefficients and infinitely many equivalent classes.

##  2 Uniform large deviation principles

In this section, we will establish a ULDP for the laws of the solutions of the
stochastic differential equations (1.2) under some mild conditions. Let us now
recall the notion of ULDP from [9]. Let $({\mathcal{E}},\rho)$ be a Polish
space and let ${\mathcal{E}}_{0}$ be some topological space that will be used
for indexing. We say a function $I:{\mathcal{E}}\rightarrow[0,+\infty]$ is a
rate function if it has a compact level sets; i.e., for all $M<\infty$,
$\\{\varphi:I(\varphi)\leq M\\}$ is compact. For any $\varphi\in{\mathcal{E}}$
and $K\subset{\mathcal{E}}$, we denote

${\rm dist}(\varphi,K)\doteq\inf_{\varphi\in K}\rho(\varphi,\varphi).$

For $x\in{\mathcal{E}}_{0}$, we write $I_{x}(\cdot)$ to indicate the
dependence of the rate function on the value $x$. Let $\mathscr{K}$ be a
collection of all compact subsets of ${\mathcal{E}}_{0}$ and
$\Phi_{x}(s)\doteq\\{\varphi:I_{x}(\varphi)\leq s\\}$.

######  Definition 2.1.

A family of rate functions $\\{I_{x}\\}_{x\in{\mathcal{E}}_{0}}$ on
${\mathcal{E}}$ has compact level sets on compact sets of ${\mathcal{E}}_{0}$
if, for any $K\in\mathscr{K}$ and for every $s\in[0,\infty)$, $\cup_{x\in
K}\Phi_{x}(s)$ is a compact subset of ${\mathcal{E}}$.

######  Definition 2.2 (Freidlin-Wentzell ULDP).

A family of ${\mathcal{E}}$-valued random variables
$\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ indexed by
$x\in{\mathcal{E}}_{0}$ is said to satisfy a ULDP with respect to the rate
function $I_{x},\ x\in{\mathcal{E}}_{0}$, uniformly over $\mathscr{K}$, if

(i) LDP lower bound: For any $K\in\mathscr{K},{\delta}>0,{\gamma}>0$ and
$s_{0}>0$, there exists ${\varepsilon}_{0}>0$ such that

$\mathbb{P}\big{(}\rho(X^{{\varepsilon},x},\varphi)<{\delta}\big{)}\geq{\rm
exp}\big{(}-{\varepsilon}^{-1}(I_{x}(\varphi)+{\gamma})\big{)},$ (2.1)

for all ${\varepsilon}\in(0,{\varepsilon}_{0}]$, $x\in K$ and
$\varphi\in\Phi_{x}(s_{0})$;

(ii) LDP upper bound: For any $K\in\mathscr{K},{\delta}>0,{\gamma}>0$ and
$s_{0}>0$, there exists ${\varepsilon}_{0}>0$ such that

$\mathbb{P}\big{(}{\rm
dist}(X^{{\varepsilon},x},\Phi_{x}(s))\geq{\delta}\big{)}\leq{\rm
exp}\big{(}-{\varepsilon}^{-1}(s-{\gamma})\big{)},$ (2.2)

for all ${\varepsilon}\in(0,{\varepsilon}_{0}]$, $s\leq s_{0}$ and $x\in K$.

The uniform Laplace principle was introduced in Definition 1.11 in [2].

######  Definition 2.3.

A family of ${\mathcal{E}}$-valued random variables
$\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ indexed by
$x\in{\mathcal{E}}_{0}$ is said to satisfy a uniform Laplace principle with
respect to the rate function $I_{x},\ x\in{\mathcal{E}}_{0}$, uniformly over
$\mathscr{K}$, if for any $K\in\mathscr{K}$ and any bounded, continuous
$h:{\mathcal{E}}\rightarrow\mathbb{R}$,

$\lim_{{\varepsilon}\rightarrow 0}\sup_{x\in K}\Big{|}{\varepsilon}{\rm
log}\mathbb{E}{\rm
exp}(-{\varepsilon}^{-1}h(X^{{\varepsilon},x}))+\inf_{\varphi\in{\mathcal{E}}}\\{h(\varphi)+I_{x}(\varphi)\\}\Big{|}=0.$
(2.3)

###  2.1 The main result on ULDP

Throughout, we will use the following notation. Let
$(\mathbb{R}^{d},\langle\cdot,\cdot\rangle,|\cdot|)$ be the _d_ -dimensional
Euclidean space with the inner product $\langle\cdot,\cdot\rangle$ which
induces the norm $|\cdot|$. The norm $\|\cdot\|$ stands for the Hilbert-
Schmidt norm $\|\sigma\|^{2}:=\sum_{i=1}^{d}{\sum_{j=1}^{m}{\sigma_{ij}^{2}}}$
for any $d\times m$-matrix
$\sigma=(\sigma_{ij})\in\mathbb{R}^{d}\otimes\mathbb{R}^{m}$. $\sigma^{*}$
stands for the transpose of the matrix $\sigma$.

Let $(\Omega,\mathcal{F},\\{{\mathcal{F}}_{t}\\}_{t\geq 0},\mathbb{P})$ be a
complete probability space with a filtration $\\{{\mathcal{F}}_{t}\\}_{t\geq
0}$ satisfying the usual conditions and $\\{B(t)\\}_{t\geq 0}$ an _m_
-dimensional Brownian motion on this probability space. Fix $T\in(0,\infty)$.
Consider the following stochastic differential equations:

$dX^{\varepsilon}(t)=b(X^{\varepsilon}(t))\,dt+\sqrt{{\varepsilon}}\sigma(X^{\varepsilon}(t))dB(t),\
t\in[0,T],\quad x(0)=x\in\mathbb{R}^{d},$ (2.4)

where $\sigma:\mathbb{R}^{d}\ni
x\mapsto\sigma(x)\in\mathbb{R}^{d}\otimes\mathbb{R}^{m}$ and
$b:\mathbb{R}^{d}\ni x\mapsto b(x)\in\mathbb{R}^{d}$ are continuous. In the
following, we use the notation $X^{{\varepsilon},x}$ to indicate the solution
of (2.4) starting from $x$.

Let us now introduce the following assumptions.

######  Assumption 2.1.

Let ${\varepsilon}_{0}\in(0,1)$. For arbitrary $R>0$, if $|x|\vee|y|\leq R$,
there exists $L_{R}>0$ such that the following locally monotonicity condition

$2\langle x-y,b(x)-b(y)\rangle+\|\sigma(x)-\sigma(y)\|^{2}\leq L_{R}|x-y|^{2}$
(2.5)

holds for $|x-y|\leq{\varepsilon}_{0}$.

Note that Assumption 2.1 holds if $b$ and $\sigma$ are locally Lipschitz
continuous.

######  Assumption 2.2.

There exist a Lyapunov function $V\in C^{2}(\mathbb{R}^{d};\mathbb{R}_{+})$
and $\theta>0,~{}\eta>0$ such that

$\lim_{|x|\rightarrow+\infty}V(x)=+\infty,$ (2.6) $J(x):=\langle b(x),\nabla
V(x)\rangle+\frac{\theta}{2}\text{Trace}\big{(}\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\big{)}+\frac{|\sigma^{*}(x)\cdot\nabla
V(x)|^{2}}{\eta V(x)}\leq C\big{(}1+V(x)\big{)},$ (2.7)

and

$\text{Trace}\big{(}\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\big{)}\geq-M-CV(x).$
(2.8)

Here $\nabla V$ and $\nabla^{2}V$ stand for the gradient vector and Hessian
matrix of the function $V$, respectively, $C,\ M>0$ are some fixed constants.

The next result gives the existence and uniqueness of the solution of SDE
(2.4). Its proof is classical. The existence and uniqueness of a local
solution can be established using the Assumption 2.1 and the continuity of $b$
and ${\sigma}$. Furthermore, one can show that the solution is global using
the Lyapunov function $V$, (2.6) and (2.7) (cf. [17, Theorem 3.5, p.75]).

######  Proposition 2.1.

For any $0<{\varepsilon}<1$, under Assumptions 2.1 and 2.2, there exists a
unique solution to Eq. (2.4) defined on $[0,+\infty)$.

For each $h\in L^{2}([0,T],\mathbb{R}^{m})$, consider the so called skeleton
equation:

$dX_{x}^{h}(t)=b(X_{x}^{h}(t))dt+\sigma(X_{x}^{h}(t))h(t)dt,$ (2.9)

with the initial value $X_{x}^{h}(0)=x$. We have the following result:

######  Proposition 2.2.

Under Assumptions 2.1 and 2.2, there exists a unique solution to Eq. (2.9).

The proof of this proposition is similar to that of Proposition 2.1, so we
omit it here.

For any $x\in\mathbb{R}^{d}$ and $f\in C([0,T],\mathbb{R}^{d})$, we define

$I_{x}(f)=\inf_{\left\\{h\in L^{2}([0,T];\mathbb{R}^{m}):\
f=X_{x}^{h}\right\\}}\left\\{\frac{1}{2}\int_{0}^{T}|h(s)|^{2}ds\right\\},$
(2.10)

with the convention $\inf\\{\emptyset\\}=\infty$, here $X_{x}^{h}\in
C([0,T],\mathbb{R}^{d})$ solves Eq. (2.9).

We now formulate the main result on ULDP.

######  Theorem 2.1.

For ${\varepsilon}>0$, let $X^{{\varepsilon},x}$ be the solution to Eq. (2.4).
Suppose Assumptions 2.1 and 2.2 are satisfied, then $I_{x}$ defined by (2.10)
is a rate function on $C([0,T],\mathbb{R}^{d})$ and the family
$\\{I_{x},x\in\mathbb{R}^{d}\\}$ of rate functions has compact level sets on
compacts. Furthermore, $\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ satisfies
a ULDP on the space $C([0,T],\mathbb{R}^{d})$ with the rate function $I_{x}$,
uniformly over the initial value $x$ in bounded subsets of $\mathbb{R}^{d}$.

######  Remark 2.1.

Note that Freidlin and Wentzell’s criteria on ULDP valid for bounded
coefficients (${\bf A}_{1}$) (see [9, Theorem 3.1, p.135]) or linear growth
drift (${\bf A}_{2}$) (see [9, Theorem 1.1, p.86] ) are corollaries of Theorem
2.1 with the Lyapunov function $V(x)=|x|^{2}+1$.

For the study of the asymptotics of the invariant measures of the diffusion
process $\\{X^{\varepsilon,x}\\}$, Freidlin assumes the dissipative condition
(see [12, p.556] and [9, p.110]):

$\langle b(x),x\rangle\leq-\gamma|x|,\ {\rm for}\ \gamma>0,\ |x|\gg 1.$ (2.11)

In this case, the above Lyapunov function is mostly used in
Freidlin–Wentzell’s proof of ULDP in $\mathbb{R}^{d}$ (see also subsection 5.2
in [36] and (2.3) in [15]).

The proof of Theorem 2.1 will be given in subsection 2.3 below.

In the sequel, the symbol $C$ will denote a positive generic constant whose
value may change from place to place.

###  2.2 A Sufficient Condition for ULDP

In this section we recall the criteria obtained in [1] for proving the uniform
Laplace principle. and we will provide a sufficient condition to verify the
criteria.

Let ${\mathcal{E}}=C([0,T],\mathbb{R}^{d})$. $\rho(\cdot,\cdot)$ stands for
the uniform metric in the space ${\mathcal{E}}$ and
${\mathcal{E}}_{0}=\mathbb{R}^{d}$. Recall that $\mathscr{K}$ is a collection
of all compact subsets of $\mathbb{R}^{d}$.

The following result shows that a uniform Laplace principle implies the ULDP.
Its proof can be found in [2, Proposition 14] and [27, 26].

######  Proposition 2.3.

Let $I_{x}$ be a family of rate functions on ${\mathcal{E}}$ parameterized by
$x$ in ${\mathcal{E}}_{0}$ and assume that this family has compact level sets
on compacts. Suppose that the family of ${\mathcal{E}}$-valued random
variables $\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ satisfies a uniform
Laplace principle with rate function $I_{x}$ uniformly over any element in
$\mathscr{K}$. Then $\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ satisfies a
uniform LDP with rate function $I_{x}$ uniformly over any element in
$\mathscr{K}$.

Let

$S^{N}:=\\{h\in
L^{2}([0,T],\mathbb{R}^{m}):|h|_{L^{2}([0,T],\mathbb{R}^{m})}^{2}\leq N\\},$

and

$\tilde{S}^{N}:=\\{\phi:\phi\ \text{is}\ \mathbb{R}^{m}\text{-valued}\
{\mathcal{F}}_{t}\text{-predictable\ process\ such\ that}\ \phi(\omega)\in
S^{N},\ \mathbb{P}\text{-}a.s.\\}.$

$S^{N}$ will be endowed with the weak topology on
$L^{2}([0,T],\mathbb{R}^{m})$, under which $S^{N}$ is a compact Polish space.

For any ${\varepsilon}>0$, let
${\Gamma}^{{\varepsilon}}:{\mathcal{E}}_{0}\times
C([0,T],\mathbb{R}^{m})\rightarrow{\mathcal{E}}$ be a measurable mapping. Set
$X^{{\varepsilon},x}:={\Gamma}^{{\varepsilon}}(x,B(\cdot))$. The following
result was proved in [1].

######  Theorem 2.2 (A Criteria of Budhiraja-Dupuis).

Suppose that there exists a measurable map
${\Gamma}^{0}:{\mathcal{E}}_{0}\times
C([0,T],\mathbb{R}^{m})\rightarrow{\mathcal{E}}$ and let

$I_{x}(f)=\inf_{\left\\{h\in L^{2}([0,T];\mathbb{R}^{m}):\
f={\Gamma}^{0}(x,\int_{0}^{\cdot}h(s)ds)\right\\}}\big{\\{}\frac{1}{2}\int_{0}^{T}|h(s)|^{2}ds\big{\\}}.$
(2.12)

Suppose that for all $f\in{\mathcal{E}}$, $x\mapsto I_{x}(f)$ is a lower semi-
continuous (l.s.c.) map from ${\mathcal{E}}_{0}$ to $[0,\infty]$ and the
following conditions hold:

(a) for every $N<\infty$ and $K\in\mathscr{K}$, the set

$\Lambda_{N,K}\doteq\big{\\{}{\Gamma}^{0}(x,\int_{0}^{\cdot}h(s)ds):h\in
S^{N},x\in K\big{\\}}$

is a compact subset of ${\mathcal{E}}$;

(b) for every $N<\infty$ and any families
$\\{h^{{\varepsilon}}\\}\subset\tilde{S}^{N}$ and
$\\{x^{{\varepsilon}}\\}\subset{\mathcal{E}}_{0}$ satisfying that
$x^{{\varepsilon}}\rightarrow x$ and $h^{{\varepsilon}}$ converges in law to
some element $h$ as ${\varepsilon}\rightarrow 0$,
${\Gamma}^{{\varepsilon}}\big{(}x^{{\varepsilon}},B(\cdot)+\frac{1}{\sqrt{{\varepsilon}}}\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$
converges in law to ${\Gamma}^{0}(x,\int_{0}^{\cdot}h(s)ds)$ as
${\varepsilon}\rightarrow 0$.

Then for all $x\in{\mathcal{E}}_{0}$, $I_{x}$ is a rate function on
${\mathcal{E}}$, the family $\\{I_{x},x\in{\mathcal{E}}_{0}\\}$ of rate
functions has compact level sets on compacts and
$\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ satisfies a uniform Laplace
principle with respect to rate function $I_{x}$ uniformly over $\mathscr{K}$.

Next we present a sufficient condition for verifying the assumptions in
Theorem 2.2. It is a modification of Theorem 3.2 in [21].

######  Theorem 2.3.

Suppose that there exists a measurable map
${\Gamma}^{0}:{\mathcal{E}}_{0}\times
C([0,T],\mathbb{R}^{m})\rightarrow{\mathcal{E}}$ such that

(i) for every $N<+\infty$, $x_{n}\rightarrow x$ and any family
$\\{h_{n},n\in\mathbb{N}\\}\subset S^{N}$ converging weakly to some element
$h$ as $n\rightarrow\infty$,
${\Gamma}^{0}\big{(}x_{n},\int_{0}^{\cdot}{h_{n}}(s)ds\big{)}$ converges to
${\Gamma}^{0}\big{(}x,\int_{0}^{\cdot}{h(s)ds}\big{)}$ in the space
$C([0,T],\mathbb{R}^{d})$;

(ii) for every $N<+\infty$,
$\\{x^{{\varepsilon}},{\varepsilon}>0\\}\subset\\{x:|x|\leq N\\}$ and any
family $\\{h^{{\varepsilon}},{\varepsilon}>0\\}\subset\tilde{S}^{N}$ and any
$\delta>0$,

$\lim_{{\varepsilon}\rightarrow
0}\mathbb{P}\big{(}\rho(Y^{{\varepsilon},x^{\varepsilon}},Z^{{\varepsilon},x^{\varepsilon}})>\delta\big{)}=0,$

where
$Y^{{\varepsilon},x^{\varepsilon}}={\Gamma}^{{\varepsilon}}\big{(}x^{\varepsilon},B(\cdot)+\frac{1}{\sqrt{{\varepsilon}}}\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$
and
$Z^{{\varepsilon},x^{\varepsilon}}={\Gamma}^{0}\big{(}x^{{\varepsilon}},\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$.

Then for all $x\in{\mathcal{E}}_{0}$, $I_{x}$ defined by (2.12) is a rate
function on ${\mathcal{E}}$, the family $\\{I_{x},x\in{\mathcal{E}}_{0}\\}$ of
rate functions has compact level sets on compacts and
$\\{X^{{\varepsilon},x}\\}_{{\varepsilon}>0}$ satisfies a uniform Laplace
principle with the rate function $I_{x}$ uniformly over $\mathscr{K}$.

Proof  We will show that the conditions in Theorem 2.2 are fulfilled.
Condition (a) in Theorem 2.2 follows from condition (i) because $S^{N}$ and
$K$ are both compact sets. Fix $f\in{\mathcal{E}}$ and $x_{n}\rightarrow x$ in
${\mathcal{E}}_{0}$, we prove that $I_{x}(f)\leq\varliminf_{x_{n}\rightarrow
x}I_{x_{n}}(f)$. Without loss of generality, we assume that
$\lim_{x_{n}\rightarrow x}I_{x_{n}}(f)=\varliminf_{x_{n}\rightarrow
x}I_{x_{n}}(f)<M-1<\infty$ for some $M>0$. By the definition of $I_{x_{n}}$,
there exists $h_{n}\in S^{2M}$ satisfies
$f={\Gamma}^{0}(x_{n},\int_{0}^{\cdot}h_{n}(s)ds)$ such that

$I_{x_{n}}(f)\geq\frac{1}{2}\int_{0}^{T}|h_{n}(s)|^{2}ds-\frac{1}{n}.$

By the compactness of $S^{2M}$, there exist $h\in S^{2M}$ such that $h_{n}$
(take a subsequence if necessary) converging weakly to $h$. Thus, condition
(i) implies that $f={\Gamma}^{0}(x,\int_{0}^{\cdot}h(s)ds)$, then

$\lim_{x_{n}\rightarrow
x}I_{x_{n}}(f)\geq\frac{1}{2}\int_{0}^{T}|h(s)|^{2}ds\geq I_{x}(f).$

Theorefore, for all $f\in{\mathcal{E}}$, $x\mapsto I_{x}(f)$ is a lower semi-
continuous map from ${\mathcal{E}}_{0}$ to $[0,\infty]$.

Condition (ii) implies that for any bounded, uniformly continuous function
$F(\cdot)$ on ${\mathcal{E}}$,

$\lim_{{\varepsilon}\rightarrow
0}\mathbb{E}[|F(Y^{{\varepsilon},x^{\varepsilon}})-F(Z^{{\varepsilon},x^{\varepsilon}})|]=0.$
(2.13)

Because the mapping ${\Gamma}^{0}$ is continuous by condition (i) and
$(x^{\varepsilon},h^{\varepsilon})$ converge in law to $(x,h)$, then
$Z^{{\varepsilon},x^{\varepsilon}}$ converges in law to
${\Gamma}^{0}(x,\int_{0}^{\cdot}h(s)ds)$ in the space ${\mathcal{E}}$.
Combined with (2.13) we see that condition (b) holds. The proof is complete. ∎

###  2.3 Proof of Theorem 2.1

Let $S^{N}$ and $\tilde{S}^{N}$ be defined as in Section 3. According to
Proposition 2.2, there exists a measurable mapping ${\Gamma}^{0}$ from
$\mathbb{R}^{d}\times C([0,T],\mathbb{R}^{m})$ to $C([0,T],\mathbb{R}^{d})$
such that $X_{x}^{h}={\Gamma}^{0}\big{(}x,\int_{0}^{\cdot}{h(s)ds}\big{)}$ for
$x\in\mathbb{R}^{d}$ and $h\in L^{2}([0,T],\mathbb{R}^{m})$.

