# Exit-problem for a class of non-Markov processes with path dependency

Ashot Aleksian  Université Jean Monnet, Institut Camille Jordan, 23, rue du
docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.  Aline
Kurtzmann  Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR
7502, Vandoeuvre-lès-Nancy, F-54506, France.  Julian Tugaut  Université Jean
Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301,
42023 Saint-Étienne Cedex 2, France.

###### Abstract

We study the exit-time of a self-interacting diffusion from an open domain
$G\subset\mathbb{R}^{d}$. In particular, we consider the equation

$\differential{X_{t}}=-\left(\nabla V(X_{t})+\frac{1}{t}\int_{0}^{t}\nabla
F(X_{t}-X_{s})\differential{s}\right)\differential{t}+\sigma\differential{W_{t}}.$

We are interested in small-noise ($\sigma\to 0$) behaviour of exit-time from
the potentials’ domain of attraction. In this work rather weak assumptions on
potentials $V$ and $F$, and on domain $G$ are considered. In particular, we do
not assume $V$ nor $F$ to be either convex or concave, which covers a wide
range of self-attracting and self-avoiding stochastic processes possibly
moving in a complex multi-well landscape. The Large Deviation Principle for
Self-interacting diffusion with generalized initial conditions is established.
The main result of the paper states that, under some assumptions on potentials
$V$ and $F$, and on domain $G$, Kramers’ type law for the exit-time holds.
Finally, we provide a result concerning the exit-location of the diffusion.

_Keywords_ : Self-interacting diffusion, exit-time, Kramers’ law, Freidlin-
Wentzell theory, metastability, large deviations.

_AMS 2010 Subject Classification_ : 60K35, 60F10, 60H10.

###### Contents

  1. 1 Introduction
    1. 1.1 State of the art
    2. 1.2 Some remarks on dynamics and initial condition
    3. 1.3 Outline of the paper
    4. 1.4 List of notations
  2. 2 Large Deviation Principle
    1. 2.1 Establishing the LDP for the SID
    2. 2.2 Results related to the LDP
    3. 2.3 Compactness results
  3. 3 Exit-time
    1. 3.1 Auxiliary results
      1. 3.1.1 Initial descent to the point of attraction
      2. 3.1.2 Convergence of deterministic process towards $a$
      3. 3.1.3 Attraction of stochastic process towards $a$
      4. 3.1.4 Behaviour in the annulus between $B_{\rho}(a)$ and $\partial G$
      5. 3.1.5 Stabilization of the occupation measure
      6. 3.1.6 Exit before nearing $a$
      7. 3.1.7 Control of dynamics for small time intervals
      8. 3.1.8 Uniform lower bound for probability of exit from $G$
    2. 3.2 Proof of the main theorem
      1. 3.2.1 Kramers’ law
      2. 3.2.2 Exit-location
    3. 3.3 Proofs of auxiliary lemmas
      1. 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction
      2. 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards $a$
      3. 3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards $a$
      4. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between $B_{\rho}(a)$ and $\partial G$
      5. 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure
      6. 3.3.6 Proof of Lemma 3.6: Exit before nearing $a$
      7. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals
      8. 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from $G$
  4. 4 Generalization

##  1 Introduction

In this paper we consider a stochastic process
$(X_{t}^{\sigma},\;t\\!\geq\\!0)$ living in $\mathbb{R}^{d}$ defined by a
stochastic differential equation that includes interaction of the process with
its own passed trajectory in the drift term. Consider the following Self-
Interacting Diffusion (SID):

$\begin{cases}\differential{X^{\sigma}_{t}}\\!\\!\\!\\!&=-\nabla
V(X^{\sigma}_{t})\differential{t}-\nabla
F*\mu^{\sigma}_{t}(X^{\sigma}_{t})\differential{t}+\sigma\differential{W_{t}},\\\
\mu^{\sigma}_{t}&=\frac{1}{t}\int_{0}^{t}\delta_{X^{\sigma}_{s}}\differential{s},\\\
X^{\sigma}_{0}&=x_{0}\in\mathbb{R}^{d}\;\text{a.s.}\end{cases}$ (1.1)

$V:\mathbb{R}^{d}\to\mathbb{R}$ is called confinement potential and represents
general geometry of the space. $F:\mathbb{R}^{d}\to\mathbb{R}$ is called
interaction potential. Its convolution with $\mu^{\sigma}_{t}$, which is
called empirical measure, represents interaction term $\nabla
F*\mu^{\sigma}_{t}=\frac{1}{t}\int_{0}^{t}\nabla F(\cdot-
X_{s}^{\sigma})\differential{s}$. We also precise that
$(W_{t},\;t\\!\geq\\!0)$ denotes the standard $d$-dimensional Brownian motion,
$\delta_{x}$ is the Dirac measure on $\mathbb{R}^{d}$ concentrated at point
$x$ and $\sigma>0$ is a parameter controlling the noise of the system.