By the Yamada-Watanabe theorem, the existence of a unique strong solution of
Eq. (2.4) and Assumption 2.1 implies that for every ${\varepsilon}>0$, there
exists a measurable mapping ${\Gamma}^{{\varepsilon}}:\mathbb{R}^{d}\times
C([0,T],\mathbb{R}^{m})\rightarrow C([0,T],\mathbb{R}^{d})$ such that

$X^{{\varepsilon},x}={\Gamma}^{{\varepsilon}}(x,B(\cdot)),$

and applying the Girsanov theorem, for any $N>0$ and
$h^{{\varepsilon}}\in\tilde{S}^{N}$,

$Y^{{\varepsilon},x}:={\Gamma}^{{\varepsilon}}\big{(}x,B(\cdot)+\frac{1}{\sqrt{{\varepsilon}}}\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$
(2.14)

is the solution of the following SDE

$Y^{{\varepsilon},x}(t)=x+\int_{0}^{t}b(Y^{{\varepsilon}}(s))ds+\int_{0}^{t}\sigma(Y^{{\varepsilon}}(s))h^{\varepsilon}(s)ds+\sqrt{{\varepsilon}}\int_{0}^{t}\sigma(Y^{{\varepsilon}}(s))dB(s).$
(2.15)

By virtue of the Proposition 2.3 and Theorem 2.3, to prove Theorem 2.1, we
need to verify the conditions (i) and (ii) in Theorem 2.3 for the measurable
maps ${\Gamma}^{\varepsilon}$ and ${\Gamma}^{0}$.

The verification of Conditions (i) and (ii) is similar to the proof of
Proposition 3.1 in [33], we here only give a sketch.

Proof of condition (i): Let $x_{n}\rightarrow x$ and
$\\{h_{n}\\}_{n\in\mathbb{N}}\subset S^{N}$ converges in the weak topology to
$h$ as $n\rightarrow\infty$.

Define
$W_{x_{n}}^{h_{n}}(t)=e^{-\eta\int_{0}^{t}|h_{n}(s)|^{2}ds}V(X_{x_{n}}^{h_{n}}(t))$,
where $\theta>0,\eta>0$ and $V$ are in Assumption 2.2. Apply the chain rule
and (2.7), (2.8), we get

$\displaystyle W_{x_{n}}^{h_{n}}(t)$ (2.16) $\displaystyle\leq$ $\displaystyle
V(x_{n})+\int_{0}^{t}e^{-\eta\int_{0}^{s}|h_{n}(r)|^{2}dr}\big{[}\langle\nabla
V(X_{x_{n}}^{h_{n}}(s)),b(X_{x_{n}}^{h_{n}}(s))\rangle+\frac{|\sigma^{*}(X_{x_{n}}^{h_{n}}(s))\cdot\nabla
V(X_{x_{n}}^{h_{n}}(s))|^{2}}{\eta V(X_{x_{n}}^{h_{n}}(s))}\big{]}ds$
$\displaystyle\leq$ $\displaystyle
V(x_{n})+\int_{0}^{t}e^{-\eta\int_{0}^{s}|h_{n}(r)|^{2}dr}[C(1+V(X_{x_{n}}^{h_{n}}(s)))+\frac{\theta}{2}(M+CV(X_{x_{n}}^{h_{n}}(s)))]ds$
$\displaystyle\leq$ $\displaystyle
V(x_{n})+C\int_{0}^{t}1+W_{x_{n}}^{h_{n}}(s)\ ds.$

Applying the Gronwall inequality, the above inequality (2.16) yields
$\sup_{n\in\mathbb{N}}\sup_{t\in[0,T]}V(X_{x_{n}}^{h_{n}}(t))<\infty$. Thus,

$\sup_{n\in\mathbb{N}}\sup_{t\in[0,T]}|X_{x_{n}}^{h_{n}}(t)|\leq L$ (2.17)

for some constant $L>0$.

Next, using the Arzela-Ascoli theorem, we can show that
$\\{X_{x_{n}}^{h_{n}},n\in\mathbb{N}\\}$ is pre-compact in the space
$C([0,T],\mathbb{R}^{d})$ and

$X_{x_{n}}^{h_{n}}\rightarrow\tilde{X}\quad\text{in}~{}C([0,T],\mathbb{R}^{d}),$

for some $\tilde{X}$. Then, the uniqueness of skeleton equation (2.9) implies
$\tilde{X}=X_{x}^{h}$. Therefore, condition (i) holds.

Proof of condition (ii): Let
$\\{x^{{\varepsilon}},{\varepsilon}>0\\}\subset\\{x:|x|\leq N\\}$ and a family
$\\{h^{{\varepsilon}},{\varepsilon}>0\\}\subset\tilde{S}^{N}$, we need to
prove that
$\rho(Y^{{\varepsilon},x^{\varepsilon}},Z^{{\varepsilon},x^{\varepsilon}})\rightarrow
0$ in probability as ${\varepsilon}\rightarrow 0$, where
$Y^{{\varepsilon},x^{\varepsilon}}={\Gamma}^{{\varepsilon}}\big{(}x^{\varepsilon},B(\cdot)+\frac{1}{\sqrt{{\varepsilon}}}\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$
and
$Z^{{\varepsilon},x^{\varepsilon}}={\Gamma}^{0}\big{(}x^{{\varepsilon}},\int_{0}^{\cdot}{h^{{\varepsilon}}(s)ds}\big{)}$.

Recall that ${\varepsilon}_{0}$ is the constant appeared in Assumption 2.1.
For $R>0$, $0<p\leq{\varepsilon}_{0}$, define

$\tau^{{\varepsilon}}_{R}=\inf\\{t\geq
0:|Y^{{\varepsilon},x^{\varepsilon}}(t)|\geq
R\\},\quad\tau^{\varepsilon}_{p}=\inf\\{t\geq
0:|Y^{{\varepsilon},x^{\varepsilon}}(t)-Z^{{\varepsilon},x^{\varepsilon}}(t)|^{2}\geq
p\\}.$

From the proof of (2.16) we also see that there exists a constant $L>0$ such
that

$\sup_{{\varepsilon}>0}\sup_{t\in[0,T]}|Z^{{\varepsilon},x^{\varepsilon}}(t)|\leq
L.$

Applying the Itô formula to
$e^{-\int_{0}^{t}(L_{R}+|h^{\varepsilon}(s)|^{2})ds}|Y^{{\varepsilon},x^{\varepsilon}}(t)-Z^{{\varepsilon},x^{\varepsilon}}(t)|^{2}$,
replacing $t$ by the stopping time
$\hat{\tau}^{\varepsilon}:=T\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p}$
and taking expectation, we obtain

$\displaystyle\mathbb{E}e^{-\int_{0}^{\hat{\tau}^{\varepsilon}}(L_{R}+|h^{\varepsilon}(s)|^{2})ds}|Y^{{\varepsilon},x^{\varepsilon}}(\hat{\tau}^{\varepsilon})-Z^{{\varepsilon},,x^{\varepsilon}}(\hat{\tau}^{\varepsilon})|^{2}\leq{\varepsilon}\mathbb{E}\int_{0}^{\hat{\tau}^{\varepsilon}}\|{\sigma}(Y^{{\varepsilon},x^{\varepsilon}}(s))\|^{2}ds.$
(2.18)

Thus,

$\lim_{{\varepsilon}\rightarrow
0}\mathbb{P}(|Y^{{\varepsilon},x^{\varepsilon}}(\hat{\tau}^{\varepsilon})-Z^{{\varepsilon},x^{\varepsilon}}(\hat{\tau}^{\varepsilon})|^{2}\geq
p)=0.$ (2.19)

Noting that

$\\{\tau^{\varepsilon}_{p}\leq
T\wedge\tau^{{\varepsilon}}_{R}\\}\subset\\{\sup_{s\leq
T\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p}}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|^{2}\geq
p\\}\subset\\{|Y^{{\varepsilon},x^{\varepsilon}}(\hat{\tau}^{\varepsilon})-Z^{{\varepsilon},x^{\varepsilon}}(\hat{\tau}^{\varepsilon})|^{2}\geq
p\\},$

it follows from (2.19) that

$\lim_{{\varepsilon}\rightarrow
0}\mathbb{P}\big{(}\tau^{\varepsilon}_{p}\leq\tau^{{\varepsilon}}_{R}\wedge
T\big{)}\leq\lim_{{\varepsilon}\rightarrow 0}\mathbb{P}\big{(}\sup_{s\leq
T\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p}}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|^{2}\geq
p\big{)}=0.$ (2.20)

Next, we prove

$\lim_{R\rightarrow\infty}\sup_{{\varepsilon}\in(0,\frac{\theta}{2})}\mathbb{P}\big{(}\tau^{\varepsilon}_{R}\leq
T\wedge\tau^{\varepsilon}_{p}\big{)}=0.$ (2.21)

Applying the Itô formula to
$U^{{\varepsilon},h^{\varepsilon}}(t)=e^{-\eta\int_{0}^{t}|h^{\varepsilon}(s)|^{2}ds}V(Y^{{\varepsilon},x^{\varepsilon}}(t))$
gives

$\displaystyle\mathbb{E}U^{{\varepsilon},h^{\varepsilon}}(t\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p})$
(2.22) $\displaystyle\leq$ $\displaystyle
V(x^{\varepsilon})+\mathbb{E}\int_{0}^{t\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p}}e^{-\eta\int_{0}^{s}|h^{\varepsilon}(r)|^{2}dr}\big{[}\langle\nabla
V(Y^{{\varepsilon},x^{\varepsilon}}(s)),b(Y^{{\varepsilon},x^{\varepsilon}}(s))\rangle$
$\displaystyle+\frac{|\sigma^{*}(Y^{{\varepsilon},x^{\varepsilon}}(s))\cdot\nabla
V(Y^{{\varepsilon},x^{\varepsilon}}(s))|^{2}}{\eta
V(Y^{{\varepsilon},x^{\varepsilon}}(s))}+{\varepsilon}\cdot\text{Trace}\big{(}\nabla^{2}V(Y^{{\varepsilon},x^{\varepsilon}}(s))\sigma(Y^{{\varepsilon},x^{\varepsilon}}(s))\sigma^{*}(Y^{{\varepsilon},x^{\varepsilon}}(s))\big{)}\big{]}ds$
$\displaystyle\leq$ $\displaystyle
V(x^{\varepsilon})+\mathbb{E}\int_{0}^{t\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p}}e^{-\eta\int_{0}^{s}|h^{\varepsilon}(r)|^{2}dr}\big{[}C(1+V(Y^{{\varepsilon},x^{\varepsilon}}(s)))+(\frac{\theta}{2}-{\varepsilon})(CV(Y^{{\varepsilon},x^{\varepsilon}}(s))+M\big{]}ds$
$\displaystyle\leq$ $\displaystyle
V(x^{\varepsilon})+C\mathbb{E}\int_{0}^{t}\big{(}1+U^{{\varepsilon},h^{\varepsilon}}(s\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p})\big{)}ds.$

Applying the Gronwall inequality again, (2.22) yields
$\mathbb{E}U^{{\varepsilon},h^{\varepsilon}}(T\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{p})\leq(V(x^{\varepsilon})+C)e^{CT}$.
From the above inequality, we deduce that .

$\mathbb{P}\big{(}\tau^{\varepsilon}_{R}\leq
T\wedge\tau^{\varepsilon}_{p}\big{)}\leq\frac{(C+\sup_{{\varepsilon}\in(0,\frac{\theta}{2})}V(x^{\varepsilon}))e^{CT+{\eta}N}}{\inf_{|x|\geq
R}V(x)}$

Let $R\rightarrow\infty$, we obtain (2.21) since
$\sup_{{\varepsilon}>0}|x^{\varepsilon}|\leq N$.

For arbitrary $\delta>0$, we have

$\displaystyle\mathbb{P}\big{(}\sup_{0\leq s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta}\big{)}$
$\displaystyle=$ $\displaystyle\mathbb{P}\big{(}\sup_{0\leq s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta},\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{{\delta}^{2}}>T\big{)}$
$\displaystyle+\mathbb{P}\big{(}\sup_{0\leq s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta},\tau^{\varepsilon}_{R}\leq
T\wedge\tau^{\varepsilon}_{{\delta}^{2}}\big{)}$
$\displaystyle+\mathbb{P}\big{(}\sup_{0\leq s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta},\tau^{\varepsilon}_{{\delta}^{2}}\leq\tau^{\varepsilon}_{R}\wedge
T\big{)}$ $\displaystyle\leq$ $\displaystyle\mathbb{P}\big{(}\sup_{0\leq s\leq
T\wedge\tau^{\varepsilon}_{R}\wedge\tau^{\varepsilon}_{{\delta}^{2}}}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|^{2}\geq{\delta}^{2}\big{)}+\mathbb{P}\big{(}\tau^{\varepsilon}_{R}\leq
T\wedge\tau^{\varepsilon}_{{\delta}^{2}}\big{)}+\mathbb{P}\big{(}\tau^{\varepsilon}_{{\delta}^{2}}\leq\tau^{\varepsilon}_{R}\wedge
T\big{)}.$

(2.20) (with $p={\delta}^{2}$) implies that

$\displaystyle\lim_{{\varepsilon}\rightarrow 0}\mathbb{P}\big{(}\sup_{0\leq
s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta}\big{)}\leq\sup_{{\varepsilon}\in(0,\frac{\theta}{2})}\mathbb{P}\big{(}\tau^{\varepsilon}_{R}\leq
T\wedge\tau^{\varepsilon}_{{\delta}^{2}}\big{)}.$

Let $R\rightarrow\infty$ and (2.21) to get

$\lim_{{\varepsilon}\rightarrow 0}\mathbb{P}\big{(}\sup_{0\leq s\leq
T}|Y^{{\varepsilon},x^{\varepsilon}}(s)-Z^{{\varepsilon},x^{\varepsilon}}(s)|\geq{\delta}\big{)}=0.$

Therefore, condition (ii) holds.

######  Remark 2.2.

If the diffusion processes $X^{{\varepsilon}}$ lives in a domain $D$ with
boundary, to apply Theorem 2.1 one only needs to slightly adjust the
conditions on $V$ in Assumption 2.2 to the condition that $V\in
C^{2}(D;\mathbb{R}_{+})$ and $\lim_{\underset{|x|\rightarrow\infty}{x\in
D}}V(x)=+\infty$.

##  3 The concentration of limiting measures and metastability

In this section, we will apply the ULDP obtained in Section 2 to describe the
support of the limiting measures of the invariant measures of the system
(1.2).

From now on we denote $C([0,T],\mathbb{R}^{d})$ by ${\bf C}_{T}$. ${\bf
C}_{T}^{x}$ (${\bf AC}_{T}^{x}$) will denote the space of (absolutely)
continuous functions on $[0,T]$ started from $x$ with values in
$\mathbb{R}^{d}$.

_Quasipotential_ , introduced by Freidlin and Wentzell (see, e.g., [9, p.90 or
p.142]), is a very useful notion and is defined by

$\mathbb{V}(x,y):=\inf\big{\\{}I_{x}^{T}(\varphi):\varphi(0)=x,\
\varphi(T)=y,\ T>0\big{\\}},\>x,\ y\in\mathbb{R}^{d}.$

where $I_{x}^{T}(\cdot)$ is the rate function defined in (2.10). We can also
define $\mathbb{V}$ on pairs of subsets in $\mathbb{R}^{d}$ as follows:

$\mathbb{V}(D_{0},D):=\inf\big{\\{}I_{x}^{T}(\varphi):\varphi(0)\in
D_{0},\varphi(T)\in D,T>0\big{\\}},\>D_{0},D\subset\mathbb{R}^{d}.$

We first assume that the noise is non-degenerate:

$\beta^{*}\sigma(x)\sigma^{*}(x)\beta>0\ {\rm for\ all}\ x\in\mathbb{R}^{d}\
{\rm and}\ \beta\in\mathbb{R}^{d}\backslash\\{0\\}.$ (3.1)

Under this assumption, the rate function has the following expression

$I_{x}^{T}(\varphi)=\left\\{\begin{array}[]{ll}\int\limits_{0}^{T}L(\varphi(t),\dot{\varphi}(t)){\rm
d}t,&\hbox{if $\varphi\in{\bf AC}_{T}^{x}$,}\\\
+\infty,&\hbox{otherwise,}\end{array}\right.$ (3.2)

where
$L(u,\beta)=\frac{1}{2}\big{(}\beta-b(u)\big{)}^{*}\left(\sigma(u)\sigma^{*}(u)\right)^{-1}\big{(}\beta-b(u)\big{)},u,\beta\in\mathbb{R}^{d}$.
It follows from [9, Lemma 2.3, p.94] or [36, Lemma 2.5] that

######  Lemma 3.1.

Suppose that (3.1) holds. Then for any $R>0$ there exists $L=L(R)>0$ such that
for any $x,y\in\bar{B}_{R}(O)$, the smooth function
$\varphi(t)=x+\frac{t}{|y-x|}(y-x),\ t\in[0,|y-x|]$ satisfies

$I_{x}^{|y-x|}(\varphi)\leq L|y-x|$

which implies that $\mathbb{V}(x,y)$ is continuous.

Consider now the small noise perturbation

$\begin{cases}X_{\varepsilon}^{(2)}(t)=b\big{(}X_{\varepsilon}(t),X_{\varepsilon}^{(1)}(t)\big{)}+\sqrt{{\varepsilon}}{\sigma}\big{(}X_{\varepsilon}(t),X_{\varepsilon}^{(1)}(t)\big{)}\dot{B}(t),\\\
(X_{\varepsilon},X_{\varepsilon}^{(1)})(0)=(x_{1},x_{2})=:x\in\mathbb{R}^{2}\end{cases}$
(3.3)

of single degree of freedom second-order differential equation:

$\begin{cases}X_{\varepsilon}^{(2)}(t)=b\big{(}X_{\varepsilon}(t),X_{\varepsilon}^{(1)}(t)\big{)},\\\
(X_{\varepsilon},X_{\varepsilon}^{(1)})(0)=(x_{1},x_{2})=:x\in\mathbb{R}^{2}\end{cases}$
(3.4)

where $X_{\varepsilon}^{(i)}$ stands for the derivative of order $i$, the
coefficients $b$ and ${\sigma}$ are real-valued, locally Lipschitz continuous
functions on $\mathbb{R}^{2}$ and ${\sigma}(x)\neq 0$ for all
$x\in\mathbb{R}^{2}$.

Let
$Y_{\varepsilon}:=(Y_{\varepsilon}^{1},Y_{\varepsilon}^{2}):=(X_{\varepsilon},X_{\varepsilon}^{(1)})$.
Then (3.3) is equivalent to the following system of SDEs:

$\begin{cases}dY_{\varepsilon}^{1}(t)=Y_{\varepsilon}^{2}(t)dt,\\\
dY_{\varepsilon}^{2}(t)=b\big{(}Y_{\varepsilon}^{1}(t),Y_{\varepsilon}^{2}(t)\big{)}dt+\sqrt{{\varepsilon}}{\sigma}\big{(}Y_{\varepsilon}^{1}(t),Y_{\varepsilon}^{2}(t)\big{)}dB(t),\\\
(Y_{\varepsilon}^{1},Y_{\varepsilon}^{2})(0)=(x_{1},x_{2}).\\\ \end{cases}$
(3.5)

In this case, the rate function for the large deviation of $Y_{\varepsilon}$
can be written as

$I_{x}^{T}(\psi)=\left\\{\begin{array}[]{ll}\frac{1}{2}\int_{0}^{T}\Big{|}\frac{\varphi^{(2)}(s)-b(\varphi(s),\varphi^{(1)}(s))}{{\sigma}(\varphi(s),\varphi^{(1)}(s))}\Big{|}^{2}ds&\hbox{if
$\psi=(\varphi,\varphi^{(1)})\in{\bf AC}_{T}^{x}$,}\\\
+\infty,&\hbox{otherwise,}\end{array}\right.$ (3.6)

and the corresponding quasipotential can be expressed as

$\mathbb{V}(x,y)=\inf\big{\\{}I_{x}^{T}(\psi):\psi(0)=(\varphi,\varphi^{(1)})(0)=x,\
\psi(T)=(\varphi,\varphi^{(1)})(T)=y,\ T>0\big{\\}},\>x,\ y\in\mathbb{R}^{2}.$

Lemma 3.1 implies that for any $x,y\in\bar{B}_{R}(O)$, there exists a sampling
orbit $\varphi(t)$ connecting $x$ and $y$ and spending the time $T(x,y)=|y-x|$
such that $I_{x}^{|y-x|}(\varphi)\leq L|y-x|$. In particular, $T(x,y)\leq 2R$.
The continuity of $\mathbb{V}$ and the boundedness of spent time of the above
constructed sampling orbit play an important role in estimating rare
probability of certain subsets of (1.2) with non-degenerate noise. However,
the driving noise of (3.5) is degenerate. We cannot prove the version of Lemma
3.1 for system (3.5) because $\varphi(t)$ is not a sampling function as
defined in (3.6), instead prove the following weaker version, which is
sufficient to study the limiting measures of stochastic van der Pol equation
and stochastic van der Pol Duffing equation in Section 4.

######  Lemma 3.2.

Suppose that the rate function $I_{x}^{T}$ is of the form (3.6). Then
(i) for any $(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ and $T>0$, there
exists $\psi^{x}\in C_{T}$ with $\psi^{x}(0)=x$ and $\psi^{x}(T)=y$ such that
$I^{T}_{x}(\psi^{x})<\infty$. Consequently, for any
$(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$, $\mathbb{V}(x,y)<\infty;$
(ii) the quasipotential $\mathbb{V}$ is upper semi-continuous on
$\mathbb{R}^{2}\times\mathbb{R}^{2}$, that is, for any
$(x,y)\in\mathbb{R}^{2}\times\mathbb{R}^{2}$,

$\displaystyle\lim_{\delta\searrow 0}\sup_{(\tilde{x},\tilde{y})\in
B_{\delta}((x,y))\setminus(x,y)}\mathbb{V}(\tilde{x},\tilde{y})\leq\mathbb{V}(x,y),$
(3.7)

in particular, $\mathbb{V}$ is continuous at any equilibrium of (3.5);
(iii) for given $(x_{0},y_{0})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$ and
$\delta,\eta>0$, suppose that $\mathbb{V}(x,y_{0})<\eta$ for all
$x\in\overline{B}_{\delta}(x_{0}):=\\{x\in\mathbb{R}^{2}:|x-x_{0}|\leq\delta\\}$.
Then there exists a constant $T_{0}\in(0,\infty)$ such that, for any
$x\in\overline{B}_{\delta}(x_{0})$, there exist $T^{x}\in(0,T_{0}]$ and
$\psi^{x}\in C_{T^{x}}$ with $\psi^{x}(0)=x$ and $\psi^{x}(T^{x})=y_{0}$ such
that $I^{T^{x}}_{x}(\psi^{x})<\eta$.

Proof  Fix $x=(x_{1},x_{2}),y=(y_{1},y_{2})\in\mathbb{R}^{2}$ and $T>0$. Let
$f_{1}(t)=x_{1}+x_{2}t$ and $f_{2}(t)=y_{1}+y_{2}(t-T)$. We choose a cut-off
function ${\alpha}\in C^{\infty}(\mathbb{R})$:

$\displaystyle{\alpha}(t)=\left\\{\begin{array}[]{ll}1,&t\in[0,\frac{T}{3}]\\\
>0,&t\in(\frac{T}{3},\frac{2T}{3})\\\
0,&t\in[\frac{2T}{3},T].\end{array}\right.$ (3.11)

Then $\varphi={\alpha}f_{1}+(1-{\alpha})f_{2}$ satisfies
$(\varphi,\varphi^{(1)})(0)=x,(\varphi,\varphi^{(1)})(T)=y$. By the
assumptions that the coefficients $b$ and ${\sigma}$ are real-valued, locally
Lipschitz continuous functions on $\mathbb{R}^{2}$ and ${\sigma}(x)\neq 0$ for
all $x\in\mathbb{R}^{2}$, it is easy to see that
$I_{x}^{T}((\varphi,\varphi^{(1)}))<\infty$. The proof of (i) is complete.

We are now in the position to prove (ii). Notice that, for any
$\tilde{x},\tilde{y}\in\mathbb{R}^{2}$, $\mathbb{V}(\tilde{x},\tilde{y})\geq
0$, and that if $U$ is a equilibrium of (3.5) and $x,y\in U$, then
$\mathbb{V}(x,y)=0$. Once we prove $\eqref{eq Part 1}$, the claim that
$\mathbb{V}$ is continuous at any equilibrium of (3.5) holds. In the
following, we prove (3.7).

To this end, we first introduce the following two functions, which will be
used later.

For $j>0$, let

$\displaystyle\theta^{-}_{j}(t)=\left\\{\begin{array}[]{ll}0,&\hbox{$t<0$;}\\\
-t,&\hbox{$t\in[0,j]$;}\\\ t-2j,&\hbox{$t\in(j,2j+\sqrt{2}j]$;}\\\
0,&\hbox{$t>2j+\sqrt{2}j$,}\end{array}\right.$ (3.16)

and

$\displaystyle\theta^{+}_{j}(t)=\left\\{\begin{array}[]{ll}0,&\hbox{$t<0$;}\\\
t,&\hbox{$t\in[0,j]$;}\\\ 2j-t,&\hbox{$t\in(j,2j+\sqrt{2}j]$;}\\\
0,&\hbox{$t>2j+\sqrt{2}j$.}\end{array}\right.$ (3.21)

We remark that

$\displaystyle\int_{0}^{2j+\sqrt{2}j}\theta^{-}_{j}(t)dt=0\text{ and
}\int_{0}^{2j+\sqrt{2}j}\int_{0}^{s}\theta^{-}_{j}(t)dtds=-\frac{1}{3}(3+2\sqrt{2})j^{3},$
(3.22) $\displaystyle\int_{0}^{2j+\sqrt{2}j}\theta^{+}_{j}(t)dt=0\text{ and
}\int_{0}^{2j+\sqrt{2}j}\int_{0}^{s}\theta^{+}_{j}(t)dtds=\frac{1}{3}(3+2\sqrt{2})j^{3},$
(3.23)

and

$\displaystyle\sup_{s\in[0,2j+\sqrt{2}j]}\Big{|}\int_{0}^{s}\theta^{-}_{j}(t)dt\Big{|}=\sup_{s\in[0,2j+\sqrt{2}j]}\Big{|}\int_{0}^{s}\theta^{+}_{j}(t)dt\Big{|}=j^{2}.$
(3.24)

Now we prove (3.7).

Since $\mathbb{V}(x,y)<\infty$, for any $\epsilon>0$, there exist $T_{0}>0$
and
$\psi=(\varphi,\varphi^{(1)})=\\{(\varphi(t),\varphi^{(1)}(t)),t\in[0,T_{0}]\\}\in{\bf
AC}_{T_{0}}^{x}$ satisfying

  1. 1.

$\psi(0)=x$, $\psi(T_{0})=y$;

  2. 2.

letting
$h^{\sigma}(s)=\frac{\varphi^{(2)}(s)-b(\varphi(s),\varphi^{(1)}(s))}{{\sigma}(\varphi(s),\varphi^{(1)}(s))},s\in[0,T_{0}]$,
we have

$\displaystyle\frac{1}{2}\int_{0}^{T_{0}}\Big{|}h^{\sigma}(s)\Big{|}^{2}ds-\epsilon\leq\mathbb{V}(x,y).$
(3.25)

Let us remark that

$\displaystyle\varphi^{(1)}(t)=x_{2}+\int_{0}^{t}\varphi^{(2)}(s)ds,\
t\in[0,T_{0}];$ (3.26)
$\displaystyle\varphi(t)=x_{1}+\int_{0}^{t}\varphi^{(1)}(s)ds,\
t\in[0,T_{0}];$ (3.27) $\displaystyle\varphi(0)=x_{1};\ \varphi(T_{0})=y_{1};\
\varphi^{(1)}(0)=x_{2};\ \varphi^{(1)}(T_{0})=y_{2}.$ (3.28)

For any $1\geq\delta>0$ and any $\tilde{x}=(\tilde{x}_{1},\tilde{x}_{2})\in
B_{\delta}(x),\tilde{y}=(\tilde{y}_{1},\tilde{y}_{2})\in B_{\delta}(y)$. We
define

$\displaystyle\zeta_{2}(t)=\left\\{\begin{array}[]{ll}{\rm
sgn}(x_{2}-\tilde{x}_{2}),&\hbox{$t\in[0,|x_{2}-\tilde{x}_{2}|)$;}\\\
\varphi^{(2)}(t-|x_{2}-\tilde{x}_{2}|),&\hbox{$t\in[|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|)$;}\\\
{\rm
sgn}(\tilde{y}_{2}-y_{2}),&\hbox{$t\in[T_{0}+|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|]$.}\end{array}\right.$
(3.32)

The proof of (3.7) is divided into three cases: Case 1: $y_{2}>0$; Case 2:
$y_{2}<0$; Case 3: $y_{2}=0$.

Case 1: $y_{2}>0$.

Without loss of generality, we choose $0<\delta\leq 1$ small enough such that
$\tilde{y}_{2}\geq\frac{y_{2}}{2}$. Set

$y_{3}:=\tilde{x}_{1}+\frac{1}{2}|x_{2}-\tilde{x}_{2}|(x_{2}+\tilde{x}_{2})+y_{1}-x_{1}+\frac{1}{2}|y_{2}-\tilde{y}_{2}|(y_{2}+\tilde{y}_{2}).$

In the following, we divide the case into two subcases.

Subcase 1: $\tilde{y}_{1}\geq y_{3}$.

For any
$t\in(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}]$,
set

$\displaystyle\zeta_{2}(t)=0.$

Let

$\displaystyle\zeta_{1}(t)=\tilde{x}_{2}+\int_{0}^{t}\zeta_{2}(s)ds;\
\zeta(t)=\tilde{x}_{1}+\int_{0}^{t}\zeta_{1}(s)ds.$

We have

$\displaystyle\zeta_{1}(0)=\tilde{x}_{2};\ \zeta(0)=\tilde{x}_{1};$ (3.33)
$\displaystyle\zeta_{1}(|x_{2}-\tilde{x}_{2}|)=x_{2};\ \ \ \ \
\zeta(|x_{2}-\tilde{x}_{2}|)=\tilde{x}_{1}+\frac{1}{2}|x_{2}-\tilde{x}_{2}|(x_{2}+\tilde{x}_{2});$
$\displaystyle\zeta_{1}(t)=\varphi^{(1)}(t-|x_{2}-\tilde{x}_{2}|),\
t\in[|x_{2}-\tilde{x}_{2}|,|x_{2}-\tilde{x}_{2}|+T_{0}];$
$\displaystyle\zeta(t)=\zeta(|x_{2}-\tilde{x}_{2}|)+\varphi(t-|x_{2}-\tilde{x}_{2}|)-x_{1},\
t\in[|x_{2}-\tilde{x}_{2}|,|x_{2}-\tilde{x}_{2}|+T_{0}];$
$\displaystyle\zeta_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|)=\varphi^{(1)}(T_{0})=y_{2};$
$\displaystyle\zeta(T_{0}+|x_{2}-\tilde{x}_{2}|)=\zeta(|x_{2}-\tilde{x}_{2}|)+\varphi(T_{0})-x_{1}=\zeta(|x_{2}-\tilde{x}_{2}|)+y_{1}-x_{1};$
$\displaystyle\zeta_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|)=\tilde{y}_{2};$
$\displaystyle\zeta(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|)=y_{3};$
$\displaystyle\zeta_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}})=\tilde{y}_{2};$
(3.34)
$\displaystyle\zeta(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}})=\tilde{y}_{1}.$
(3.35)

Let

$\displaystyle
h(s)=\frac{\zeta_{2}(s)-b(\zeta(s),\zeta_{1}(s))}{{\sigma}(\zeta(s),\zeta_{1}(s))}.$
(3.36)

Then

$\displaystyle\left|\int_{0}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}}|h(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\right|$
$\displaystyle\leq$
$\displaystyle\int_{0}^{|x_{2}-\tilde{x}_{2}|}|h(s)|^{2}ds+\Big{|}\int_{|x_{2}-\tilde{x}_{2}|}^{T_{0}+|x_{2}-\tilde{x}_{2}|}|h(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\Big{|}$
$\displaystyle+\int_{T_{0}+|x_{2}-\tilde{x}_{2}|}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|y_{2}-\tilde{y}_{2}|}|h(s)|^{2}ds+\int_{T_{0}+|x_{2}-\tilde{x}_{2}|+|y_{2}-\tilde{y}_{2}|}^{{T_{0}+|x_{2}-\tilde{x}_{2}|+|y_{2}-\tilde{y}_{2}|}+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}}|h(s)|^{2}ds$
$\displaystyle\leq$
$\displaystyle\delta\Big{(}\frac{1+\max_{\\{|\xi_{1}|+|\xi_{2}|\leq|x_{1}|+3|x_{2}|+3\\}}|b(\xi_{1},\xi_{2})|}{\min_{\\{|\xi_{1}|+|\xi_{2}|\leq|x_{1}|+3|x_{2}|+3\\}}|\sigma(\xi_{1},\xi_{2})|}\Big{)}^{2}$
$\displaystyle+\Big{|}\int_{|x_{2}-\tilde{x}_{2}|}^{T_{0}+|x_{2}-\tilde{x}_{2}|}|h(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\Big{|}$
$\displaystyle+\delta\Big{(}\frac{1+\max_{\\{|\xi_{1}|+|\xi_{2}|\leq
2|x_{1}|+|x_{2}|+|y_{1}|+2|y_{2}|+4\\}}|b(\xi_{1},\xi_{2})|}{\min_{\\{|\xi_{1}|+|\xi_{2}|\leq
2|x_{1}|+|x_{2}|+|y_{1}|+2|y_{2}|+4\\}}|\sigma(\xi_{1},\xi_{2})|}\Big{)}^{2}$
$\displaystyle+2\delta\frac{3+|x_{2}|+|y_{2}|}{|y_{2}|}\Big{(}\frac{\max_{\\{|\xi_{1}|+|\xi_{2}|\leq
2|x_{1}|+|x_{2}|+|y_{1}|+|y_{2}|+3\\}}|b(\xi_{1},\xi_{2})|}{\min_{\\{|\xi_{1}|+|\xi_{2}|\leq
2|x_{1}|+|x_{2}|+|y_{1}|+|y_{2}|+3\\}}|\sigma(\xi_{1},\xi_{2})|}\Big{)}^{2}.$

Notice that

$\displaystyle\int_{|x_{2}-\tilde{x}_{2}|}^{T_{0}+|x_{2}-\tilde{x}_{2}|}|h(s)|^{2}ds$
$\displaystyle=$
$\displaystyle\int_{0}^{T_{0}}\Big{|}\frac{\varphi^{(2)}(s)-b(\zeta(|x_{2}-\tilde{x}_{2}|)+\varphi(s)-x_{1},\varphi^{(1)}(s))}{{\sigma}(\zeta(|x_{2}-\tilde{x}_{2}|)+\varphi(s)-x_{1},\varphi^{(1)}(s))}\Big{|}^{2}ds.$

Keeping in mind the assumptions that the coefficients $b$ and ${\sigma}$ are
real-valued, locally Lipschitz continuous functions on $\mathbb{R}^{2}$ and
${\sigma}(x)\neq 0$ for all $x\in\mathbb{R}^{2}$, it is not difficult to prove
that there exists a positive constant $C_{\delta,T_{0},\varphi,x,y}$,
independent of $\tilde{x}$ and $\tilde{y}$, such that

$\displaystyle\Big{|}\int_{|x_{2}-\tilde{x}_{2}|}^{T_{0}+|x_{2}-\tilde{x}_{2}|}|h(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\Big{|}\leq
C_{\delta,T_{0},\varphi,x,y}.$

Here $C_{\delta,T_{0},\varphi,x,y}$ satisfies

$\displaystyle\lim_{\delta\rightarrow 0}C_{\delta,T_{0},\varphi,x,y}=0.$

We can conclude that there exists a positive constant
$D_{\delta,T_{0},\varphi,x,y}$, independent of $\tilde{x}$ and $\tilde{y}$,
such that

$\displaystyle\Big{|}\int_{0}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}}|h(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\Big{|}\leq
D_{\delta,T_{0},\varphi,x,y},$ (3.37)

here

$\displaystyle\lim_{\delta\rightarrow 0}D_{\delta,T_{0},\varphi,x,y}=0.$
(3.38)

By (3.25), (3.33)–(3.37), the definition of $\mathbb{V}$ implies that

$\displaystyle\mathbb{V}(\tilde{x},\tilde{y})$ $\displaystyle\leq$
$\displaystyle
I_{\tilde{x}}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}}((\zeta,\zeta_{1}))$
(3.39) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|+\frac{\tilde{y}_{1}-y_{3}}{\tilde{y}_{2}}}|h(s)|^{2}ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds+D_{\delta,T_{0},\varphi,x,y}$
$\displaystyle\leq$
$\displaystyle\mathbb{V}(x,y)+\epsilon+D_{\delta,T_{0},\varphi,x,y}.$

Subcase 2: $\tilde{y}_{1}<y_{3}$. For
$t\in[0,T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|]$, define

$\displaystyle\widetilde{\zeta}_{2}(t)=\zeta_{2}(t)+\theta^{-}_{j_{1}}(t-(T_{0}+|x_{2}-\tilde{x}_{2}|)+2j_{1}+\sqrt{2}j_{1}).$
(3.40)

Here $j_{1}$ satisfies

$\frac{1}{3}(3+2\sqrt{2})j_{1}^{3}=y_{3}-\tilde{y}_{1}.$

Set

$\displaystyle\widetilde{\zeta}_{1}(t)=\tilde{x}_{2}+\int_{0}^{t}\widetilde{\zeta}_{2}(s)ds;\
\widetilde{\zeta}(t)=\tilde{x}_{1}+\int_{0}^{t}\widetilde{\zeta}_{1}(s)ds.$

We have

$\displaystyle\widetilde{\zeta}_{1}(0)=\tilde{x}_{2};\
\widetilde{\zeta}(0)=\tilde{x}_{1};$
$\displaystyle\widetilde{\zeta}_{1}(|x_{2}-\tilde{x}_{2}|)=x_{2};\ \ \ \ \
\widetilde{\zeta}(|x_{2}-\tilde{x}_{2}|)=\tilde{x}_{1}+\frac{1}{2}|x_{2}-\tilde{x}_{2}|(x_{2}+\tilde{x}_{2});$
$\displaystyle\widetilde{\zeta}_{1}(t)=\varphi^{(1)}(t-|x_{2}-\tilde{x}_{2}|),\
t\in[|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|-2j_{1}-\sqrt{2}j_{1}];$
$\displaystyle\widetilde{\zeta}(t)=\widetilde{\zeta}(|x_{2}-\tilde{x}_{2}|)+\varphi(t-|x_{2}-\tilde{x}_{2}|)-x_{1},\
t\in[|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|-2j_{1}-\sqrt{2}j_{1}].$

For
$t\in[T_{0}+|x_{2}-\tilde{x}_{2}|-2j_{1}-\sqrt{2}j_{1},T_{0}+|x_{2}-\tilde{x}_{2}|]$,

$\displaystyle\widetilde{\zeta}_{1}(t)=\varphi^{(1)}(t-|x_{2}-\tilde{x}_{2}|)+\int_{0}^{t}\theta^{-}_{j_{1}}(s-(T_{0}+|x_{2}-\tilde{x}_{2}|)+2j_{1}+\sqrt{2}j_{1})ds;$
$\displaystyle\widetilde{\zeta}(t)=\widetilde{\zeta}(|x_{2}-\tilde{x}_{2}|)+\varphi(t-|x_{2}-\tilde{x}_{2}|)-x_{1}+\int_{0}^{t}\int_{0}^{s}\theta^{-}_{j_{1}}(l-(T_{0}+|x_{2}-\tilde{x}_{2}|)+2j_{1}+\sqrt{2}j_{1})dlds.$

By (3.22), we have

$\displaystyle\widetilde{\zeta}_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|)=y_{2};$
$\displaystyle\widetilde{\zeta}(T_{0}+|x_{2}-\tilde{x}_{2}|)=\widetilde{\zeta}(|x_{2}-\tilde{x}_{2}|)+y_{1}-x_{1}+\tilde{y}_{1}-y_{3}.$

Hence

$\displaystyle\widetilde{\zeta}_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|)=\tilde{y}_{2};$
$\displaystyle\widetilde{\zeta}(T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|)=\tilde{y}_{1}.$

Let

$\displaystyle\widetilde{h}(s)=\frac{\widetilde{\zeta}_{2}(s)-b(\widetilde{\zeta}(s),\widetilde{\zeta}_{1}(s))}{{\sigma}(\widetilde{\zeta}(s),\widetilde{\zeta}_{1}(s))}.$
(3.41)

Using similar arguments as proving (3.37), we can get that there exists a
positive constant $\widetilde{D}_{\delta,T_{0},\varphi,x,y}$, independent of
$\tilde{x}$ and $\tilde{y}$, such that

$\displaystyle\Big{|}\int_{0}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|}|\widetilde{h}(s)|^{2}ds-\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds\Big{|}\leq\widetilde{D}_{\delta,T_{0},\varphi,x,y},$
(3.42)

here

$\displaystyle\lim_{\delta\rightarrow
0}\widetilde{D}_{\delta,T_{0},\varphi,x,y}=0.$ (3.43)

Similar to the proof of (3.39), we have

$\displaystyle\mathbb{V}(\tilde{x},\tilde{y})$ $\displaystyle\leq$
$\displaystyle
I_{\tilde{x}}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|}((\widetilde{\zeta},\widetilde{\zeta}_{1}))$
(3.44) $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T_{0}+|x_{2}-\tilde{x}_{2}|+|\tilde{y}_{2}-y_{2}|}|\widetilde{h}(s)|^{2}ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{0}^{T_{0}}|h^{\sigma}(s)|^{2}ds+\widetilde{D}_{\delta,T_{0},\varphi,x,y}$
$\displaystyle\leq$
$\displaystyle\mathbb{V}(x,y)+\epsilon+\widetilde{D}_{\delta,T_{0},\varphi,x,y}.$

The arbitrary of $\epsilon$, (3.38), (3.39), (3.43), and (3.44) imply that if
$y_{2}>0$ then (3.7) holds.