In this paper we establish the Kramers’ type law for the exit-time from an
open bounded domain of attraction $G\subset\mathbb{R}^{d}$ of a Self-
interacting diffusion (1.1) with general assumptions on confinement and
interaction. Moreover, assume that inside the domain $G$ there is a unique
point of attraction $a$. The precise assumptions on potentials $V$ and $F$,
and domain $G$ that we need for the main theorem will be presented and
discussed later (the baseline assumptions are given on pages A-1 and A-2,
whereas their generalised version is presented in Section 4). The following
theorem is the main result of the paper.

######  Theorem 1.1.

Let Assumptions A-1 and A-2 be fulfilled. Let the process $X^{\sigma}$ be the
unique strong solution of the system (1.1) with initial condition $x_{0}\in
G$. Let $\tau_{G}^{\sigma}:=\inf\\{t:X_{t}^{\sigma}\notin G\\}$ denote the
first time when $X^{\sigma}$ exits the domain $G$. Let $H:=\inf_{x\in\partial
G}\\{V(x)+F(x-a)-V(a)\\}$ be the height of the effective potential. Then, the
following two results hold:

  1. i)

Kramers’ law: for any $\delta>0$

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}\left(e^{\frac{2(H-\delta)}{\sigma^{2}}}<\tau_{G}^{\sigma}<e^{\frac{2(H+\delta)}{\sigma^{2}}}\right)=1;$
(1.2)

  2. ii)

Exit-location: for any closed set $N\subset\partial G$ such that $\inf_{z\in
N}\\{V(z)+F(z-a)-V(a)\\}>H$ the following limit holds:

$\lim_{\sigma\to 0}\mathbf{P}_{x_{0}}(X^{\sigma}_{\tau_{G}^{\sigma}}\in N)=0.$
(1.3)

###  1.1 State of the art

Similar systems with path-interaction behaviour have already been studied by
numerous researchers for more than 30 years. In most of the papers the long-
time behaviour of the process is considered. One of the first mathematical
descriptions of such a process under the name of Self-avoiding random walk is
presented by J.R. Norris, L.C.G. Rogers, and D. Williams in [NRW87]. Some
stochastic properties of the process along with some long-time behaviour
results in a particular setting were shown. The main difference between the
system considered there and ours is that in [NRW87] there is no
renormalization of the interaction term with time. In [DR92] R.T. Durrett and
L.C.G. Rogers introduce a similar to the previous paper system that aims to
model the shape of a growing polymer:

$\differential{X_{t}}=\left(\int_{0}^{t}f(X_{t}-X_{s})\differential{s}\right)\differential{t}+\differential{W_{t}}.$

Given the physical interpretation of the process, the authors called this
model “the Brownian Polymers”. In this paper asymptotic bounds in dimension
$d=1$ with some assumptions on interaction function were presented, along with
some conjectures on more precise long-time behaviour. Among those, the authors
conjectured that, in the symmetrical, repulsive case with limited interaction
($f(-x)=-f(x)$, $xf(x)\geq 0$, and $f$ has compact support), one can show that
$X_{t}/t\xrightarrow{}0$ a.s. This repulsive case was later studied in the
paper [TTV12], where the conjecture was partially proved (with an additional
assumption on smoothness of $f$).

Another branch of models with path-interaction, but this time via renormalized
with time occupation measure
$\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\differential{s}$ leaving on a
compact manifold, started to develop with a paper of M. Benaïm, M. Ledoux, and
O. Raimond [BLR02]. This work was continued in [BR03], [BR05], and others. In
the first paper, the authors showed that the asymptotic behaviour of the
empirical measure of the process $\mu_{t}$ can be linked to some deterministic
dynamical flow. In the following paper [BR03], the authors used these
techniques to describe some convergence in law properties of the process.
Later, in [BR05], assuming interaction to be symmetric, the authors gave an
almost sure convergence result for the occupation measure $\mu_{t}$ and
described its limit points.

First step towards studying this process in the non-compact setting, which was
$\mathbb{R}^{d}$, was done by A. Kurtzmann in [Kur10]. The author considers
the following model with interaction depending on the empirical measure of the
process.

$\differential{X_{t}}=-\left(\nabla
V(X_{t})+\frac{1}{t}\int_{0}^{t}\nabla_{x}F(X_{t},X_{s})\differential{s}\right)\differential{t}+\differential{W_{t}},$

where $F$ is regular and $V$ is convex at infinity. In this paper the ergodic
properties of $X$ were studied as well as certain conditions on $V$ and $F$
that guarantee almost sure convergence of $\mu_{t}$. This work generalizes the
previous ones of Benaïm and co-authors.

In the paper of V. Kleptsyn and A. Kurtzmann [KK12] dynamic of the form (1.1)
with constant $\sigma=\sqrt{2}$ is considered. The authors proved that, in the
case of convex confinement $V$ with $\lim_{|x|\to+\infty}V(x)=+\infty$,
attractive and symmetrical interaction (potential $F$ is uniformly convex and
spherically symmetrical, i.e. $F(x)=F(|x|)$), and some other regularity
assumptions, the occupation measure $\mu_{t}$ converges (in *-weakly sense)
almost surely to a density $\rho_{\infty}$. It means that there exists a
unique density $\rho_{\infty}:\mathbb{R}^{d}\to\mathbb{R}_{+}$ such that

$\mu_{t}=\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\differential{s}\xrightarrow[t\to\infty]{*-\text{weakly}}\rho_{\infty}(x)\differential{x}\quad\text{a.s.}$

Moreover, $\rho_{\infty}$ is a function satisfying the following functional
equation:

$\rho_{\infty}=\Pi(\rho_{\infty}):=\frac{\exp{-V-F*\rho_{\infty}}}{\int_{\mathbb{R}^{d}}\exp{-V(y)-F*\rho_{\infty}(y)}\differential{y}}.$

That brings us to the second part of the paper, the exit-time problem. The
main goal of this work is to study first exit-time of $X^{\sigma}$, given by
(1.1), from some bounded regular domain $G\subset\mathbb{R}^{d}$, i.e., the
following stopping time $\tau_{G}^{\sigma}:=\inf\\{t\geq
0:X_{t}^{\sigma}\notin G\\}$. In particular, we are interested in exits driven
by the Brownian motion with small noise $\sigma>0$. Similar problem for Itô
diffusion was introduced and solved by M. Freidlin and A. Wentzell. The
techniques that they used in number of papers, starting with [VF70] and
presented in their book [FW12], are known under the name of Freidlin-Wentzell
theory (see also [DZ10, §5.6]). Consider an Itô diffusion in gradient form:

$\differential{X_{t}}=-\nabla
V(X_{t})\differential{t}+\sigma\differential{W_{t}}.$

Then, the main result of the Freidlin-Wentzell theory is that, given a domain
$G\subset\mathbb{R}^{d}$ with only one attraction point of marginal
deterministic process (with $\sigma=0$) named $a\in G$ and some other
regularity assumptions on $G$ and $V$, the so-called Kramers’ type law for
exit-time $\tau_{G}^{\sigma}$ can be established. Namely, if we denote
$H:=\inf_{z\in\partial G}\\{V(z)-V(a)\\}$, then for any starting point
$x_{0}\in G$ and for any $\delta>0$

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}(e^{\frac{2(H-\delta)}{\sigma^{2}}}<\tau^{\sigma}_{G}<e^{\frac{2(H+\delta)}{\sigma^{2}}})=1.$
(1.4)

It means that, with decreasing $\sigma>0$, the exit-time from stable (or
positively invariant) by $-\nabla V$ domain $G$ grows exponentially with rate
that depends on the height $H$ of potential $V$ inside the domain $G$.

A similar question was posed by S. Herrmann, P. Imkeller, and D. Peithmann for
McKean-Vlasov type diffusion in [HIP08]. In particular, they considered a so-
called Self-stabilizing diffusion (SSD) of the form

$\begin{cases}\differential{X^{\sigma}_{t}}\\!\\!\\!\\!&=-\nabla
V(X^{\sigma}_{t})\differential{t}-\nabla
F*\nu^{\sigma}_{t}(X^{\sigma}_{t})\differential{t}+\sigma\differential{W_{t}},\\\
\nu^{\sigma}_{t}&=\mathcal{L}(X_{t}^{\sigma}),\\\
X^{\sigma}_{0}&=x_{0}\in\mathbb{R}^{d}\;\text{a.s.},\end{cases}$ (1.5)

where $\mathcal{L}$ denotes the law of random variable $X_{t}^{\sigma}$. Apart
from the obvious parallels that can be drawn between equations (1.1) and (1.5)
that define SID and SSD respectively, the following fact makes self-
stabilizing case interesting for the current paper. As was shown in [CGM08],
in the case of convex confinement and interaction and under some other
regularity assumptions, the law $\nu_{t}$ converges to the unique probability
measure that is solution of $\nu=\Pi(\nu)$, which is the same behaviour that
was established in [KK12].

Following the Freidlin-Wentzell theory techniques, in [HIP08] the authors
first establish a large deviation principle for SSD with general assumptions
on $V$ and $F$. After that, they had to restrict themselves to the case of
convex confinement and convex interaction potentials in order to achieve,
under some stability and regularity assumptions on $G$, the Kramers’ type law
(1.4). Assuming $F(0)=0$, the rate $H$ in this case have the form
$H=\inf_{z\in\partial G}\\{V(z)-V(a)+F(z-a)\\}$.