Case 2: Using similar arguments as proving Case 1, if $y_{2}<0$ then (3.7)
holds.

Now we consider Case 3: $y_{2}=0$.

Set

$\displaystyle
y_{4}:=\tilde{x}_{1}+\frac{1}{2}|x_{2}-\tilde{x}_{2}|(x_{2}+\tilde{x}_{2})+y_{1}-x_{1}+\frac{1}{2}|\tilde{y}_{2}|^{2}{\rm
sgn}(\tilde{y}_{2}),$

and ${\rm sgn}_{1}={\rm sgn}(\tilde{y}_{1}-y_{4})$.

Define

$\displaystyle\bar{\zeta}_{2}(t)=\left\\{\begin{array}[]{ll}{\rm
sgn}(x_{2}-\tilde{x}_{2}),&\hbox{$t\in[0,|x_{2}-\tilde{x}_{2}|)$;}\\\
\varphi^{(2)}(t-|x_{2}-\tilde{x}_{2}|),&\hbox{$t\in[|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|)$;}\\\
\theta^{{\rm
sgn}_{1}}_{j_{2}}(t-|x_{2}-\tilde{x}_{2}|),&\hbox{$t\in[T_{0}+|x_{2}-\tilde{x}_{2}|,T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2})$;}\\\
{\rm
sgn}(\tilde{y}_{2}-y_{2}),&\hbox{$t\in[T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2},T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2}+|\tilde{y}_{2}-y_{2}|]$.}\end{array}\right.$
(3.49)

Here $j_{2}$ satisfies

$\displaystyle\frac{1}{3}(3+2\sqrt{2})j_{2}^{3}=|\tilde{y}_{1}-y_{4}|.$ (3.50)

Set

$\displaystyle\bar{\zeta}_{1}(t)=\tilde{x}_{2}+\int_{0}^{t}\bar{\zeta}_{2}(s)ds;\
\bar{\zeta}(t)=\tilde{x}_{1}+\int_{0}^{t}\bar{\zeta}_{1}(s)ds.$

Using similar arguments as proving Case 1, we have

$\displaystyle\bar{\zeta}_{1}(0)=\tilde{x}_{2};\
\bar{\zeta}(0)=\tilde{x}_{1};$
$\displaystyle\bar{\zeta}_{1}(T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2}+|\tilde{y}_{2}-y_{2}|)=\tilde{y}_{2};$
$\displaystyle\bar{\zeta}(T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2}+|\tilde{y}_{2}-y_{2}|)=\tilde{y}_{1}.$

Let

$\displaystyle\bar{h}(s)=\frac{\bar{\zeta}_{2}(s)-b(\bar{\zeta}(s),\bar{\zeta}_{1}(s))}{{\sigma}(\bar{\zeta}(s),\bar{\zeta}_{1}(s))},\
t\in[0,T_{0}+|x_{2}-\tilde{x}_{2}|+2j_{2}+\sqrt{2}j_{2}+|\tilde{y}_{2}-y_{2}|].$
(3.51)

Again using similar arguments as in the proof of Case 1, for Case 3, we can
prove (3.7), completing the proof of (3.7).

The proof of (ii) is complete.

Now we prove (iii). Fix $(x_{0},y_{0})\in\mathbb{R}^{2}\times\mathbb{R}^{2}$
and $\delta,\eta>0$, and suppose that $\mathbb{V}(x,y_{0})<\eta$ for all
$x\in\overline{B}_{\delta}(x_{0})$. The proof of (ii) implies that for any
$z\in\overline{B}_{\delta}(x_{0})$, there exist constants
$\delta_{z},T_{z}\in(0,\infty)$ such that, for any $k\in B_{\delta_{z}}(z)$,
there exists a $T^{k}\in(0,T_{z}]$ and $\psi^{k}\in C_{T^{k}}$ with
$\psi^{k}(0)=k$ and $\psi^{k}(T^{k})=y_{0}$ such that
$I^{T^{k}}_{k}(\psi^{k})<\eta$. Applying the Heine-Borel Covering theorem, it
is not difficult to see that (iii) holds.

The proof of Lemma 3.2 is complete. ∎ For $x\in\mathbb{R}^{d}$, let
$\Psi_{t}(x)=\Psi(t,x)$ denote the solution of the system $\dot{Y}=b(Y)$ with
$Y(0)=x$. We now recall the definitions of $\alpha$, $\omega$ limit sets:

$\alpha(x)=\\{p\in\mathbb{R}^{d}:\mbox{there exists a
sequence}\,\,t_{n}\,\,\mbox{ such that}\,\,t_{n}\rightarrow-\infty\,\,\mbox{
and that}\,\,\Psi(t_{n},x)\rightarrow p\\};$
$\omega(x)=\\{p\in\mathbb{R}^{d}:\mbox{there exists a
sequence}\,\,t_{n}\,\,\mbox{ such that}\,\,t_{n}\rightarrow+\infty\,\,\mbox{
and that}\,\,\Psi(t_{n},x)\rightarrow p\\}.$

The Birkhoff center, $B(\Psi)$, of $\Psi$ is defined to be the closure of all
periodic orbits of $\Psi$. By [4, Theorem 3.1], ${\rm supp(}\mu)\subset
B(\Psi)$ for any limiting measure $\mu$ of the invariant measures of the
system (1.2). The following concepts are adopted from [9, Chapter 6]. We
define an equivalent relation $\sim$ between points of $\mathbb{R}^{d}$:

$x\sim y\Leftrightarrow\mathbb{V}(x,y)=0=\mathbb{V}(y,x).$

A set $K$ is called an equivalent class if $\mathbb{V}(x,y)=0$ for any $x,y\in
K$. We note that if the quasipotential $\mathbb{V}$ is continuous at any point
of a limit set of the system $\dot{x}=b(x)$ then it is an equivalent class.
The precise meaning of the assumption (${\bf A}_{4}$) in Section 1 is the
following:
There exists a finite number of compact equivalent classes
$K_{1},K_{2},\cdots,K_{l}$ such that
(1) if $x\in K_{i}$ and $y\notin K_{i}$, then $x\nsim y$;
(2) every $\omega-$ limit set(also $\alpha-$ limit set) of the dynamical
system $\dot{x}=b(x)$ is contained in one of the $K_{i}$. The condition (1)
means that each equivalent class $K_{i}$ is maximal with respect to the
relation $\sim$.

Let $L:=\\{K_{1},K_{2},\cdots,K_{l}\\}$. For simplicity, we denote
$L=\\{1,2,\cdots,l\\}$. Let $W\subset L$. A graph consisting of arrows
$m\rightarrow n(m\in L\backslash W,n\in L,n\neq m)$ is called a $W$-graph if
it satisfies the following conditions:
(1) every point $m\in L\backslash W$ is the initial point of exactly one
arrow;
(2) there are no closed cycles in the graph. We denote by $G(W)$ the set of
$W$-graphs. For each $i\in L$, define

$W(K_{i}):={\rm min}_{g\in G(i)}\sum_{(m\rightarrow n)\in
g}\mathbb{V}(K_{m},K_{n}).$

Suppose that (${\bf A}_{3}$) and (3.1) hold. Then it follows from Khasminskii
[17, Chapter 4] that the system (1.2) admits a unique invariant measure
$\mu^{\varepsilon}$ for $0<{\varepsilon}\leq{\varepsilon}_{0}$. Moreover, the
condition

$\langle b(x),\nabla V\rangle\leq-\gamma,\ {\rm for}\ |x|\gg 1,$ (3.52)

implies that the system $\dot{x}=b(x)$ is dissipative. Furthermore, if
Assumptions 2.1, 2.2 and (${\bf A}_{4}$) are satisfied, then ULDP, the Lemmas
on estimating the escape time from a domain in [9, Chapter 6] and
Khasminskii’s formula (4.14) in [17, Theorem 3.5, p.75] hold. Following the
proofs of [9, Theorem 4.1, p.166], [9, Theorem 4.2, p.167], we deduce the
following:

######  Theorem 3.1.

Suppose that Assumptions 2.1, 2.2, (${\bf A}_{3}$), (${\bf A}_{4}$) and (3.1)
are satisfied. Then for any $\gamma>0$ there exist
$0<{\varepsilon}^{*}<{\varepsilon}_{0}$ and $\rho>0$ (which can be chosen
arbitrarily small) such that when $0<{\varepsilon}<{\varepsilon}_{*}$ and
$i\in L$,

$\exp\\{-{\varepsilon}^{-1}\left(W(K_{i})-{\rm min}_{j\in
L}W(K_{j})+\gamma\right)\\}\leq\mu^{{\varepsilon}}(B(K_{i},\rho))\leq\exp\\{-{\varepsilon}^{-1}\left(W(K_{i})-{\rm
min}_{j\in L}W(K_{j})-\gamma\right)\\},$

where $B(K_{i},\rho)$ denotes the $\rho$-neighborhood of $K_{i}$. Let

$\nu={\rm min}_{j\in L}\\{W(K_{j})\\}\ {\rm and}\ L_{0}:=\\{i_{1},\
i_{2},\cdots,\ i_{k}\\}=\\{i\in L:W(K_{i})=\nu\\}.$

If $\mu^{{\varepsilon}_{k}}$ converges weakly to $\mu$ as
${\varepsilon}_{k}\rightarrow 0$, then the support of $\mu$, ${\rm
supp(}\mu)$, is contained in $\bigcup_{j=1}^{k}K_{i_{j}}$. In particular, if
$L_{0}=\\{i_{0}\\}$ and there exists only one normalized invariant measure
$\mu_{0}$ of the system $\dot{x}=b(x)$ whose support is contained in
$K_{i_{0}}$, then $\mu^{{\varepsilon}}$ converges weakly to $\mu_{0}$ as
${\varepsilon}\rightarrow 0$.

In view of Theorem 2.1, following the proofs in the paper [15] by Hwang and
Sheu we obtain the following result which is applicable to unbounded
coefficients.

######  Theorem 3.2.

Suppose that Assumptions 2.1, 2.2, (${\bf A}_{3}$), (${\bf A}_{4}$) and (3.1)
are satisfied. Then for any $\alpha>0,\exists\ \delta>0$ such that for any
compact set $F$ there is $0<{\varepsilon}^{*}_{0}<{\varepsilon}^{*}$, the
following holds

$\left|\mathbb{E}_{x}[f(X^{{\varepsilon}}(T))]-\int
f(y)\mu^{{\varepsilon}}(dy)\right|\leq\|f\|_{\infty}\exp(-{\varepsilon}^{-1}\delta)$

where

$x\in F,\quad\quad f\in C_{b}(\mathbb{R}^{d}),\quad\quad
0<{\varepsilon}\leq{\varepsilon}_{0}^{*},\quad\quad
T=\exp\left({\varepsilon}^{-1}(\Lambda+\alpha)\right)\ {\rm with}$
$\Lambda=\nu-{\rm min}\\{\mathbb{V}(g):g\in G(\\{i,j\\}),i,j\in L,i\neq j\\}.$

A subset $\mathcal{R}\subset\mathbb{R}^{d}$ is said to be a repeller (an
attractor) for the solution flow $\Psi$ of $\dot{x}=b(x)$ provided: (i)
$\mathcal{R}$ is nonempty, compact and invariant; and (ii) $\mathcal{R}$ has a
neighborhood $N\subset\mathbb{R}^{d}$, called a fundamental neighborhood of
$\mathcal{R}$, such that $\displaystyle\lim_{t\rightarrow-\infty}{\rm
dist}\big{(}\Psi_{t}(x),\mathcal{R}\big{)}=0$
($\displaystyle\lim_{t\rightarrow+\infty}{\rm
dist}\big{(}\Psi_{t}(x),\mathcal{R}\big{)}=0$) uniformly in $x\in N$. In the
case of attractor, we call $\mathcal{R}$ Lyapunov stable.

Substituting (${\bf A}_{1}$) by the Assumptions 2.1 and 2.2, following the
same proof of [36, Theorem 3.1, Corollary 3.3, p.78] we obtain the following
two results, which significantly extend the scope of applications.

######  Theorem 3.3.

Suppose that Assumptions 2.1, 2.2, (${\bf A}_{3}$) and (3.1) hold. Suppose
that there exist $m$ points $x_{1},x_{2},\cdots,x_{m}$ in $\mathbb{R}^{d}$, an
attractor $\mathcal{A}=:\mathcal{S}_{m+1}$ and compact equivalent classes
$\mathcal{S}_{1},\mathcal{S}_{2},\cdots,\mathcal{S}_{m}$ such that
$\mathcal{S}_{i}\supset\alpha(x_{i}),i=1,2,\cdots,m$,
$\\{\mathcal{A},\mathcal{S}_{i},i=1,2,\cdots,m\\}$ are pairwise disjoint,
$\omega(x_{i})\subset\mathcal{S}_{i+1},i=1,2,\cdots,m$. Then there exist
neighborhoods $U_{i}$ of $\mathcal{S}_{i},i=1,2,\cdots,m$, $\kappa>0$ and
$0<\varepsilon_{*}\leq{\varepsilon}_{0}$ such that for any
${\varepsilon}\in(0,\varepsilon_{*})$, we have

$\mu^{{\varepsilon}}(U_{i})\leq\exp\\{-\kappa/{\varepsilon}\\}.$

As a result, if $\mu^{{\varepsilon}_{j}}$ converges weakly $\mu$ as
${\varepsilon}_{j}\rightarrow 0$, then $\mu(U_{i})=0,i=1,2,\cdots,m$.

######  Remark 3.1.

Note that if $m=1$ and $\mathcal{S}_{1}$ is a repeller of $\dot{x}=b(x)$ then
its dissipation implies that there must be an attractor $\mathcal{S}_{2}$ such
that $\dot{x}=b(x)$ admits a connecting orbit connecting $\mathcal{S}_{1}$ and
$\mathcal{S}_{2}$. Therefore, limiting measure always stays away from any
repelling equivalent class.

##  4 Applications

In this part, we will provide several interesting models to give the precise
description of the support of the limiting measures of the invariant measures
of corresponding SDEs (1.2) using the results in the previous sections.

###  4.1 The realization of Freidlin’s two phase portraits of dynamical
systems

In the book [9, Figure 12, p.151], Freidlin and Wentzell depicted the phase
portraits of some dynamical system (see Figure 1 below) in which there exist
exactly four equivalent classes. They designed the following transitive
difficulty matrix

$\ \big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times
4}:=\left(\begin{array}[]{ccccc}&0&0&9&9\\\ &1&0&9&9\\\ &7&6&0&6\\\
&1&0&0&0\end{array}\right)$

and used it to illustrate their estimates of the transition probabilities
between the equivalent classes, the concentration of the limiting measures of
invariant measures (see [9, p.170]), the exit position on the boundary of a
domain (see [9, p.175]) and the asymptotics of the mathematical expectation of
the exit time from a domain (see [9, p.177]). However, they did not provide
examples of stochastic planar dynamical systems that have the phase portraits
in Figure 1, which is a nontrivial matter. They did not carry out the
computation of the matrix $\big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times 4}$
for concrete systems either, which is actually a difficult task for a given
system of SDEs.

In the following, we first present a planar polynomial system to realize the
global phase portraits of Figure 1 and then precisely compute the transition
difficulty matrix $\big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times 4}$. As a
consequence, we are able to identify the supports of the limiting measures of
the invariant measures of the perturbed systems.

######  Example 4.1.

Let $x:=(x_{1},x_{2})$ and consider the following deterministic planar
polynomial system

$\frac{dx}{dt}=b_{1}(x):=\left(x_{2}-F^{\prime}(x_{1})\big{(}H(x_{1},x_{2})+1\big{)},-F^{\prime}(x_{1})-x_{2}\big{(}H(x_{1},x_{2})+1\big{)}\right)$
(4.1)

where

$F(x_{1}):=\frac{x_{1}^{4}}{4}+\frac{x_{1}^{3}}{3}-x_{1}^{2},\
F^{\prime}(x_{1})=x_{1}(x_{1}-1)(x_{1}+2),\
H(x_{1},x_{2}):=\frac{x_{2}^{2}}{2}+F(x_{1}).$

By the chain rule,

$\frac{dH(x_{1},x_{2})}{dt}=-\big{(}H(x_{1},x_{2})+1\big{)}|\nabla H|^{2}.$

Therefore, the equilibria of the system (4.1) are

$K_{1}(-2,0),\ K_{3}(1,0),\ K_{4}(0,0)$

and (4.1) admits the unique limit cycle

$K_{2}:H(x_{1},x_{2})+1=0.$

By linearized technique, we can prove that $K_{1}$ is an unstable node,
$K_{4}$ is a saddle, $K_{3}$ is a stable focus and $K_{2}$ is a stable limit
cycle. Applying the LaSalle invariant principle to the interior and exterior
of $K_{2}$, respectively, we get the global portraits depicted in Figure 1,

Figure 1: The phase portrait of system (4.1).

which is the realization of [9, Figure12, p.151]. It is easy to see that

$H(K_{1})=-\frac{8}{3}={\rm
min}\\{H(x_{1},x_{2})|(x_{1},x_{2})\in\mathbb{R}^{2}\\},\ H(K_{2})=-1,\
H(K_{3})=-\frac{5}{12},\ H(K_{4})=0.$

Now we consider the random perturbation of (4.1) by an additive noise:

$dx^{\varepsilon}(t)=b_{1}(x^{\varepsilon}(t))dt+\sqrt{{\varepsilon}}dB(t),$
(4.2)

where $\\{B(t)\\}_{t\geq 0}$ is a $2$-dimensional Brownian motion. To
characterize the support of the limiting measures of the invariant measures of
the system (4.2), we verify the various conditions listed in the theorems in
Section 3. The Assumptions 2.1 and the condition (${\bf A}_{4}$) are obviously
satisfied. Thus, we only need to verify the Assumptions 2.2 and the condition
(${\bf A}_{3}$). Define a Lyapunov function
$V(x_{1},x_{2})=H(x_{1},x_{2})+3>0$. Clearly,
$\lim_{|(x_{1},x_{2})|\rightarrow\infty}V(x_{1},x_{2})=\infty$ and

$\text{Trace}\big{(}\sigma^{*}(x_{1},x_{2})\nabla^{2}V(x_{1},x_{2})\sigma(x_{1},x_{2})\big{)}=3x_{1}^{2}+2x_{1}-1>-2$

which means that (2.8) holds. Besides, the left-hand of (2.7) with $\theta=2$
and $\eta=1$ is

$-(V(x_{1},x_{2})-2)|\nabla V|^{2}+(3x_{1}^{2}+2x_{1}-1)+\frac{|\nabla
V|^{2}}{V}=:J_{1}+J_{2}+J_{3}.$

We can verify that

$\lim_{|(x_{1},x_{2})|\rightarrow\infty}J_{1}=-\infty,\
\lim_{|(x_{1},x_{2})|\rightarrow\infty}\frac{J_{i}}{J_{1}}=0\,\,{\rm
for}\,\,i=2,3.$

Thus there exists a positive constant $C$ such that

$J_{1}+J_{2}+J_{3}\leq\big{(}\frac{1}{2}J_{1}+C\big{)}\rightarrow-\infty\ {\rm
as}\ |(x_{1},x_{2})|\rightarrow\infty$

which implies that both (2.7) and (${\bf A}_{3}$) hold. Thus we have proved
that Assumptions 2.1 and 2.2, (${\bf A}_{3}$) and (${\bf A}_{4}$) are
satisfied.

From the phase portraits for (4.1) in Figure 1, we see that

$\ \big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times
4}:=\left(\begin{array}[]{ccccc}&0&0&a&a\\\ &b&0&a&a\\\ &b+c&c&0&c\\\
&b&0&0&0\end{array}\right)$

where

$a:=\mathbb{V}(K_{2},K_{4}),\ b:=\mathbb{V}(K_{2},K_{1}),\
c:=\mathbb{V}(K_{3},K_{4}).$

We shall prove that

$\mathbb{V}(K_{2},K_{4})=1,\ \mathbb{V}(K_{2},K_{1})=\frac{25}{9},\
\mathbb{V}(K_{3},K_{4})=\frac{95}{144}.$ (4.3)

Let

$U(x_{1},x_{2}):=\frac{H^{2}(x_{1},x_{2})}{2}+H(x_{1},x_{2}),\
l(x_{1},x_{2}):=(x_{2},-x_{1}(x_{1}-1)(x_{1}+2)).$

Then $\langle\nabla U(x_{1},x_{2}),l(x_{1},x_{2})\rangle\equiv 0$ and (4.1)
can be rewritten as a quasipotential system

$\frac{dx}{dt}=b_{1}(x)=-\nabla U(x_{1},x_{2})+l(x_{1},x_{2}).$ (4.4)

Note that

$U(K_{1})=\frac{8}{9},\ U(K_{2})=-\frac{1}{2},\ U(K_{3})=-\frac{95}{288},\
U(K_{4})=0.$

Following the arguments from Section 3 of [9, Chapter 4], we proceed as
follows. For any $T>0$ and $\varphi\in{\bf AC}_{X}^{T}$ with $\varphi_{T}=Y$,
we have

$\displaystyle I_{X}^{T}(\varphi)=$
$\displaystyle\frac{1}{2}\int_{0}^{T}|\dot{\varphi}_{s}+\nabla
U(\varphi_{s})-l(\varphi_{s})|^{2}ds$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}|\dot{\varphi}_{s}-\nabla
U(\varphi_{s})-l(\varphi_{s})|^{2}ds+2\int_{0}^{T}\langle\dot{\varphi}_{s},\nabla
U(\varphi_{s})\rangle ds$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{T}|\dot{\varphi}_{s}-\nabla
U(\varphi_{s})-l(\varphi_{s})|^{2}ds+2(U(Y)-U(X))$ $\displaystyle\geq$
$\displaystyle 2(U(Y)-U(X)).$

This shows that for any $X,Y\in\mathbb{R}^{2}$

$I_{X}^{T}(\varphi)=\frac{1}{2}\int_{0}^{T}|\dot{\varphi}_{s}-\nabla
U(\varphi_{s})-l(\varphi_{s})|^{2}+2(U(Y)-U(X))$ (4.5)

and

$\mathbb{V}(X,Y)\geq 2(U(Y)-U(X)).$ (4.6)

In particular, letting $Y=K_{1}$ and $X$ lie in the interior of $K_{2}$, we
have

$\mathbb{V}(X,K_{1})\geq 2\Big{(}\frac{8}{9}-U(X)\Big{)}$

from which and the continuity of $\mathbb{V}(X,K_{1})$ it follows that

$\mathbb{V}(K_{2},K_{1})\geq
2\Big{(}\frac{8}{9}-U(K_{2})\Big{)}=\frac{25}{9}.$

In order to study the extremals of $I(\varphi)$ from $K_{2}$ to $K_{1}$, we
have to consider the following equations for the extremals

$\frac{dX}{dt}=\nabla U(x_{1},x_{2})+l(x_{1},x_{2})$ (4.7)

whose equilibria still are

$K_{1}(-2,0),\ K_{3}(1,0),\ K_{4}(0,0)$

and $K_{2}$ is the limit cycle of (4.7). Its phase portraits are drawn in
Figure 2 below.

Figure 2: The phase portrait of system (4.7).

Let $X\neq K_{1}$ be a point lying inside $K_{2}$ and denote by
$\varphi_{t},t\in\mathbb{R}$ the solution of (4.7) passing through $X$. From
Figure 2, $\alpha(X)=K_{2}$ and $\omega(X)=K_{1}$. For any $T>0$,

$2(U(\varphi_{T})-U(\varphi_{-T}))=I(\varphi)\geq\mathbb{V}(\varphi_{-T},\varphi_{T}).$

Let $T\rightarrow\infty$ to obtain that

$\frac{25}{9}=2(U(K_{1})-U(K_{2}))\geq\mathbb{V}(K_{2},K_{1}).$

This proves that $b=\mathbb{V}(K_{2},K_{1})=\frac{25}{9}$.

Fix $Y=K_{4}$ in (4.6), we get

$\mathbb{V}(X,K_{4})\geq 2(U(K_{4})-U(X))=-2U(X).$

From the continuity of $\mathbb{V}(X,K_{4})$ it follows that

$\mathbb{V}(K_{2},K_{4})\geq-2U(K_{2})=1.$

Choose $X_{0}\neq K_{4}$ in the stable manifold of $K_{4}$ illustrated in
Figure 2 and let $\varphi_{t},t\in\mathbb{R}$ be the solution of (4.7) passing
through $X_{0}$. From Figure 2, $\alpha(X_{0})=K_{2}$ and
$\omega(X_{0})=K_{4}$. For any $T>0$,

$2(U(\varphi_{T})-U(\varphi_{-T}))=I(\varphi)\geq\mathbb{V}(\varphi_{-T},\varphi_{T}).$

Letting $T\rightarrow\infty$, we have $1\geq\mathbb{V}(K_{2},K_{4})$, which
means that $\mathbb{V}(K_{2},K_{4})=1$. Analogously, we can prove that
$\mathbb{V}(K_{3},K_{4})=\frac{95}{144}$.

Finally, from (4.3) we conclude that

$\ \big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times
4}:=\left(\begin{array}[]{ccccc}&0&0&1&1\\\ &\frac{25}{9}&0&1&1\\\
&\frac{55}{16}&\frac{95}{144}&0&\frac{95}{144}\\\
&\frac{25}{9}&0&0&0\end{array}\right).$

Let $\mu^{\varepsilon}$ be the invariant measure of the solution of the SDEs
(4.2) for $0<\varepsilon\ll 1$. Then for any $\varepsilon_{i}\rightarrow 0$
such that $\mu^{\varepsilon_{i}}$ converges weakly to $\mu$, Theorem 3.3 and
Remark 3.1 imply that $\mu(K_{1})=\mu(K_{4})=0$. Because
$\mathbb{V}(K_{3},K_{2})=\frac{95}{144}<1=\mathbb{V}(K_{2},K_{3})$, it follows
from Theorem 3.1 that $\mu^{\varepsilon}$ converges weakly to the unique
ergodic measure
$\mu_{K_{2}}(\cdot):=\frac{1}{T}\int_{0}^{T}\delta_{\varphi_{t}(X^{*})}(\cdot)dt$
supported on $K_{2}$ as $\varepsilon\rightarrow 0$, where $X^{*}\in K_{2}$ and
$T$ is the period of the limit cycle $K_{2}$. The density for
$\mu_{K_{2}}(\cdot)$ is

$m_{K_{2}}(X)=\frac{1}{T}\frac{1}{|\nabla H(X)|},\ X:=(x_{1},x_{2})\in K_{2}.$

Consider the following Cauchy problem $\mathbb{R}_{+}\times\mathbb{R}^{2}$:

$\begin{cases}\frac{\partial u^{\epsilon}(t,X)}{\partial
t}=\frac{\epsilon}{2}\Delta u^{\epsilon}(t,X)+\langle b(X),\nabla
u^{\epsilon}(t,X)\rangle,\\\ u^{\epsilon}(0,X)=g(X)\in
C_{b}(\mathbb{R}^{2}).\end{cases}$ (4.8)

Denote by $\mathcal{B}(K_{2})$ and $\mathcal{B}(K_{3})$ the attracting domains
of $K_{2}$ and $K_{3}$, respectively. Define

$g_{K_{2}}:=\frac{1}{T}\oint_{K_{2}}g(X)\frac{1}{|\nabla H(X)|}dl.$

Here the integral is respect to arc measure. Then from the discussion above
and [12, Theorem 2.2], we derive the following.

######  Proposition 4.1.

$\mathbb{V}(K_{3},K_{2})=\frac{95}{144},\mathbb{V}(K_{2},K_{3})=1$ and
$\mu^{\varepsilon}$ converges weakly to the unique ergodic measure
$\mu_{K_{2}}(\cdot)$ as $\varepsilon\rightarrow 0$.

Let $\lim_{\epsilon\rightarrow 0}T(\epsilon)=+\infty$. Then we have
(1) If $X\in\mathcal{B}(K_{3})$, then

$\lim_{\epsilon\rightarrow
0}u^{\epsilon}(T(\epsilon),X)=\begin{cases}g(K_{3}),&\text{if
~{}$\limsup_{\epsilon\rightarrow 0}\epsilon\log
T(\epsilon)<\frac{95}{144}$},\\\ g_{K_{2}},&\text{if
~{}$\liminf_{\epsilon\rightarrow 0}\epsilon\log
T(\epsilon)>\frac{95}{144}$}.\end{cases}$

(2) If $X\in\mathcal{B}(K_{2})$, then

$\lim_{\epsilon\rightarrow 0}u^{\epsilon}(T(\epsilon),X)=g_{K_{2}}.$

######  Example 4.2.

Let $X:=(x_{1},x_{2})$ and consider the following deterministic planar
polynomial system

$\frac{dX}{dt}=b_{2}(X):=\left(x_{2}-F^{\prime}(x_{1})\big{(}H(x_{1},x_{2})+1\big{)},-F^{\prime}(x_{1})-x_{2}\big{(}H(x_{1},x_{2})+1\big{)}\right)$
(4.9)

where

$F(x_{1}):=\frac{x_{1}^{4}}{4}-\frac{x_{1}^{3}}{3}-x_{1}^{2},\
F^{\prime}(x_{1})=x_{1}(x_{1}+1)(x_{1}-2),\
H(x_{1},x_{2}):=\frac{x_{2}^{2}}{2}+F(x_{1})$

and its random perturbed system

$dX^{\varepsilon}(t)=b_{2}(X^{\varepsilon}(t))dt+\sqrt{{\varepsilon}}dB(t).$
(4.10)

The system (4.9) admits three equilibria

$K_{1}(-1,0),\ K_{3}(0,0),\ K_{4}(2,0)$

and the stable limit cycle

$K_{2}:H(x_{1},x_{2})+1=0.$

The global phase portraits of (4.9) are depicted in Figure 3 below,

Figure 3: The phase portrait of system (4.9).

which is the realization of [12, Figure 1(a), p.557]. In the same way as in
Example 4.1, we can show that the transition difficulty matrix is given by

$\ \big{(}\mathbb{V}(K_{i},K_{j})\big{)}_{4\times
4}:=\left(\begin{array}[]{ccccc}&0&\frac{95}{144}&\frac{95}{144}&\frac{55}{16}\\\
&1&0&1&\frac{25}{9}\\\ &0&0&0&\frac{25}{9}\\\ &1&0&1&0\end{array}\right).$

Let $\mu^{\varepsilon}$ be the invariant measure of the SDEs (4.10) for
$0<\varepsilon\ll 1$. Then $\mu^{\varepsilon}$ converges weakly to the unique
ergodic measure
$\mu_{K_{2}}(\cdot):=\frac{1}{T}\int_{0}^{T}\delta_{\varphi_{t}(X^{*})}(\cdot)dt$
supported on $K_{2}$ as $\varepsilon\rightarrow 0$, where $X^{*}\in K_{2}$ and
$T$ is the period of the limit cycle $K_{2}$. Furthermore, the same result as
Proposition 4.1 holds with $K_{3}$ replaced by $K_{1}$.

###  4.2 Stochastic van der Pol equation and single diode circuit

######  Example 4.3.

van der Pol (1927) established the following well known triode circuit
equation

$X^{(2)}+(X^{2}-1)X^{(1)}+X=0$

which is equivalent to

$\begin{cases}X^{(1)}=Y,\\\ Y^{(1)}=-[(X^{2}-1)Y+X]\end{cases}$ (4.11)

and now called the van der Pol equation. The Birkhoff center is composed of
the origin $O$ and the limit cycle $\Gamma$, where $O$ is a repeller and
$\Gamma$ attracts all points except the origin $O$. The global portraits of
(4.11) are shown in Figure 4, see [37].

Figure 4: The unique limit cycle.

Consider the random perturbation of the van der Pol equation:

$X_{\varepsilon}^{(2)}+(X_{\varepsilon}^{2}-1)X_{\varepsilon}^{(1)}+X_{\varepsilon}=\sqrt{\varepsilon}\sigma(X_{\varepsilon},X_{\varepsilon}^{(1)})\dot{B}_{t},\
\sigma(X_{\varepsilon},X_{\varepsilon}^{(1)})>0.$ (4.12)

Let
$Y_{\varepsilon}:=(Y_{\varepsilon}^{1},Y_{\varepsilon}^{2}):=(X_{\varepsilon},X_{\varepsilon}^{(1)})$.
Then (4.12) is equivalent to the following system of SDEs:

$\begin{cases}dY_{\varepsilon}^{1}(t)=Y_{\varepsilon}^{2}(t)dt,\\\
dY_{\varepsilon}^{2}(t)=-[(Y_{\varepsilon}^{1}(t)^{2}-1)Y_{\varepsilon}^{2}(t)+Y_{\varepsilon}^{1}(t)]dt+\sqrt{{\varepsilon}}{\sigma}\big{(}Y_{\varepsilon}^{1}(t),Y_{\varepsilon}^{2}(t)\big{)}dB(t),\\\
(Y_{\varepsilon}^{1},Y_{\varepsilon}^{2})(0)=(x_{1},x_{2})=:x\in\mathbb{R}^{2}.\end{cases}$
(4.13)

If the first equation of (4.13) were also perturbed by a Brownian motion,
independent of $B(t)$, then one could prove that the limiting measure of
invariant measures for corresponding system (4.13) is the Haar measure of
$\Gamma$ by Theorem 3.3 and Remark 3.1, the readers are referred to the
discussion below. However, for system (4.13), which has interesting physical
background, we are not able to prove that its quasipotential $\mathbb{V}(x,y)$
is continuous on $\mathbb{R}^{2}\times\mathbb{R}^{2}$, so it is not
straightforward to get the above conclusion. But, with the help of Lemma 3.2,
we still can prove the same result.

Let $b(Y^{1},Y^{2})$ and $(0,\sigma(Y^{1},Y^{2}))^{*}$ denote the drift and
diffusion coefficient of (4.13), respectively. According to Nevelson’s, as in
[17, p.82], we construct the following Lyapunov function for (4.13)

$V(Y^{1},Y^{2})=\frac{\sqrt{5}-1}{24}(Y^{1})^{4}+\frac{1}{2}\left(Y^{2}+\frac{(Y^{1})^{3}}{3}-\frac{\sqrt{5}+1}{2}Y^{1}\right)^{2}.$
(4.14)

We have

$V_{Y^{1}}=\frac{(Y^{1})^{5}}{3}-\frac{3\sqrt{5}+5}{6}(Y^{1})^{3}+(Y^{1})^{2}Y^{2}+\frac{\sqrt{5}+3}{2}Y^{1}-\frac{\sqrt{5}+1}{2}Y^{2};$
$V_{Y^{2}}=\frac{(Y^{1})^{3}}{3}-\frac{\sqrt{5}+1}{2}Y^{1}+Y^{2};$ $\langle
b(Y^{1},Y^{2}),\nabla
V(Y^{1},Y^{2})\rangle=-\frac{(Y^{1})^{4}}{3}+\frac{\sqrt{5}+1}{2}(Y^{1})^{2}-\frac{\sqrt{5}-1}{2}(Y^{2})^{2}.$

We have the following result

######  Proposition 4.2.

Suppose that ${\sigma}(Y^{1},Y^{2})$ is a positive, locally Lipschitz
continuous function satisfying ${\sigma}^{2}(Y^{1},Y^{2})\leq
c_{1}((Y^{1})^{4}+(Y^{2})^{2}+1)$, $c_{1}>0$. Then there exists
$\varepsilon_{0}>0$ such that for any given
$\varepsilon\in(0,\varepsilon_{0}]$, the set $\mathcal{J^{\varepsilon}}$ of
invariant measures of the system (4.13) is nonempty and
$\mathcal{J}:=\bigcup_{0<\varepsilon\leq\varepsilon_{0}}\mathcal{J^{\varepsilon}}$
is tight. Furthermore, let
$\mu^{\varepsilon_{i}}\in\mathcal{J}^{\varepsilon_{i}}$ such that
$\mu^{\varepsilon_{i}}$ converges weakly to $\mu$ as
$\varepsilon_{i}\rightarrow 0$, then $\mu=\mu_{\Gamma}$, where $\Gamma$ is the
unique limit cycle of the deterministic van der Pol equation (4.11)
illustrated in Figure 4 and $\mu_{\Gamma}$ is its corresponding Haar measure
supported on $\Gamma$.

Proof  Let $0<\varepsilon\leq\varepsilon_{0}:=\frac{1}{6c_{1}}$. By the
assumption of $\sigma$, there exists a positive constant $R$ such that for
$(Y^{1})^{2}+(Y^{2})^{2}\geq R,$

$\displaystyle\mathcal{L}^{\varepsilon}V(Y^{1},Y^{2})$ $\displaystyle:=$
$\displaystyle\langle b(Y^{1},Y^{2}),\nabla
V(Y^{1},Y^{2})\rangle+\frac{\varepsilon}{2}{\sigma}^{2}(Y^{1},Y^{2})$
$\displaystyle\leq$
$\displaystyle-\frac{(Y^{1})^{4}}{3}+\frac{\sqrt{5}+1}{2}(Y^{1})^{2}-\frac{\sqrt{5}-1}{2}(Y^{2})^{2}+\frac{\varepsilon
c_{1}}{2}\left((Y^{1})^{4}+(Y^{2})^{2}+1\right)$ $\displaystyle\leq$
$\displaystyle-\frac{(Y^{1})^{4}}{4}+\frac{\sqrt{5}+1}{2}(Y^{1})^{2}-\frac{(Y^{2})^{2}}{4}+\frac{1}{12}$
$\displaystyle\leq$
$\displaystyle-\frac{1}{5}\left((Y^{1})^{4}+(Y^{2})^{2}\right),$

where $\mathcal{L}^{\varepsilon}$ denotes the generator of the system (4.13).
Denote by $\mathcal{J^{\varepsilon}}$ the set of all invariant measures of
(4.13) for a given $0<\varepsilon\leq\varepsilon_{0}$. By [4, Theorem 3.1],
$\mathcal{J^{\varepsilon}}$ is nonempty and
$\mathcal{J}:=\bigcup_{0<\varepsilon\leq\varepsilon_{0}}\mathcal{J^{\varepsilon}}$
is tight. Furthermore, let
$\mu^{\varepsilon_{i}}\in\mathcal{J}^{\varepsilon_{i}}$ such that
$\mu^{\varepsilon_{i}}$ converges weakly to $\mu$ as
$\varepsilon_{i}\rightarrow 0$. Then $\mu$ is an invariant measure of the
deterministic van der Pol equation (4.11) and its support is contained in the
Birkhoff center $\Gamma\cup\\{O\\}$. The origin of the deterministic van der
Pol equation is a repeller and the unique limit cycle $\Gamma$ is stable,
which is illustrated in Figure 4 (see [37]).

In order to see that the system (4.13) admits ULDP, we are going to check that
the Lyapunov function $V(Y^{1},Y^{2})$ constructed above satisfies the
Assumption 2.2. (2.6) and (2.8) are obvious because

$\text{Trace}\big{(}(0,\sigma(Y^{1},Y^{2}))\nabla^{2}V(Y^{1},Y^{2})(0,\sigma(Y^{1},Y^{2}))^{*}\big{)}=\sigma^{2}(Y^{1},Y^{2})>0.$

For any given $0<\varepsilon\leq\varepsilon_{0}$, let
$\theta=\frac{\varepsilon}{2}$ and $\eta=\frac{8}{\varepsilon}$ in (2.7), then
we have the following estimate for the left-hand of (2.7):

$J\leq\mathcal{L}^{\varepsilon}V(Y^{1},Y^{2})\leq-\frac{1}{5}\left((Y^{1})^{4}+(Y^{2})^{2}\right),\
{\rm for}\ (Y^{1})^{2}+(Y^{2})^{2}\geq R$

which implies that (2.7) holds. Applying Theorem 2.1, we conclude that the
system (4.13) admits ULDP.