In [Tug12] and [Tug16], J. Tugaut studies the exit-time problem for SSD in
convex landscape with convex interaction as it was presented in [HIP08]. In
these papers he establishes the Kramers’ type law avoiding using large
deviations principle. Instead, the author uses various analytical and coupling
methods to deal with the problem, thus simplifying calculations. This work was
continued in [Tug18] and [Tug19] where the same techniques were used to
establish the Kramers’ type law in the case of confinement potential $V$ that
is of the double-well form. However, exit-time for the SSD with general
assumptions (as in Freidlin-Wentzell theory for Itô diffusion) is still an
open problem.

Exit-time problem was also studied for the case of self-interacting diffusion.
In [AdMKT23], A. Aleksian, P. Del Moral, A. Kurtzmann, and J. Tugaut prove
Kramers’ type law for SID in which both interaction and confinement potentials
$V$ and $F$ are convex. This nice property of potentials was used by the
authors in order to prove this exit-time result by applying analytical and
coupling techniques similar to the ones used in [Tug16]. Since the goal of the
current paper is to establish some exit-time result for SID with more general,
than in [AdMKT23], assumptions, we have to use a different approach. In our
case, that consists in proving the Large Deviation Principle and restoring the
logic of Freidlin-Wentzell theory for SID.

###  1.2 Some remarks on dynamics and initial condition

Let us introduce some notation and notions that will be useful to describe
laconically later derivations in this paper.

First, let us define the marginal deterministic system corresponding to (1.1).
By $X^{0}$ we will denote the solution of the following deterministic system

$\begin{cases}\frac{\differential}{\differential{t}}X^{0}_{t}\\!\\!\\!\\!&=-\nabla
V(X_{t}^{0})-\frac{1}{t}\int_{0}^{t}\nabla
F(X^{0}_{t}-X^{0}_{s})\differential{s},\\\
X^{0}_{0}&=x_{0}\in\mathbb{R}^{d}.\end{cases}$ (1.6)

Note that, moreover, we can generalize the initial conditions of diffusion
(1.1). This process is non-Markov, but, at the same time, knowing only the
past trajectory of the process, we can construct its subsequent path. We will
describe the notion of previous path using three parameters. $t_{0}\geq 0$
will represent its time-length, $\mu_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d})$
will represent the occupation measure of the past trajectory and
$x_{0}\in\mathbb{R}^{d}$ its end point starting from which we continue the
dynamic. We contract initial conditions into a vector
$\mathbf{x}=(t_{0},\mu_{0},x_{0})\in\mathfrak{X}=[0,\infty]\\!\times\\!\mathcal{P}_{2}(\mathbb{R}^{d})\\!\times\\!\mathbb{R}^{d}$,
and, in order to be able to operate with the three components of $\mathbf{x}$
separately, we introduce the following projection mappings. Let
$\mathrm{T}:\mathfrak{X}\to[0;\infty]$ be the projection to the first
coordinate returning initial time $t_{0}$. For the initial empirical measure,
let $\mathrm{M}:\mathfrak{X}\to\mathcal{P}_{2}(\mathbb{R}^{d})$ be the mapping
that returns the second coordinate of $\mathbf{x}\in\mathfrak{X}$. Finally,
for the the starting point of the process, respectively, define the mapping
$\mathrm{P}:\mathfrak{X}\to\mathbb{R}^{d}$ that returns the third coordinate
of $\mathbf{x}\in\mathfrak{X}$.

Following the classical notation for diffusions, we introduce the following
system of equations:

$\begin{cases}\differential{X^{\mathbf{x},\sigma}_{t}}\\!\\!\\!\\!&=-\nabla
V(X^{\mathbf{x},\sigma}_{t})\differential{t}-\nabla
F*\mu^{\mathbf{x},\sigma}_{t}(X^{\mathbf{x},\sigma}_{t})\differential{t}+\sigma\differential{W_{t}},\\\
\mu^{\mathbf{x},\sigma}_{t}&=\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\mathrm{M}{\mathbf{x}}+\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\delta_{X^{\mathbf{x},\sigma}_{s}}\differential{s},\\\
X^{\mathbf{x},\sigma}_{0}&=\mathrm{P}{\mathbf{x}}\;\text{a.s.}\end{cases}$
(1.7)

As before, we define marginal deterministic system that corresponds to the
equation $\eqref{eq:SID_main_sys_gen}$ as

$\begin{cases}\frac{\differential{}}{\differential{t}}X^{\mathbf{x},0}_{t}\\!\\!\\!\\!&=-\nabla
V(X^{\mathbf{x},0}_{t})-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\int_{\mathbb{R}^{d}}\nabla
F(X^{\mathbf{x},0}_{t}-s)\mathrm{M}{\mathbf{x}}(\differential{s})\\\
&\quad-\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\nabla
F(X^{\mathbf{x},0}_{t}-X^{\mathbf{x},0}_{s})\differential{s},\\\
X^{\mathbf{x},0}_{0}&=\mathrm{P}{\mathbf{x}}.\end{cases}$ (1.8)