Finally, we shall use Lemma 3.2 and the idea to prove [36, Theorem 3.3] to get
that there is no concentration for limiting measures on the repeller $O(0,0)$.
This will prove that when $\varepsilon\rightarrow 0$ the invariant measure
$\mu^{\varepsilon}$ of (4.13) converges weakly to the Haar measure
$\mu_{\Gamma}$.

In order to keep the same notation as the proof of [36, Theorem 3.1], we let
$\mathcal{R}=O(0,0)$.

From the proof of [36, Lemma 4.1], we know that [36, Lemma 4.1] still holds as
soon as ULDP is satisfied. Thus, applying this lemma, we get that there exist
$\delta_{1},\ \delta^{*}_{2}\in(0,\delta),\ \delta_{1}>\delta^{*}_{2}$ and
$s_{0}>0$ such that
$\mathbb{V}(\partial\mathcal{R}_{\delta_{1}},\partial\mathcal{R}_{\delta^{*}_{2}})\geq
s_{0}$. Set $\delta_{2}=\delta^{*}_{2}/2$. Applying [36, Proposition 2.1], we
have

$T_{0}=\inf\big{\\{}u\geq
0:\Psi_{t}\big{(}\overline{\mathcal{R}_{\delta_{1}}}\backslash\mathcal{R}_{\delta_{2}}\big{)}\subset(\mathcal{R}_{\delta})^{c},t\geq
u\big{\\}}<+\infty.$

Let

$F_{0}=\\{\psi\in{\bf
C}_{T_{0}}:\psi(0)\in\overline{\mathcal{R}_{\delta_{1}}}\backslash\mathcal{R}_{\delta_{2}},\psi(T_{0})\in\overline{\mathcal{R}_{\delta_{1}}}\\}.$

Then it is a closed subset of ${\bf C}_{T_{0}}$,
$F_{0}^{0}$($:=\\{\varphi(0):\varphi\in F_{0}\\}$) is bounded and $F_{0}$ does
not contain any solution of system (4.11) by the definition of $T_{0}$. By
[36, Proposition 2.4], we have

$s_{1}:=\inf\\{I^{T_{0}}_{x}(\psi):\psi\in F_{0},\
x=\psi(0)\in\overline{\mathcal{R}_{\delta_{1}}}\backslash\mathcal{R}_{\delta_{2}}\
\\}=:I^{T_{0}}(F_{0})>0.$

Since $O$ is an equilibrium of (4.11), $\mathbb{V}(O,O)=0$ by its definition.
Applying (ii) of Lemma 3.2, we know that $\mathbb{V}(x,y)$ is continuous at
$(O,O)$. Thus, there exists $0<\delta_{3}<\delta_{2}$ such that

$\mathbb{V}(O,y),\ \mathbb{V}(x,O)<\frac{1}{20}(s_{0}\wedge s_{1})\ {\rm for\
all}\ x,y\in\overline{\mathcal{R}_{\delta_{3}}}.$ (4.15)

Using [36, Proposition 2.1], we get that

$T_{3}=\inf\big{\\{}u\geq
0:\Psi_{t}\big{(}\overline{\mathcal{R}_{\delta_{1}}}\backslash\mathcal{R}_{\delta_{3}}\big{)}\subset(\mathcal{R}_{\delta})^{c},\forall
t\geq u\big{\\}}<+\infty.$ (4.16)

Then obviously $T_{0}\leq T_{3}$.

Fix a point $y_{0}\in\partial{\mathcal{R}_{\delta_{3}}}$. Then by (4.15),
there exist $T_{2}$ and $\tilde{\psi}^{0}\in{\bf C}_{T_{2}}$ such that
$\tilde{\psi}^{0}(0)=O,\ \tilde{\psi}^{0}(T_{2})=y_{0}$ and

$I^{T_{2}}_{O}(\tilde{\psi}^{0})<\frac{1}{20}(s_{0}\wedge s_{1}).$ (4.17)

Besides, from (iii) of Lemma 3.2 it follows that there exists $T_{1}$ such
that for any $x\in\overline{\mathcal{R}_{\delta_{3}}}$, there exist $T^{x}\leq
T_{1}$ and $\tilde{\psi}^{x}\in{\bf C}_{T^{x}}$ with $\tilde{\psi}^{x}(0)=x,\
\tilde{\psi}^{x}(T^{x})=O$ satisfying

$I^{T^{x}}_{x}(\tilde{\psi}^{x})<\frac{1}{20}(s_{0}\wedge s_{1}).$ (4.18)

Let $T=T_{1}+T_{2}+T_{3}$. Then for each
$x\in\overline{\mathcal{R}_{\delta_{2}}}$, if
$x\in\overline{\mathcal{R}_{\delta_{2}}}\backslash\mathcal{R}_{\delta_{3}}$,
$\psi^{x}:=\Psi_{\cdot}(x)|_{[0,T]}$; for $x\in\mathcal{R}_{\delta_{3}}$,

$\psi^{x}:=\left\\{\begin{array}[]{ll}\tilde{\psi}^{x}(t),&t\in[0,T^{x}],\\\
\tilde{\psi}^{0}(t-T^{x}),&t\in[T^{x},T^{x}+T_{2}],\\\
\Psi_{t-T^{x}-T_{2}}(y_{0}),&t\in[T^{x}+T_{2},T].\end{array}\right.$

From (4.17), (4.18) and (4.16) it follows immediately that

$I^{T}_{x}(\psi^{x})\leq\frac{1}{10}(s_{0}\wedge s_{1}),\ {\rm and}$ (4.19)
$\psi^{x}(T)\in(\mathcal{R}_{\delta})^{c}.$ (4.20)

The remaining proof is entirely the same as that of [36, Theorem 3.1] if the
notation $I^{T}_{x}$ here is replaced by $S_{T}^{x}$ there. Thus, we conclude
that there exists a neighborhood $U_{0}$ of $\mathcal{R}$, $\kappa>0$ and
$\varepsilon_{*}>0$ such that for any ${\varepsilon}\in(0,\varepsilon_{*})$,
we have

$\mu^{{\varepsilon}}(U_{0})\leq\exp\\{-\kappa/{\varepsilon}\\}.$

This completes the proof.

∎

######  Remark 4.1.

The Lyapunov function (4.14) can serve us to prove both stochastically and
periodically forced van der Pol equation

$X_{\varepsilon}^{(2)}+(X_{\varepsilon}^{2}-1)X_{\varepsilon}^{(1)}+X_{\varepsilon}=\sin\omega
t+\sqrt{\varepsilon}\sigma(X_{\varepsilon},X_{\varepsilon}^{(1)})\dot{B}_{t},\
\sigma(X_{\varepsilon},X_{\varepsilon}^{(1)})>0$

admits a periodic stationary distribution under the same condition on the
diffusion coefficient, which is again new as far as we know.

######  Example 4.4.

It is important to investigate the system behavior of random perturbation of
voltage in electric circuits for circuit performance optimization, fault
diagnosis, robustness analysis, etc., see for example [35]. The following is
the single diode circuit taken from Moser [23], see Figure 5. Here the
rectangle is the symbol for the nonlinear characteristic, given by $i=f(v)$.
During the operation of the circuit, the voltage $E$ is usually randomly
perturbed by external noise, component variations and temperature changes. Let
$L$ and $C$ be positive constants. Then the stochastic diode circuit equation
reads as

Figure 5: Single Diode Circuit.

$\begin{cases}Ldi_{t}=\left(E-Ri_{t}-v_{t}\right)dt+\sqrt{\varepsilon}\sigma(v_{t},i_{t})dB_{t},\
i,v\in\mathbb{R},\\\ Cdv_{t}=(i_{t}-f(v_{t}))dt,\
\sigma(v_{t},i_{t})>0\end{cases}$ (4.21)

where the noise term in the first equation represents the random perturbation
to the voltage $E$, $i_{t}$ and $v_{t}$ are the current and the voltage in the
circuit at the time $t$, respectively, $f$ is continuously differentiable.

Using the homeomorphic transformation
$H:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ defined by

$H(v,i):\begin{cases}v=v,\\\ w=C^{-1}(i-f(v)),\end{cases}$ (4.22)

the system (4.21) is transformed into the following stochastic second-order
equation

$\begin{cases}dv=wdt,\\\
dw=-C^{-1}[\left(f^{\prime}(v)+\frac{CR}{L}\right)w+\frac{R}{L}f(v)+\frac{v-E}{L}]dt+\frac{\sqrt{\varepsilon}}{LC}\sigma\big{(}v,Cw+f(v)\big{)}d{B}_{t}.\end{cases}$
(4.23)

For $x,y,p,q\in\mathbb{R}^{2}$, let

$\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}:=\\{(v(\cdot),i(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\ \
C\dot{v}(\cdot)=i(\cdot)-f(v(\cdot))\\},$ $\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x}:=\\{(v(\cdot),i(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\
(v(0),i(0))=x,\ C\dot{v}(\cdot)=i(\cdot)-f(v(\cdot))\\},$ $\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}:=\\{(v(\cdot),i(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\
(v(0),i(0))=x,\ (v(T),i(T))=y,\ C\dot{v}(\cdot)=i(\cdot)-f(v(\cdot))\\},$
$\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}:=\\{(v(\cdot),w(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\ \
w(\cdot)=\dot{v}(\cdot)\\},$ $\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}^{p}:=\\{(v(\cdot),w(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\
(v(0),w(0))=p,\ w(\cdot)=\dot{v}(\cdot)\\},$ $\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}^{p,q}:=\\{(v(\cdot),w(\cdot))\in\mathbf{A}\mathbf{C}_{T}|\
(v(0),w(0))=p,\ (v(T),w(T))=q,\ w(\cdot)=\dot{v}(\cdot)\\}.$

The homeomorphic transformation (4.22) induces the mappings
$\mathcal{H}_{T}:\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}\rightarrow\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}$, $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x}\rightarrow\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{H(x)}$
and $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}\rightarrow\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}^{H(x),H(y)}$. It is easy to check that
$\mathcal{H}_{T}:\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}\rightarrow\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}$ is a one-to-one mapping. Therefore, both
$\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x}\rightarrow\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{H(x)}$
and $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}\rightarrow\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}^{H(x),H(y)}$ are also one-to-one. Now let us define the
norm on the space ${\bf AC}_{T}$ by

$\|f\|:=\|f\|_{C^{0}}+\|f^{\prime}\|_{L^{1}}\ {\rm for}\ f\in{\bf AC}_{T}$

where $\|f\|_{C^{0}}:={\rm max}_{0\leq t\leq T}|f(t)|$ and
$\|f^{\prime}\|_{L^{1}}:=\int_{0}^{T}|f^{\prime}(t)|dt$. Thus we have

######  Lemma 4.1.

The space $({\bf AC}_{T},\|\cdot\|)$ is a Banach space. The metric spaces
$\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}^{x}$, $\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}$, $\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{p}$ and
$\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{p,q}$ are all complete under the metric
induced by $\|\cdot\|$.

Proof  Let $\\{f_{n}\\}\subset{\bf AC}_{T}$ be a Cauchy sequence. Then both
$\\{f_{n}\\}\subset{\bf C}_{T}$ and $\\{f_{n}^{\prime}\\}\subset L^{1}([0,T])$
are Cauchy sequences. Therefore, from the completeness of the spaces $({\bf
C}_{T},\|\cdot\|_{C^{0}})$ and $(L^{1}([0,T]),\|\cdot\|_{L^{1}})$ it follows
that there exist $f\in{\bf C}_{T}$ and $g\in L^{1}([0,T])$ such that
$\|f_{n}-f\|_{C^{0}}\rightarrow 0$ and
$\|f_{n}^{\prime}-g\|_{L^{1}}\rightarrow 0$ as $n\rightarrow\infty$. Applying
the Newton-Leibnitz formula, we get that

$f_{n}(t)=f_{n}(0)+\int_{0}^{t}f_{n}^{\prime}(s)ds,\ {\rm for\ all}\
t\in[0,T].$

Letting $n\rightarrow\infty$ in the above equality, we have that

$f(t)=f(0)+\int_{0}^{t}g(s)ds,\ {\rm for\ all}\ t\in[0,T],$

which implies that $f\in{\bf AC}_{T}$. This proves that the space $({\bf
AC}_{T},\|\cdot\|)$ is a Banach space.

Because the spaces $\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}^{x}$, $\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}$, $\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{p}$ and
$\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{p,q}$ are closed in ${\bf AC}_{T}$, they
are all complete. ∎

Using the definition given in (2.10), the rate function of system (4.21) is
given by

$\displaystyle\mathcal{I}^{T}_{(v,i)}(v(\cdot),i(\cdot))$
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{2}\int\limits_{0}^{T}\Big{|}\frac{L\dot{i}(s)-(E-v(s)-Ri(s))}{L^{2}C^{2}\sigma(v(s),i(s))}\Big{|}^{2}ds,&\hbox{if}\begin{array}[]{c}(v(\cdot),i(\cdot))\in\mathbf{A}^{\rm
20}\mathbf{C}_{T}^{(v,i)},\\\ \end{array}\\\
+\infty,&\hbox{otherwise.}\end{array}\right.$ (4.27)
$\displaystyle=\left\\{\begin{array}[]{ll}\frac{1}{2}\int\limits_{0}^{T}\Big{|}\frac{LC\dot{w}(s)+Lf^{\prime}(v(s))w(s)-(E-v(s)-RCw(s)-Rf(v(s)))}{L^{2}C^{2}\sigma(v(s),Cw(s)+f(v(s)))}\Big{|}^{2}ds,&\hbox{if}\
(v(\cdot),w(\cdot))\in\mathbf{A}^{\rm 22}\mathbf{C}_{T}^{(v,w)},\\\
+\infty,&\hbox{otherwise.}\end{array}\right.$ (4.30)

Let $\mathcal{J}^{T}_{(v,w)}(v(\cdot),w(\cdot))$ denote the rate function of
the second-order equation (4.23). Denote by $\mathbb{V}^{20}$ and
$\mathbb{V}^{22}$ the quasipotentials of (4.21) and (4.23), respectively. Then
we summarize the relations between the solutions, the rate functions, the
quasipotentials and invariant measures between (4.21) and (4.23) as follows.

######  Lemma 4.2.

(i) Let $(v(t,\omega,v_{0},i_{0}),i(t,\omega,v_{0},i_{0}))$ and
$(v(t,\omega,v_{0},w_{0}),w(t,\omega,v_{0},w_{0}))$ be the solutions of (4.21)
and (4.23) passing through $(v_{0},i_{0})$ and $(v_{0},w_{0})$, respectively.
Then

$\begin{cases}(v(t,\omega,H(v_{0},i_{0})),w(t,\omega,H(v_{0},i_{0})))=H(v(t,\omega,v_{0},i_{0}),i(t,\omega,v_{0},i_{0})),\\\
{\rm for}\ {\rm all}\
(t,\omega,(v_{0},i_{0}))\in\mathbb{R}_{+}\times\Omega\times\mathbb{R}^{2}.\end{cases}$
(4.31)

(ii) The mapping $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}\rightarrow\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}$ is a
homeomorphism. Besides, $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x}\rightarrow\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{H(x)}$
and $\mathcal{H}_{T}:\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}^{x,y}\rightarrow\mathbf{A^{{\rm
22}}}\mathbf{C}_{T}^{H(x),H(y)}$ are homeomorphisms for all
$x,y\in\mathbb{R}^{2}$ and $T>0$.

(iii)
$\mathcal{J}^{T}_{H(v,i)}(\mathcal{H}_{T}(v(\cdot),i(\cdot)))=\mathcal{I}^{T}_{(v,i)}(v(\cdot),i(\cdot))$
for all $(v,i)\in\mathbb{R}^{2}$ and $(v(\cdot),i(\cdot))\in\mathbf{A}^{\rm
20}\mathbf{C}_{T}^{(v,i)}.$

(iv) $\mathbb{V}^{22}(H(x),H(y))=\mathbb{V}^{20}(x,y)$ for all
$x,y\in\mathbb{R}^{2}$.

(v) If $\mu^{\varepsilon}$ is an invariant measure of (4.21) if and only if
$\mu^{\varepsilon}\circ H^{-1}$ is the invariant measure of (4.23).

Proof  (i) follows immediately from the transformation $H$ and the Itô
formula. (iii) follows from (4.27) and (4.30).

In order to prove (ii), let
$\\{(v_{n}(\cdot),i_{n}(\cdot))\\}_{n=0}^{\infty}\subset\mathbf{A^{{\rm
20}}}\mathbf{C}_{T}$ such that $\|v_{n}(\cdot)-v_{0}(\cdot)\|\rightarrow 0$
and $\|i_{n}(\cdot)-i_{0}(\cdot)\|\rightarrow 0$ as $n\rightarrow\infty$, then
we need to prove that $\|w_{n}(\cdot)-w_{0}(\cdot)\|\rightarrow 0$ as
$\rightarrow\infty$ with $w_{n}(\cdot)=C^{-1}(i_{n}(\cdot)-f(v_{n}(\cdot)))$
and $w_{0}(\cdot)=C^{-1}(i_{0}(\cdot)-f(v_{0}(\cdot)))$. In fact, it follows
from $\lim_{n\rightarrow\infty}\|v_{n}(\cdot)-v_{0}(\cdot)\|=0$ that there
exist constants $K>0,L>0$ such that $|v_{n}(t)|\leq K$ for all $t\in[0,T]$ and
$n=0,1,2,\cdots$ and $|f^{\prime}(s)|\leq L$ for all $|s|\leq K$.

$\displaystyle\|w_{n}(\cdot)-w_{0}(\cdot)\|$ $\displaystyle\leq
C^{-1}\|i_{n}(\cdot)-i_{0}(\cdot)\|+C^{-1}\|f(v_{n}(\cdot))-f(v_{0}(\cdot))\|_{C^{0}}$
$\displaystyle+C^{-1}\|f^{\prime}(v_{n}(\cdot))v_{n}^{\prime}(\cdot)-f^{\prime}(v_{0}(\cdot))v_{0}^{\prime}(\cdot)\|_{L^{1}}$
$\displaystyle\leq
C^{-1}\|i_{n}(\cdot)-i_{0}(\cdot)\|+C^{-1}L\|v_{n}(\cdot)-v_{0}(\cdot)\|_{C^{0}}$
$\displaystyle+C^{-1}\|f^{\prime}(v_{n}(\cdot))(v_{n}^{\prime}(\cdot)-v_{0}^{\prime}(\cdot))+(f^{\prime}(v_{n}(\cdot))-f^{\prime}(v_{0}(\cdot)))v_{0}^{\prime}(\cdot)\|_{L^{1}}$
$\displaystyle\leq
C^{-1}\|i_{n}(\cdot)-i_{0}(\cdot)\|+C^{-1}L\|v_{n}(\cdot)-v_{0}(\cdot)\|$
$\displaystyle+\|f^{\prime}(v_{n}(\cdot))-f^{\prime}(v_{0}(\cdot))\|_{C^{0}}\|v_{0}^{\prime}(\cdot)\|_{L^{1}}\rightarrow
0\ {\rm as}\ n\rightarrow\infty.$

This proves that
$\lim_{n\rightarrow\infty}\mathcal{H}_{T}(v_{n}(\cdot),i_{n}(\cdot))=\mathcal{H}_{T}(v_{0}(\cdot),i_{0}(\cdot))$.
Similarly, we can show that

$\lim_{n\rightarrow\infty}\mathcal{H}_{T}^{-1}(v_{n}(\cdot),w_{n}(\cdot))=\mathcal{H}_{T}^{-1}(v_{0}(\cdot),w_{0}(\cdot)).$

Now we prove (iv). For any fixed points $x,y\in\mathbb{R}^{2}$, by the
definition of quasipotential,

$\displaystyle\mathbb{V}^{22}(H(x),H(y))$ $\displaystyle={\rm
inf}\\{\mathcal{J}^{T}_{H(x)}(v(\cdot),w(\cdot))|\
(v(\cdot),w(\cdot))\in\mathbf{A^{{\rm 22}}}\mathbf{C}_{T}^{H(x),H(y)},\
T>0\\}$ $\displaystyle={\rm
inf}\\{\mathcal{J}^{T}_{H(x)}(\mathcal{H}_{T}(v(\cdot),i(\cdot)))|\
(v(\cdot),i(\cdot))\in\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}^{x,y},\ T>0\\}$
$\displaystyle={\rm inf}\\{\mathcal{I}^{T}_{x}((v(\cdot),i(\cdot)))|\
(v(\cdot),i(\cdot))\in\mathbf{A^{{\rm 20}}}\mathbf{C}_{T}^{x,y},\ T>0\\}$
$\displaystyle=\mathbb{V}^{20}(x,y).$

The second and third equalities have used (ii) and (iii) respectively. (iv) is
proved.