Hereafter, we will drop one part of the subscript or both (initial conditions
and $\sigma$) if it is clear which one of the equations is meant. Though, to
avoid confusion we will always use the subscript $0$ to emphasize that we
consider specifically the deterministic system. For $\mathbf{x}$ such that
$\mathrm{T}{\mathbf{x}}=\infty$ we naturally extend the definition of the
processes (1.7) and (1.8) as

$\begin{cases}\differential{X^{\mathbf{x},\sigma}_{t}}\\!\\!\\!\\!&=-\nabla
V(X^{\mathbf{x},\sigma}_{t})\differential{t}-\nabla
F*\mathrm{M}{\mathbf{x}}(X^{\mathbf{x},\sigma}_{t})\differential{t}+\sigma\differential{W_{t}},\\\
X^{\mathbf{x},\sigma}_{0}&=\mathrm{P}{\mathbf{x}}\;\text{a.s.}\end{cases}$
(1.9)

Despite that introduction of $\mathbf{x}$ aims to describe a notion of
“previous path” of the process, the set $\mathfrak{X}$ is defined to be more
general. Not only we allow $t_{0}$ to be equal to infinity, but also $\mu_{0}$
belongs to $\mathcal{P}_{2}(\mathbb{R}^{d})$, which, in general, does not
restrict it to be an empirical measure of any path.

Note that topology naturally defined on $[0;\infty]$ by open intervals in
usual sense and intervals of the form $[0,x)$, $(x;\infty]$ is metrizable. Let
us denote $d_{[0,\infty]}$ some metric on this space. We also equip the set
$\mathcal{P}_{2}(\mathbb{R}^{d})$ with Wasserstein-2 metric (see e.g. [Vil09,
Definition 6.1]). Thus, the Cartesian product $\mathfrak{X}$ is a complete
separable metric space with the following metric

$d_{\mathfrak{X}}(\mathbf{x}_{1},\mathbf{x}_{2})=\max(d_{[0,\infty]}(\mathrm{T}{\mathbf{x}_{1}},\mathrm{T}{\mathbf{x}_{2}});\mathbb{W}_{2}(\mathrm{M}{\mathbf{x}_{1}},\mathrm{M}{\mathbf{x}_{2}});|\mathrm{P}{\mathbf{x}_{1}}-\mathrm{P}{\mathbf{x}_{2}}|),$

where $\mathbb{W}_{2}$ is the Wasserstein distance (see references in §2.1).

Existence and uniqueness result for stochastic differential equations that
depend on their path is standard for the case of Lipschitz continuous drift
term (see e.g. [RW00, Theorem 11.2]). It is worth noting that introducing
$\mathbf{x}\in\mathfrak{X}$ does not complicate matters in any way, as long as
$\nabla V$ and $\nabla F$ remain Lipschitz continuous. Assumption A-1 ensures
that we are working in this framework.

###  1.3 Outline of the paper

Section 2 is dedicated to the Large deviation principle. First, we define what
we mean by the large deviation principle (LDP), then we establish, under the
set of assumptions A-1, LDP for self-interacting diffusion in Theorem 2.2.
Next, some useful (for exit-time problem) results related to LDP are
presented.

Section 3 deals with the exit-time problem. First, we present the set of
assumptions A-2 that contains assumptions on domain $G$, its boundary and
stability properties. We discuss the role that each of these assumptions play
in later proof and present some examples of confinement and interaction
potentials along with possible domains $G$ that one can choose. Then, we
present the main result of the paper that is Theorem 1.1. This result is
followed by Section 3.1 where we present auxiliary lemmas that are later used
in Section 3.2 to prove the main Theorem 1.1. These lemmas are later proved in
Section 3.3.

Section 4 is focused on the generalization of Assumptions A-1 and A-2. In this
section, we examine the scenario of a locally Lipschitz continuous drift term
($\nabla V$ and $\nabla F$) and an unbounded domain $G$ from which we want to
exit.

###  1.4 List of notations

The paper contains number of technical results. To make them more reader-
friendly, we present here a list of notions that are used later in the paper.