(v) follows from (4.31) and [3, Theorem 2.6]. ∎

Take $\ f(v)=(v-E)^{3}-(v-E)$ and replace $v-E$ by $v$. Then the unperturbed
equation of (4.23) is the following Liénard equation

$\begin{cases}\dot{v}=w,\\\
\dot{w}=-C^{-1}[\left(3v^{2}-1+\frac{CR}{L}\right)w+\frac{R}{L}(v^{3}-v)+\frac{v}{L}].\end{cases}$
(4.32)

Let $V(v,i)=Li^{2}+Cv^{2}$ be a Lyapunov function for (4.21) and assume that
${\sigma}$ is a locally Lipschitz continuous function satisfying
$0<{\sigma}^{2}(v,i)\leq c_{1}(v^{4}+i^{2}+1)$, $c_{1}>0$. Then, the left-hand
side of the condition (2.7) with $\theta=\frac{L}{4c_{1}}$ and
$\eta=\frac{16c_{1}}{L}$ is

$\displaystyle J$
$\displaystyle:=-2Ri^{2}+2Ei-2v(v-E)^{3}+2v(v-E)+\frac{\theta}{L}{\sigma}^{2}(v,i)+\frac{4i^{2}{\sigma}^{2}(v,i)}{{\eta}V(v,i)}$
$\displaystyle\leq-2Ri^{2}+2Ei-2v(v-E)^{3}+2v(v-E)+\frac{1}{2}(i^{2}+v^{4}+1),$
(4.33)

for all $(v,i)\in\mathbb{R}^{2}$. Since
$\lim_{|v|\rightarrow\infty}\frac{-2v(v-E)^{3}+2v(v-E)}{v^{4}}=-2$, (4.2)
implies that $J\leq C^{*}(1+V(v,i)$ for some large constant $C^{*}$, that is,
(2.7) holds. On the other hand, the conditions (2.6) and (2.8) are obviously
satisfied. It follows from Theorem 2.1 that (4.21) admits ULDP.

Let $\varepsilon_{0}=\frac{L}{4c_{1}}(R\wedge 1)$. It can also be seen from
(4.2) that there exist positive constants $k,M$ such that

$\displaystyle\mathcal{L}^{\varepsilon}V(v,i)$
$\displaystyle:=-2Ri^{2}+2Ei-2v(v-E)^{3}+2v(v-E)+\frac{\varepsilon}{L}{\sigma}^{2}(v,i)$
$\displaystyle\leq-k(i^{2}+v^{4})\ {\rm for}\
0<\varepsilon\leq\varepsilon_{0},i^{2}+v^{2}\geq M,$

where $\mathcal{L}^{\varepsilon}$ is the generator of the system (4.21).
Denote by $\mathcal{J^{\varepsilon}}$ the set of all invariant measures of
(4.21) for a given $0<\varepsilon\leq\varepsilon_{0}$. By [4, Theorem 3.1],
$\mathcal{J^{\varepsilon}}$ is nonempty and
$\mathcal{J}:=\bigcup_{0<\varepsilon\leq\varepsilon_{0}}\mathcal{J^{\varepsilon}}$
is tight. Furthermore, let
$\mu^{\varepsilon_{i}}\in\mathcal{J}^{\varepsilon_{i}}$ such that
$\mu^{\varepsilon_{i}}$ converges weakly to $\mu$ as
$\varepsilon_{i}\rightarrow 0$. Then $\mu$ is an invariant measure of the
unperturbed equations of (4.21):

$\begin{cases}L\frac{di}{dt}=E-Ri-v,\\\
C\frac{dv}{dt}=i-(v-E)^{3}+(v-E)\end{cases}$ (4.34)

and its support is contained in the Birkhoff center of (4.34) (see [4]).

The details are contained in the following result.

######  Proposition 4.3.

Suppose that ${\sigma}(v,i)$ is a locally Lipschitz, positive continuous
function with ${\sigma}^{2}(v,i)\leq c_{1}(v^{4}+i^{2}+1)$, $c_{1}>0$. Then
there exists $\varepsilon_{0}>0$ such that for any given
$\varepsilon\in(0,\varepsilon_{0}]$, the set $\mathcal{J^{\varepsilon}}$ of
invariant measures of the system (4.21) is nonempty and
$\mathcal{J}:=\bigcup_{0<\varepsilon\leq\varepsilon_{0}}\mathcal{J^{\varepsilon}}$
is tight. Furthermore, let
$\mu^{\varepsilon_{i}}\in\mathcal{J}^{\varepsilon_{i}}$ such that
$\mu^{\varepsilon_{i}}$ converges weakly to $\mu$ as
$\varepsilon_{i}\rightarrow 0$. Then $\mu$ is an invariant measure of (4.34)
and its support is contained in the Birkhoff center of the system (4.34). More
precisely,
(i) if $\frac{CR}{L}\geq 1$ and $0<R\leq 1$, then when $\varepsilon\rightarrow
0$ the invariant measure $\mu^{\varepsilon}$ of (4.21) converges weakly to
$\delta_{(E,0)}(\cdot)$;
(ii) if $\frac{CR}{L}\geq 1$ and $R>1$, then when $\varepsilon\rightarrow 0$
the invariant measure $\mu^{\varepsilon}$ of (4.21) converges weakly to
$\lambda\delta_{(E+\sqrt{1-R^{-1}},-R^{-1}\sqrt{1-R^{-1}})}(\cdot)+(1-\lambda)\delta_{(E-\sqrt{1-R^{-1}},R^{-1}\sqrt{1-R^{-1}})}(\cdot)$
with $0\leq\lambda\leq 1$;
(iii) if $\frac{CR}{L}<1$ and $0<R\leq 1$, then when $\varepsilon\rightarrow
0$ the invariant measure $\mu^{\varepsilon}$ of (4.21) converges weakly to
$\mu_{\Gamma}$, where $\mu_{\Gamma}$ is the Haar measure supported on the
unique limit cycle $\Gamma$, see Figure 4.

Proof  As discussed above, (4.21) admits ULDP under the condition on
$\sigma(v,i)$. It is easy to see from (4.22), (4.34) and (4.32) that
$E^{*}(v^{*},i^{*})$ is an equilibrium of (4.34) if and only if
$F^{*}(v^{*},0)$ is an equilibrium of (4.32). By (iv) of Lemma 4.2, the
continuity of $H$ and (ii) of Lemma 3.2,

$\lim_{x\rightarrow E^{*}}\mathbb{V}^{20}(x,E^{*})=\lim_{x\rightarrow
E^{*}}\mathbb{V}^{22}(H(x),F^{*})=\lim_{p\rightarrow
F^{*}}\mathbb{V}^{22}(p,F^{*})=\mathbb{V}^{22}(F^{*},F^{*})=0.$

Similarly, $\lim_{y\rightarrow E^{*}}\mathbb{V}^{20}(E^{*},y)=0$. Therefore,
for any $\eta>0$, there exists $\delta_{1}>0$ such that
$\mathbb{V}^{22}(p,F^{*})<\eta$ for all
$p\in\overline{B}_{\delta_{1}}(F^{*})$. By (iii) of Lemma 3.2, there exists
$T^{*}>0$ such that for any $p\in\overline{B}_{\delta_{1}}(F^{*})$, there
exist $T^{p}\leq T^{*}$ and $\psi^{p}\in{\bf C}_{T^{p}}$ with $\psi^{p}(0)=p$
and $\psi^{p}(T^{p})=F^{*}$ satisfying
$\mathcal{J}^{T^{p}}_{p}(\psi^{p})<\eta$. Since $H$ is a homeomorphism on the
plan $\mathbb{R}^{2}$, $H^{-1}(B_{\delta_{1}}(F^{*}))$ is a neighborhood of
$E^{*}$. Choose $\delta>0$ sufficiently small so that
$\overline{B}_{\delta}(E^{*})\subset H^{-1}(B_{\delta_{1}}(F^{*}))$. For any
$x\in\overline{B}_{\delta}(E^{*})$, $p:=H(x)\in B_{\delta_{1}}(F^{*})$. Thus,
let $T^{x}=T^{p}$ and $\phi^{x}=\mathcal{H}_{T^{x}}^{-1}(\psi^{H(x)})$. Then
by (iii) of Lemma 4.2,
$\mathcal{I}_{x}^{T^{x}}(\phi^{x})=\mathcal{J}^{T^{p}}_{p}(\psi^{p})<\eta$ and
$T^{x}\leq T^{*}$ for any $x\in\overline{B}_{\delta}(E^{*})$.

Suppose that $\frac{CR}{L}\geq 1$. If $0<R\leq 1$, then $(E,0)$ is the unique
equilibrium of (4.34). Assume that $R>1$. Then the equilibria of (4.34) are
$\mathcal{S}(E,0)$, $\mathcal{A}(E+\sqrt{1-R^{-1}},-R^{-1}\sqrt{1-R^{-1}})$
and $\mathcal{B}(E-\sqrt{1-R^{-1}}),R^{-1}\sqrt{1-R^{-1}})$, $\mathcal{S}$ is
a saddle, $\mathcal{A}$ and $\mathcal{B}$ are asymptotically stable. This
shows that the equilibria of (4.34) is finite in the both cases. It follows
from [23, Theorem 3.3] that all solutions of (4.34) are convergent to
equilibria. Therefore the Birkhoff center of (4.34) consists of equilibria.
The global portraits of (4.34) in this case are drawn in Figure 6.

Figure 6: The phase portraits of system (4.34).

(i) In this case, $(E,0)$ is the Birkhoff center of (4.34). Then the
conclusion follows immediately.

(iii) Suppose that $\frac{CR}{L}<1$ and $0<R\leq 1$. Then $(E,0)$ is the
unique equilibrium of (4.34) which is a repeller. Since (4.34) is equivalent
to the Liénard system (4.32) which has the unique limit cycle $\Gamma$
illustrated in Figure 4, see [37]. Therefore, the system (4.34) admits a
unique limit cycle, still denoted by $\Gamma$. The Birkhoff center of (4.34)
is composed of the repeller $(E,0)$ and the unique stable limit cycle
$\Gamma$. Because (4.21) admits ULDP, based on the results obtained in the
first paragraph and preceded in the same manner as in the proof of Proposition
4.2, we can conclude that there is no concentration for limiting measures on
the repeller $(E,0)$. This proves that when $\varepsilon\rightarrow 0$ the
invariant measure $\mu^{\varepsilon}$ of (4.21) converges weakly to the Haar
measure $\mu_{\Gamma}$.

(ii) Suppose $\frac{CR}{L}\geq 1$ and $R>1$. Then (4.21) admits ULDP. We shall
show that the limiting measures will not concentrate on the saddle
$\mathcal{S}(E,0)$.

Let $\delta^{\prime}<\frac{1}{2}\sqrt{(1-R^{-1})(1+R^{-2})}$ such that
$\mathcal{A}_{\delta^{\prime}}$ is a fundamental neighborhood of
$\mathcal{A}$. Note that [36, Lemma 4.2] still holds as soon as ULDP is
satisfied. Thus by Theorem 2.1 and [36, Lemma 4.2], there exist
$0<\delta_{3}<\delta_{2}<\delta^{*}_{1}<\delta_{1}<\delta^{\prime}$ and
$s_{0}>0$ such that
$\mathbb{V}\big{(}\partial\mathcal{A}_{\delta_{2}},\partial\mathcal{A}_{\delta^{*}_{1}}\big{)}\geq
s_{0}$. Since $\mathcal{A}$ is an attractor, we obtain

$T_{0}:=\inf\big{\\{}u\geq
0:\Psi_{t}\big{(}\overline{\mathcal{A}_{\delta_{1}}}\big{)}\subset\mathcal{A}_{\delta_{3}},t\geq
u\big{\\}}<+\infty.$

The set

$F_{0}=\\{\varphi\in{\bf
C}_{T_{0}}:\varphi(0)\in\overline{\mathcal{A}_{\delta_{1}}},\varphi(T_{0})\in(\mathcal{A}_{\delta_{2}})^{c}\\}$

is a closed subset of ${\bf C}_{T_{0}}$.
$F_{0}^{0}=\overline{\mathcal{A}_{\delta_{1}}}$ is compact and $F_{0}$ does
not contain any solution of system (4.34). Thus, by Theorem 2.1 and [36,
Proposition, 2.4] we have

$s_{1}:=\inf\\{\mathcal{I}^{T_{0}}_{x}(\varphi):\varphi\in F_{0},\
x=\varphi(0)\in\overline{\mathcal{A}_{\delta_{1}}}\ \\}>0.$

We claim that there exist $T_{1}>0$ and $\tilde{\psi}\in{\bf C}_{T_{1}}$ with
$\tilde{\psi}(0)=\mathcal{S}$ and $\tilde{\psi}(T_{1})=\mathcal{A}$ such that

$\mathcal{I}_{\mathcal{S}}^{T_{1}}(\tilde{\psi})<\frac{1}{5}(s_{0}\wedge
s_{1}).$ (4.35)

Since $\mathcal{S}$ is an equilibrium of (4.21), by the definition of
quasipotential, $\mathbb{V}^{20}(\mathcal{S},\mathcal{S})=0$. The result in
the first paragraph with $E^{*}=\mathcal{S}$ and
$\eta=\frac{1}{10}(s_{0}\wedge s_{1})$ shows that there is
$0<\delta<\delta_{3}$ such that

$\mathbb{V}^{20}(x,\mathcal{S}),\
\mathbb{V}^{20}(\mathcal{S},y)<\frac{1}{10}(s_{0}\wedge s_{1})\ {\rm for\
all}\ x,y\in\overline{B_{\delta}(\mathcal{S})}.$ (4.36)

Let

$W^{u}(\mathcal{S}):=\\{x\in\mathbb{R}^{2}:\lim_{t\rightarrow-\infty}\Psi_{t}(x)=\mathcal{S}\\}$

be the unstable manifold of $\mathcal{S}$ and

$W^{s}(\mathcal{S}):=\\{x\in\mathbb{R}^{2}:\lim_{t\rightarrow\infty}\Psi_{t}(x)=\mathcal{S}\\}$

denote the stable manifold of $\mathcal{S}$. Then $W^{s}(\mathcal{S})$
separates $\mathbb{R}^{2}$ into two parts which are the attracting domains of
$\mathcal{A}$ and $\mathcal{B}$ respectively, see Figure 6.

Choose $P\in B_{\delta}(\mathcal{S})\cap W^{u}(\mathcal{S})$ with
$P\neq\mathcal{S}$. Then
$\mathbb{V}^{20}(\mathcal{S},P)<\frac{1}{10}(s_{0}\wedge s_{1})$ by (4.36). By
the definition of quasipotential, there exists $T_{1}^{0}$ and
$\tilde{\psi}_{1}\in{\bf C}_{T_{1}^{0}}$ with
$\tilde{\psi}_{1}(0)=\mathcal{S}$ and $\tilde{\psi}_{1}(T_{1}^{0})=P$ such
that

$\mathcal{I}_{\mathcal{S}}^{T_{1}^{0}}(\tilde{\psi_{1}})<\frac{1}{10}(s_{0}\wedge
s_{1}).$ (4.37)

Similarly, since $\mathcal{A}$ is an equilibrium of (4.21) and
$\mathbb{V}^{20}(\mathcal{A},\mathcal{A})=0$. The first paragraph has shown
that $\mathbb{V}^{20}(x,\mathcal{A})$ is continuous at $x=\mathcal{A}$. Thus,
$\lim_{t\rightarrow\infty}\mathbb{V}^{20}(\Psi_{t}(P),\mathcal{A})=0$.
Therefore, there exists $T_{2}^{0}$ such that
$\mathbb{V}^{20}(\Psi_{T_{2}^{0}}(P),\mathcal{A})<\frac{1}{10}(s_{0}\wedge
s_{1})$. This implies that there exists $T_{3}^{0}$ and
$\tilde{\psi}_{3}\in{\bf C}_{T_{3}^{0}}$ with
$\tilde{\psi}_{3}(0)=\Psi_{T_{2}^{0}}(P)$ and
$\tilde{\psi}_{3}(T_{3}^{0})=\mathcal{A}$ such that

$\mathcal{I}_{\Psi_{T_{2}^{0}}(P)}^{T_{3}^{0}}(\tilde{\psi_{3}})<\frac{1}{10}(s_{0}\wedge
s_{1}).$ (4.38)

Let $T_{1}=T_{1}^{0}+T_{2}^{0}+T_{3}^{0}$ and define $\tilde{\psi}\in{\bf
C}_{T_{1}}$ by

$\tilde{\psi}(t)=\left\\{\begin{array}[]{ll}\tilde{\psi}_{1}(t),&t\in[0,T_{1}^{0}],\\\
\Psi_{t-T_{1}^{0}}(P),&t\in[T_{1}^{0},T_{1}^{0}+T_{2}^{0}],\\\
\tilde{\psi}_{3}(t-T_{1}^{0}-T_{2}^{0}),&t\in[T_{1}^{0}+T_{2}^{0},T_{1}].\end{array}\right.$

Obviously, $\tilde{\psi}(0)=\mathcal{S}$ and
$\tilde{\psi}(T_{1})=\mathcal{A}$. (4.35) follows immediately from (4.37) and
(4.38), hence the claim is true.

Choose $E^{*}=\mathcal{S}$ and $\eta=\frac{1}{10}(s_{0}\wedge s_{1})$ in the
first paragraph. Then we know that there exists $T^{*}>0$ such that for any
$x\in\overline{B}_{\delta}(\mathcal{S})$, there exist $T^{x}\leq T^{*}$ and
$\tilde{\psi}^{x}\in{\bf C}_{T^{x}}$ with $\tilde{\psi}^{x}(0)=x,\
\tilde{\psi}^{x}(T^{x})=\mathcal{S}$ such that

$I^{T^{x}}_{x}(\tilde{\psi}^{x})<\frac{1}{10}(s_{0}\wedge s_{1}).$ (4.39)

Let $T=T^{*}+T_{1}$. Define $\psi^{x}\in{\bf C}_{T}$ by

$\psi^{x}:=\left\\{\begin{array}[]{ll}\tilde{\psi}^{x}(t),&t\in[0,T^{x}],\\\
\tilde{\psi}(t-T^{x}),&t\in[T^{x},T^{x}+T_{1}],\\\
\mathcal{A},&t\in[T^{x}+T_{1},T].\end{array}\right.$

Thus, by (4.35) and (4.39), for every $x\in\bar{B}_{\delta}(\mathcal{S})$,

$\mathcal{I}_{x}^{T}(\psi^{x})\leq\frac{3}{10}(s_{0}\wedge s_{1})$ (4.40)

and

$\psi^{x}(T)=\mathcal{A}.$ (4.41)

The remaining proof is entirely the same as that of [36, Theorem 3.2] if the
notation $\mathcal{I}_{x}^{T}$ here is replaced by $S_{T}^{x}$ there. Thus, we
conclude that there exists $\varepsilon_{*}>0$ such that for any
${\varepsilon}\in(0,\varepsilon_{*})$,

$\mu^{{\varepsilon}}(B_{\delta}(\mathcal{S}))\leq\exp\\{-\frac{2}{5}(s_{0}\wedge
s_{1})/{\varepsilon}\\}.$

This completes the proof.

∎

######  Remark 4.2.

By Proposition 4.3, it remains that the case

${\rm(iv)}\ 1<R<\frac{L}{C}$

not have been considered yet. In this case, $\mathcal{S}$ is still a saddle
and we cannot give the global dynamical behavior of system (4.34). Let $M$ be
the linearized matrix of (4.34) at both equilibria $\mathcal{A}$ and
$\mathcal{B}$. Then the domain of (iv) in plane $(\frac{L}{C},R)$ is divided
into two parts by ${\rm Trace}(M)=0$ , (iv)+: ${\rm Trace}(M)>0$ and (iv)-:
${\rm Trace}(M)<0$. By calculation,

${\rm det}M=2(LC)^{-1}(R-1)>0;\ {\rm
Trace}(M)=-(LR)^{-1}\left(R^{2}+2\frac{L}{C}R-3\frac{L}{C}\right)=:=-(LR)^{-1}g\left(\frac{L}{C},R\right).$

And $g=0$ is equivalent to
$R=h\left(\frac{L}{C}\right):=-\frac{L}{C}+\sqrt{\left(\frac{L}{C}\right)^{2}+3\frac{L}{C}},$
which is the Hopf bifurcation curve and strictly increasing to $\frac{3}{2}$
as $\frac{L}{C}\rightarrow\infty$. Both $\mathcal{A}$ and $\mathcal{B}$ are
locally asymptotically stable between $R=\frac{L}{C}$ and
$R=h\left(\frac{L}{C}\right)$, and they are repelling between $R=1$ and
$R=h\left(\frac{L}{C}\right)$, see the classification Figure 7 in the
parameter $(\frac{L}{C},R)$ plane for detail. By the Poincaré and Bendixson
theorem, there exists at least a stable limit cycle on the subdomain (iv)+. By
the similar way as above, we can prove that the limit measures stay away from
the saddle $\mathcal{S}$ and both repellers $\mathcal{A}$ and $\mathcal{B}$ on
the subdomain (iv)+, therefore, its support must be contained in limit cycles.
On the subdomain (iv)-, the limiting measures stay away from the saddle
$\mathcal{S}$. Thus, its support is contained in the union of limit cycles and
stable equilibria $\mathcal{A}$ and $\mathcal{B}$. At this stage, we cannot
give the exact limiting measure or its support due to lacking of the global
dynamics for (4.34).

We conjecture that system (4.34) admits a unique stable limit cycle containing
$\mathcal{S}$, $\mathcal{A}$ and $\mathcal{B}$ in its interior on the
subdomain (iv)+. If it does hold, then the limiting measure is the arc-length
measure of the unique limit cycle. And we also conjecture that when $R$ passes
through the Hopf bifurcating line from bottom to top, the system has exactly
three limit cycles, two of which are unstable and surround $\mathcal{A}$ and
$\mathcal{B}$, respectively; the biggest limit cycle of which is stable and
surrounds three equilibria $\mathcal{S}$, $\mathcal{A}$ and $\mathcal{B}$. In
this case, the support of limiting measure is the union of $\mathcal{A}$,
$\mathcal{B}$ and the biggest limit cycle. We continue to conjecture that as
$R$ continues to increase, there must be another bifurcating line, when $R$
passes through it from bottom to top, the phase plane is divided into two
parts by the stable manifold of the saddle point $\mathcal{S}$, which are
attracting basins of $\mathcal{A}$ and $\mathcal{B}$, respectively. In this
case, the limiting measure is the convex combination of Delta measures of
$\mathcal{A}$ and $\mathcal{B}$.

Figure 7: The classification of dynamical behavior of (4.34) in $R,L,C$.

###  4.3 Random perturbation of May-Leonard model

May and Leonard [22] studied the following system in population biology

$\begin{array}[]{l}\displaystyle\frac{dx_{1}}{dt}=x_{1}(1-x_{1}-\alpha
x_{2}-\beta x_{3}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\frac{dx_{2}}{dt}=x_{2}(1-\beta x_{1}-x_{2}-\alpha
x_{3}),\\\ \vskip 6.0pt plus 2.0pt minus
2.0pt\cr\displaystyle\frac{dx_{3}}{dt}=x_{3}(1-\alpha x_{1}-\beta
x_{2}-x_{3})\end{array}$ (4.42)

on the space
$\mathbb{R}_{+}^{3}:=\\{x:=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{i}\geq
0,i=1,2,3\\}$ and classified the long-term behavior of the system according to
the sign of $\alpha+\beta-2$. We assume that the parameters satisfy
$-1<\alpha+\beta<2$. The system (4.42) has a positive equilibrium
$E:=\frac{1}{1+\alpha+\beta}(1,1,1)=:(e_{1},e_{2},e_{3})$. Consider random
perturbation of (4.42)

$\begin{array}[]{l}\displaystyle dx_{1}=x_{1}\left(1-x_{1}-\alpha x_{2}-\beta
x_{3}\right)dt+\sqrt{\varepsilon}x_{1}\sigma_{1}(x)dB_{t}^{1},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\displaystyle\displaystyle dx_{2}=x_{2}\left(1-\beta
x_{1}-x_{2}-\alpha
x_{3}\right)dt+\sqrt{\varepsilon}x_{2}\sigma_{2}(x)dB_{t}^{2},\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr\displaystyle dx_{3}=x_{3}\left(1-\alpha x_{1}-\beta
x_{2}-x_{3}\right)dt+\sqrt{\varepsilon}x_{3}\sigma_{3}(x)dB_{t}^{3}\end{array}$
(4.43)

where $\sigma(x)={\rm
Diag}(x_{1}\sigma_{1}(x),x_{2}\sigma_{2}(x),x_{3}\sigma_{3}(x))$ is defined on
$\mathbb{R}_{+}^{3}$ and $\\{B(t)=(B_{t}^{1},B_{t}^{2},B_{t}^{3})\\}_{t\geq
0}$ is a $3$-dimensional Brownian motion.

######  Proposition 4.4.

Suppose that ${\sigma}(x)$ is a locally Lipschitz continuous function
satisfying $0<\sum_{i=1}^{3}|{\sigma}_{i}(x)|^{2}\leq c_{1}$ for all $x\in{\rm
Int}\mathbb{R}_{+}^{3}$. If $-1<\alpha+\beta<2$, then the system (4.43)
satisfies a uniform large deviation principle, and moreover there exists
$\varepsilon_{0}>0$ such that for any given
$\varepsilon\in(0,\varepsilon_{0}]$, (4.43) admits a unique invariant measure
$\mu^{\varepsilon}$ supported on ${\rm Int}\mathbb{R}_{+}^{3}$ and
$\mu^{\varepsilon}$ converges weakly to $\delta_{E}(\cdot)$ as
$\varepsilon\rightarrow 0$.

Proof  Let $b(x)$ be the drift vector field of the system (4.42) and define
the Lyapunov function

$V(x)=\left(x_{1}-e_{1}-e_{1}\ln\frac{x_{1}}{e_{1}}\right)+\left(x_{2}-e_{2}-e_{2}\ln\frac{x_{2}}{e_{2}}\right)+\left(x_{3}-e_{3}-e_{3}\ln\frac{x_{3}}{e_{3}}\right).$

Then

$\lim_{x\rightarrow\partial\mathbb{R}_{+}^{3}}V(x)=+\infty,$ (4.44)

$\nabla
V(x)=\left(\frac{x_{1}-e_{1}}{x_{1}},\frac{x_{2}-e_{2}}{x_{2}},\frac{x_{3}-e_{3}}{x_{3}}\right)^{*}$,
$\nabla^{2}V(x)={\rm
Diag}\left(\frac{e_{1}}{x_{1}^{2}},\frac{e_{2}}{x_{2}^{2}},\frac{e_{3}}{x_{3}^{2}}\right)$
and

$\langle b(x),\nabla
V(x)\rangle=-\sum_{i=1}^{3}(x_{i}-e_{i})^{2}-(\alpha+\beta)\sum_{1\leq i<j\leq
3}(x_{i}-e_{i})(x_{j}-e_{j})\leq-k|x-E|^{2}$ (4.45)

for some $k>0$ because the symmetric matrix

$\left(\begin{array}[]{cccc}&1&\frac{\alpha+\beta}{2}&\frac{\alpha+\beta}{2}\\\
&\frac{\alpha+\beta}{2}&1&\frac{\alpha+\beta}{2}\\\
&\frac{\alpha+\beta}{2}&\frac{\alpha+\beta}{2}&1\\\ \end{array}\right)$

is positive definite following the assumption $-1<\alpha+\beta<2$. Note that
the limit in (4.44) is understood that there is at least $i$ such that either
$x_{i}\rightarrow+\infty$ or $x_{i}\rightarrow 0$. (4.45) implies that $E$ is
a globally asymptotically stable equilibrium of (4.42) on ${\rm
Int}\mathbb{R}_{+}^{3}$. In addition,

$0<\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)=(1+\alpha+\beta)^{-1}\sum_{i=1}^{3}|{\sigma}_{i}(x)|^{2}\leq
c_{1}(1+\alpha+\beta)^{-1}$ (4.46)

and

$|\sigma^{*}(x)\nabla
V(x)|^{2}\leq|x-E|^{2}\sum_{i=1}^{3}|{\sigma}_{i}(x)|^{2}\leq c_{1}|x-E|^{2}.$

Since $\lim_{x_{i}\rightarrow
0}\frac{x_{i}-e_{i}}{x_{i}-e_{i}-e_{i}\ln\frac{x_{i}}{e_{i}}}=0$ and
$\lim_{x_{i}\rightarrow+\infty}\frac{x_{i}-e_{i}}{x_{i}-e_{i}-e_{i}\ln\frac{x_{i}}{e_{i}}}=1$,
there exists a positive constant $c_{2}>0$ such that

$\frac{|x-E|^{2}}{V^{2}(x)}\leq c_{2}\ {\rm for\ all}\ x\in{\rm
Int}\mathbb{R}_{+}^{3}.$ (4.47)

(2.6) and (2.8) in the Assumption 2.2 follow from (4.44) and (4.46),
respectively. Combining with (4.45), (4.46) and (4.47) together, we see that
(2.7) is satisfied. According to Remark 2.2 and Theorem 2.1, (4.43) admits
ULDP.

Let

$\mathcal{L}^{\varepsilon}V(x):=\langle b(x),\nabla
V(x)\rangle+\frac{\varepsilon}{2}\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right).$

Then it follows from (4.45) and (4.46) that

$\mathcal{L}^{\varepsilon}V(x)\leq-k|x-E|^{2}+\frac{\varepsilon
c_{1}}{2(1+\alpha+\beta)}.$ (4.48)

Fix $\varepsilon\in(0,\frac{k}{c_{1}(1+\alpha+\beta)})$ and let
$r(\varepsilon):=\sqrt{\frac{\varepsilon c_{1}}{2k(1+\alpha+\beta)}}$ and
$B_{r}(E)$ denote the open ball with the center $E$ and the radius $r$. Then
$B_{r(\varepsilon)}(E)\subset
B_{(\sqrt{2}(1+\alpha+\beta))^{-1}}(E)\subset{\rm Int}\mathbb{R}_{+}^{3}$.
Define $\gamma:=\frac{k}{4(1+\alpha+\beta)^{2}}$. Then it follows from (4.48)
that

$\mathcal{L}^{\varepsilon}V(x)\leq-\gamma,\ x\in
B^{c}_{(\sqrt{2}(1+\alpha+\beta))^{-1}}(E),\
\varepsilon\in\left(0,\frac{k}{2c_{1}(1+\alpha+\beta)}\right).$ (4.49)

By [13, Theorem A] or [17, Theorem 4.1], (4.43) admits a unique invariant
measure $\mu^{\varepsilon}$ for
$0<\varepsilon<\frac{k}{2c_{1}(1+\alpha+\beta)}$. Applying [4, Theorem 3.1] to
${\rm Int}\mathbb{R}_{+}^{3}$, we obtain that
$\\{\mu^{\varepsilon}|0<\varepsilon<\frac{k}{2c_{1}(1+\alpha+\beta)}\\}$ is
tight and $\mu^{\varepsilon}$ converges weakly to $\delta_{E}(\cdot)$ as
$\varepsilon\rightarrow 0$ because the equivalent class of (4.42) on ${\rm
Int}\mathbb{R}_{+}^{3}$ is the unique equilibrium $E$. ∎

###  4.4 An unbounded example with infinite equivalent classes

In this subsection, we will provide an example whose coefficients are
unbounded and equivalent classes are infinite. Nevertheless, we can determine
its limiting measure by applying Theorems 2.1 and 3.3.

Let
$H(x_{1},x_{2}):=\frac{x_{2}^{2}}{2}+\frac{x_{1}^{4}}{4}-\frac{x_{1}^{2}}{2}\in[-\frac{1}{4},+\infty)$.
Consider the two dimensional deterministic system

$\frac{dx}{dt}=\mathcal{H}(H)-F(H)\nabla H$ (4.50)

and its random perturbation

$dx=[\mathcal{H}(H)-F(H)\nabla H]dt+\sqrt{\varepsilon}\sigma(x)dB(t)$ (4.51)

where $\mathcal{H}(H):=(\frac{\partial H}{\partial x_{2}},-\frac{\partial
H}{\partial x_{1}})^{*}$ and the $C^{1}$ function $F$ is taken to be one of
the functions $F_{i},i=1,...,4$ where

$F_{i}(s)=\begin{cases}(-1)^{i}|s|^{3},\quad&s\in[-\frac{1}{4},0),\\\
s^{5}\sin^{2}\frac{\pi}{s},\quad&s\in[0,1],\\\ \rm positive\ smooth\
function,\quad&s\in(1,2),\\\ 1,\quad&s\in[2,+\infty),\\\ \end{cases}$

for $i=1,2$, or

$F_{j}(s)=\begin{cases}(-1)^{j}|s|^{3},\quad&s\in[-\frac{1}{4},0),\\\
G(s),\quad&s\in[0,+\infty),\\\ \end{cases}$

for $j=3,4$. Here $G:[0,+\infty)\rightarrow[0,1]$ is a $C^{\infty}$ function
satisfying that $G(0)=0$ and there exist a sequence of intervals
$I_{n}=[a_{n},b_{n}]\subset(0,1]$ with $b_{n+1}<a_{n}$, $n=1,2,3,\cdots$,
$\lim_{n\rightarrow\infty}b_{n}=0$ and $b_{1}=1$ such that $G(s)=0$ for any
$s\in I=\bigcup_{n=1}^{\infty}I_{n}$, $G(s)>0$ for any
$s\in(0,+\infty)\backslash I$ and $G(s)\equiv 1$ for $s\gg 1$. Note that such
a $G$ can be constructed using cut-off functions.

For these systems, we have the following result.

######  Proposition 4.5.

Suppose that $\sigma(x)=(\sigma_{i,j}(x_{1},x_{2}))_{2\times 2}$ is a locally
Lipschitz continuous function such that

$\beta^{*}\sigma(x)\sigma^{*}(x)\beta>0\ {\rm for\ all}\ x\in\mathbb{R}^{2}\
{\rm and}\ \beta\in\mathbb{R}^{2}\backslash\\{0\\}.$ (4.52)

and that there exists a positive constant $c_{1}$ such that for all
$x\in\mathbb{R}^{2}$,

$\|\sigma(x)\|^{2}:=\sum_{i,j=1}^{2}\sigma_{i,j}^{2}(x_{1},x_{2})\leq
c_{1}(x_{1}^{4}+x_{2}^{\frac{4}{3}}+1).$ (4.53)

Then, for $i=1,3$, $\mu^{\varepsilon}$ converges weakly to $\delta_{O}(\cdot)$
as $\varepsilon\rightarrow 0$, and for $i=2,4$, $\mu^{\varepsilon}$ converges
weakly to $\lambda_{1}\delta_{(1,0)}(\cdot)+\lambda_{2}\delta_{(-1,0)}(\cdot)$
with $\lambda_{1}+\lambda_{2}=1$ as $\varepsilon\rightarrow 0$.

Proof  First, we prove that the system (4.51) admits ULDP. It suffices to
check that $V(x_{1},x_{2}):=H(x_{1},x_{2})+\frac{1}{4}$ satisfies (2.6), (2.7)
and (2.8) in the Assumption 2.2. (2.6) is obvious. We have

$\nabla V(x_{1},x_{2})=(x_{1}^{3}-x_{1},x_{2})^{*},\
\nabla^{2}V(x_{1},x_{2})={\rm Diag}(3x_{1}^{2}-1,1);$ $\langle b(x),\nabla
V(x)\rangle=-F(H)|\nabla H|^{2}\ {\rm and}\ \langle b(x),\nabla
V(x)\rangle=-|\nabla H(x)|^{2}\ {\rm if}\ |x|\gg 1;$ (4.54)
$\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)=(3x_{1}^{2}-1)(\sigma_{1,1}^{2}(x)+\sigma_{1,2}^{2}(x))+\sigma_{2,1}^{2}(x)+\sigma_{2,2}^{2}(x);$
(4.55) $|\sigma^{*}(x)\nabla V(x)|^{2}\leq|\nabla H(x)|^{2}\|\sigma(x)\|^{2}.$

$\displaystyle\limsup_{|x|\rightarrow+\infty}\frac{\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)}{|\nabla
H(x)|^{2}}$ $\displaystyle=$
$\displaystyle\limsup_{|x|\rightarrow+\infty}\frac{\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)}{x_{1}^{6}+x_{2}^{2}}$
$\displaystyle\leq$
$\displaystyle\limsup_{|x|\rightarrow+\infty}\frac{3x_{1}^{2}+1}{(x_{1}^{6}+x_{2}^{2})^{\frac{1}{3}}}\times\frac{\sigma_{1,1}^{2}(x)+\sigma_{1,2}^{2}(x)}{(x_{1}^{6}+x_{2}^{2})^{\frac{2}{3}}}+\limsup_{|x|\rightarrow+\infty}\frac{\sigma_{2,1}^{2}(x)+\sigma_{2,2}^{2}(x)}{x_{1}^{6}+x_{2}^{2}}$
$\displaystyle\leq$ $\displaystyle
3c_{1}\limsup_{|x|\rightarrow+\infty}\frac{x_{1}^{4}+x_{2}^{\frac{4}{3}}+1}{(x_{1}^{6}+x_{2}^{2})^{\frac{2}{3}}}$
$\displaystyle\leq$ $\displaystyle 6c_{1}.$

In the second inequality, we have used the condition (4.53). This implies that
there exists a positive constant $d_{1}$ such that

$\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)\leq
d_{1}(|\nabla H(x)|^{2}+1)\ {\rm for\ all}\ x\in\mathbb{R}^{2}.$ (4.56)

Besides, the inequality

$\limsup_{|x|\rightarrow+\infty}\frac{|\sigma^{*}(x)\nabla V(x)|^{2}}{|\nabla
H(x)|^{2}V(x)}\leq\limsup_{|x|\rightarrow+\infty}\frac{\|\sigma(x)\|^{2}}{V(x)}\leq
c_{1}\limsup_{|x|\rightarrow+\infty}\frac{x_{1}^{4}+x_{2}^{\frac{4}{3}}+1}{V(x)}\leq
4c_{1}$

implies that there is a positive constant $d_{2}$ such that

$\frac{|\sigma^{*}(x)\nabla V(x)|^{2}}{V(x)}\leq d_{2}(|\nabla H(x)|^{2}+1)\
{\rm for\ all}\ x\in\mathbb{R}^{2}.$ (4.57)

Take $\theta=\frac{1}{2d_{1}}$ and $\eta=4d_{2}$. Then it follows from (4.54),
(4.56) and (4.57) that the left-hand side of (2.7) is negative if $|x|\gg 1$,
which means that (2.7) holds. Finally, by (4.55) and (4.53), we have

$\displaystyle\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)$
$\displaystyle\geq$ $\displaystyle-\|\sigma^{*}(x)\|^{2}$ $\displaystyle\geq$
$\displaystyle-c_{1}(x_{1}^{4}+x_{2}^{\frac{4}{3}}+1)$ $\displaystyle\geq$
$\displaystyle-c_{1}(x_{1}^{4}+x_{2}^{2}+2)$ $\displaystyle\geq$
$\displaystyle-4c_{1}-8c_{1}V(x),$

that is, (2.8) holds. Thus, the system (4.51) admits ULDP according to Theorem
2.1.

From (4.54) and (4.56) we can see that when $0<\varepsilon\leq d_{1}^{-1}$,

$\mathcal{L}^{\varepsilon}V(x)=\langle b(x),\nabla
V(x)\rangle+\frac{\varepsilon}{2}\text{Trace}\left(\sigma^{*}(x)\nabla^{2}V(x)\sigma(x)\right)\leq-\frac{1}{3}|\nabla
H(x)|^{2}\,\,{\rm if}\ |x|\gg 1.$

Therefore, there is a positive constant $k>0$ such that

$\mathcal{L}^{\varepsilon}V(x)\leq-k(x_{1}^{6}+x_{2}^{2})\ {\rm if}\ |x|\gg
1.$

[13, Theorem A] or [17, Theorem 4.1] implies that (4.51) admits a unique
invariant measure $\mu^{\varepsilon}$ for $0<\varepsilon\leq\frac{1}{d_{1}}$
and [4, Theorem 3.1] implies that
$\\{\mu^{\varepsilon}|0<\varepsilon\leq\frac{1}{d_{1}}\\}$ is tight.