$|\cdot|$, $\langle\cdot\;;\cdot\rangle$ |  Euclidean norm and scalar product in $\mathbb{R}^{d}$
---|---
$\mathsf{Int}(\Gamma)$/$\accentset{\circ}{\Gamma}$, $\mathsf{Cl}(\Gamma)$/$\overline{\Gamma}$, $\partial\Gamma$ |  Interior, closure and boundary of a set $\Gamma\subset\mathbb{R}^{d}$
$C([0;T];\mathbb{R}^{d})$ |  The set of continuous functions on $[0;T]$ with values in $\mathbb{R}^{d}$ equipped with the norm $\|\phi\|_{\infty}:=\max_{t\in[0;T]}|\phi(t)|$
$C_{0}([0;T];\mathbb{R}^{d})$ |  Subset of $C([0;T];\mathbb{R}^{d})$ consisting of functions $\psi$ such that $\psi(0)=0$
$C^{2}(\mathbb{R}^{d};\mathbb{R})$ |  The set of functions on $\mathbb{R}^{d}$ with values in $\mathbb{R}$ with continuous partial derivatives up to order 2
$L([0;T];\mathbb{R}^{d})$ |  The set of integrable w.r.t the Lebesgue measure functions on $[0;T]$ with values in $\mathbb{R}^{d}$
$L_{2}([0;T];\mathbb{R}^{d})$ |  Subset of $L_{2}([0;T];\mathbb{R}^{d})$ consisting of functions $\psi$ such that $\int_{0}^{T}|\psi(s)|^{2}\differential{s}<\infty$
$\mathcal{P}_{2}(\mathbb{R}^{d})$ |  The set of probability measures with finite second moments, defined on $\mathbb{R}^{d}$, and equipped with the Wasserstein-2 distance $\mathbb{W}_{2}$ (see e.g. [Vil09, Definition 6.1])
$\mathcal{P}_{2}(\overline{G})$ |  The set of probability measures with finite second moment, defined on the closure $\overline{G}\subset\mathbb{R}^{d}$, where $G$ is the same domain as in Assumption A-2, page A-2
$(\mathfrak{X},d_{\mathfrak{X}})$ |  A metric space consisting of triples $\mathbf{x}=(t,\mu,x)$, $t\in[0,\infty]$, $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and $x\in\mathbb{R}^{d}$ (see Section 1.2)
$\mathrm{T}$, $\mathrm{M}$, $\mathrm{P}$ |  Projection maps on the first, second, and third coordinates of an $\mathbf{x}\in\mathfrak{X}$ (see Section 1.2)
$B_{\rho}(a):=\\{x\in\mathbb{R}^{d}:|x-a|\leq\rho\\}$, $S_{\rho}(a):=\\{x\in\mathbb{R}^{d}:|x-a|=\rho\\}$ |  A closed ball and a sphere of radius $\rho$ around $a$ in $\mathbb{R}^{d}$

$\mathbb{B}_{\rho}(\mu):=\\{\nu\in\mathcal{P}_{2}(\overline{G}):\mathbb{W}_{2}(\mu;\nu)\leq\rho\\}$ |  A closed ball of radius $\rho$ around $\mu$ in $\mathcal{P}_{2}(\overline{G})$, where $G$ will be the same domain as in Assumption A-2, page A-2
---|---
$\mathbb{B}_{\rho}(\mathbf{x}):=\\{\mathbf{y}\in\mathfrak{X}:d_{\mathfrak{X}}(\mathbf{y},\mathbf{x})\leq\rho\\}$ |  A closed ball of radius $\rho$ around $\mathbf{x}$ in $\mathfrak{X}$
$\\#$ |  A push-forward symbol (see e.g. [Vil09, p.11])

All the equations in the paper are meant to hold a.s. unless stated otherwise.

##  2 Large Deviation Principle

In this paper Large Deviations techniques are widely used. Here, by the Large
Deviation Principle (LDP) we mean the following asymptotic behaviour of
measures.

######  Definition 2.1.

Family of measures $(\nu^{\sigma})_{\sigma>0}$ defined on some Banach space
$B$ equipped with Borel sigma-algebra $\mathcal{B}$ is said to satisfy the
Large Deviation Principle with a good rate function $I$ if for any measurable
set $\Gamma\in\mathcal{B}$:

$-\inf_{x\in\accentset{\circ}{\Gamma}}I(x)\leq\liminf_{\sigma\xrightarrow{}0}\frac{\sigma^{2}}{2}\log\nu^{\sigma}(\Gamma)\leq\limsup_{\sigma\xrightarrow{}0}\frac{\sigma^{2}}{2}\log\nu^{\sigma}(\Gamma)\leq-\inf_{x\in\overline{\Gamma}}I(x),$

where $I:B\to[0,\infty]$ is a lower-semicontinuous function (this property
defines rate function) whose level sets $\\{x:I(x)\leq\alpha\\}$ are compact
subsets of $B$ for any $0\leq\alpha<\infty$ (which means by definition that
the rate function is good).

For more information about LDP, see [DZ10]. Note that our definition of LDP by
its appearance deviates from the conventional one, but they are equivalent up
to multiplication of the “conventional” rate function by $1/2$.

###  2.1 Establishing the LDP for the SID

In this section we prove LDP for Self-Interacting diffusions of the type
(1.7). First, we introduce the following group of main assumptions.

######  Assumptions A-1.

  1. i)

(regularity) Potentials $V$ and $F$ belong to the space
$C^{2}(\mathbb{R}^{d};\mathbb{R})$.

  2. ii)

(Lipschitz continuity) There exist $\text{Lip}_{\nabla V},\text{Lip}_{\nabla
F}>0$ such that $|\nabla V(x)-\nabla V(y)|\leq\text{Lip}_{\nabla V}|x-y|$ and
$|\nabla F(x)-\nabla F(y)|\leq\text{Lip}_{\nabla F}|x-y|$ for any
$x,\;y\in\mathbb{R}^{d}$.

  3. iii)

(boundedness of $\nabla F$) There exists $C_{\nabla F}>0$ such that $|\nabla
F(x)|\leq C_{\nabla F}$ for any $x\in\mathbb{R}^{d}$.

Not only the first two assumptions above help to establish the global
existence and uniqueness result for (1.1) and (1.7) easily, but they also play
an important role in proving the Large Deviation Principle. The third
assumption A-1.iii is only used in Lemma 2.4 and can be substituted, for
example, with a condition that diffusion $X^{\mathbf{x},\sigma}$ does not
leave some bounded domain in $\mathbb{R}^{d}$. Since it is indeed the case in
Section 3, Assumption A-1.iii is not as restrictive as it could seem. By the
same logic, Assumption A-1.ii can be relaxed to locally Lipschitz case
whenever the existence and uniqueness problem is resolved. We will stress
again this observations in Section 4.

Consider the following theorem that is the main result of the section.

######  Theorem 2.2.

Under Assumptions A-1.i, A-1.ii for any $\mathbf{x}\in\mathfrak{X}$ the
probability measures $(\nu^{\mathbf{x},\sigma})_{\sigma>0}$ induced on
$C([0,T];\mathbb{R}^{d})$ by the process $(X_{t}^{\mathbf{x},\sigma})_{0\leq
t\leq T}$, which is the unique solution of the system (1.7), satisfy LDP with
the following good rate function:

$\displaystyle I_{T}^{\mathbf{x}}(f)$ $\displaystyle:=$
$\displaystyle\frac{1}{4}\int_{0}^{T}\Big{|}\dot{f}(t)+\nabla
V(f(t))+\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\nabla
F*\mathrm{M}{\mathbf{x}}(f(t))$ (2.1) $\displaystyle+$
$\displaystyle\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\nabla
F(f(t)-f(s))\differential{s}\Big{|}^{2}\differential{t},\;\text{for }f\in
H_{1}^{\mathrm{P}{\mathbf{x}}}$

and $I_{T}^{\mathbf{x}}(f):=\infty$ otherwise. Here, $H_{1}^{x}$ is the space
of absolutely continuous functions with square-integrable derivatives that
start at point $x=\mathrm{P}{\mathbf{x}}$;
$H_{1}^{\mathrm{P}{\mathbf{x}}}:=\\{\mathrm{P}{\mathbf{x}}+\int_{0}^{\cdot}g(s)\differential{s}:g\in
L_{2}([0,T];\mathbb{R}^{d})\\}$.

###### Proof.

Let us define function $G:C_{0}([0,T];\mathbb{R}^{d})\to
C([0,T];\mathbb{R}^{d})$ that maps every function $g$ to the unique solution
of the following equation:

$\displaystyle f(t)=\mathrm{P}{\mathbf{x}}-\int_{0}^{t}\nabla
V(f(s))\differential{s}-\int_{0}^{t}\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\nabla
F*\mathrm{M}{\mathbf{x}}(f(s))\differential{s}$
$\displaystyle-\int_{0}^{t}\frac{1}{\mathrm{T}{\mathbf{x}}+s}\int_{0}^{s}\nabla
F(f(s)-f(u))\differential{u}\differential{s}+g(t).$

Let us show that $G$ is continuous. Fix some $\delta>0$ and two functions
$g_{1},g_{2}\in C_{0}([0,T];\mathbb{R}^{d})$ such that
$\|g_{1}-g_{2}\|_{\infty}\leq\delta$. Let $f_{1}=G(g_{1})$, $f_{2}=G(g_{2})$.
Then,

$\displaystyle|f_{1}(t)-$ $\displaystyle f_{2}(t)|\leq\int_{0}^{t}|\nabla
V(f_{1}(s))-\nabla V(f_{2}(s))|\differential{s}$
$\displaystyle\quad+\int_{0}^{t}\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\int_{\mathbb{R}^{d}}\big{|}\nabla
F(f_{1}(s)-u)-\nabla
F(f_{2}(s)-u)\big{|}\mathrm{M}{\mathbf{x}}(\differential{u})\differential{s}$
$\displaystyle\quad+\int_{0}^{t}\frac{1}{\mathrm{T}{\mathbf{x}}+s}\int_{0}^{s}\big{|}\nabla
F(f_{1}(s)-f_{1}(u))-\nabla
F(f_{2}(s)-f_{2}(u))\big{|}\differential{u}\differential{s}$
$\displaystyle\quad+|g_{1}(t)-g_{2}(t)|.$

Use Lipschitz continuity of $\nabla V$, $\nabla F$ (Assumption A-1.ii) and
get:

$\displaystyle|f_{1}(t)-f_{2}(t)|$ $\displaystyle\leq\left(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla
F}\right)\int_{0}^{t}\big{|}f_{1}(s)-f_{2}(s)\big{|}\differential{s}$
$\displaystyle\quad+\int_{0}^{t}\frac{\text{Lip}_{\nabla
F}}{\mathrm{T}{\mathbf{x}}+s}\int_{0}^{s}\big{|}f_{1}(u)-f_{2}(u)\big{|}\differential{u}\differential{s}+\delta$
$\displaystyle\leq\left(\text{Lip}_{\nabla
V}+\left(1+\frac{T}{\mathrm{T}{\mathbf{x}}}\right)\text{Lip}_{\nabla
F}\right)\int_{0}^{t}|f_{1}(s)-f_{2}(s)|\differential{s}+\delta.$

Therefore, by Grönwall’s inequality

$\|f_{1}-f_{2}\|_{\infty}\leq\delta\exp\left\\{\left(\text{Lip}_{\nabla
V}+\left(1+\frac{T}{\mathrm{T}{\mathbf{x}}}\right)\text{Lip}_{\nabla
F}\right)T\right\\};$

it means the continuity of the map $G$ for any possible
$\mathbf{x}\in\mathfrak{X}$.

By the uniqueness of the solution to (1.7), we can express
$\nu^{\mathbf{x},\sigma}$ as $\nu^{\mathbf{x},\sigma}=G_{\\#}\eta^{\sigma}$,
where $\eta^{\sigma}$ is the probability measure induced on
$C_{0}([0,T];\mathbb{R}^{d})$ by the path of Brownian motion
$W^{\sigma}=\sigma W$. LDP for the path of Brownian motion with vanishing
noise is known under the name of Schilder theorem ([DZ10, Theorem 5.2.3]).
Since $G$ is continuous, we can apply the Contraction principle (see [DZ10,
Theorem 4.2.1]) and conclude that the family of measures
$(\nu^{\mathbf{x},\sigma})_{\sigma\geq 0}$ satisfies an LDP with the good rate
function

$\displaystyle I_{T}^{\mathbf{x}}(f)$ $\displaystyle=\inf_{g\in
G^{-1}(f)}\frac{1}{4}\int^{T}_{0}|\dot{g}(t)|^{2}\differential{t}$
$\displaystyle=\begin{cases}\frac{1}{4}\int_{0}^{T}|\dot{f}(t)+\nabla
V(f(t))+\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\int_{\mathbb{R}^{d}}\nabla
F(f(t)-u)\mathrm{M}{\mathbf{x}}(\differential{u})\\\
\quad\quad+\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\nabla
F(f(t)-f(u))\differential{u}|^{2}\differential{t},\;\text{for }f\in
H_{1}^{\mathrm{P}{\mathbf{x}}},\\\ \infty,\quad\text{otherwise}.\end{cases}$

∎

######  Remark 2.3.

For simplicity of the notation, we define the rate function corresponding to
the system (1.1) as $I_{T}^{x_{0}}$, i.e.

$I_{T}^{x_{0}}:=\begin{cases}\frac{1}{4}\int_{0}^{T}|\dot{f}(t)+\nabla
V(f(t))+\frac{1}{t}\int_{0}^{t}\nabla
F(f(t)-f(s))\differential{s}|^{2}\differential{t},\;\text{for }f\in
H_{1}^{x_{0}},\\\ \infty,\quad\text{otherwise}.\end{cases}$

###  2.2 Results related to the LDP

The following lemma generalizes the large deviation principle for the case of
converging initial conditions.

######  Lemma 2.4.

Under assumptions A-1, for any sequence $\\{\mathbf{x}_{n}\\}_{1}^{\infty}$
such that
$\mathbf{x}_{n}\xrightarrow[n\to\infty]{d_{\mathfrak{X}}}\leavevmode\nobreak\
\mathbf{x}$ the following inequalities hold:

  1. i)

For any closed $\Phi\subset C([0,T];\mathbb{R}^{d})$

$\limsup_{\begin{subarray}{c}n\to\infty\\\ \sigma\to
0\end{subarray}}\frac{\sigma^{2}}{2}\log\mathbf{P}_{\mathbf{x}_{n}}(X^{\sigma}\in\Phi)\leq-\inf_{\phi\in\Phi}I_{T}^{\mathbf{x}}(\phi).$

  2. ii)

For any open $\Psi\subset C([0,T];\mathbb{R}^{d})$

$\liminf_{\begin{subarray}{c}n\to\infty\\\ \sigma\to
0\end{subarray}}\frac{\sigma^{2}}{2}\log\mathbf{P}_{\mathbf{x}_{n}}(X^{\sigma}\in\Psi)\geq-\inf_{\phi\in\Psi}I_{T}^{\mathbf{x}}(\phi).$

###### Proof.

We will show that the families of measures
$(\nu^{\mathbf{x}_{\sigma},\sigma})_{\sigma>0}$ and
$(\nu^{\mathbf{x},\sigma})_{\sigma>0}$, which are families of probability
measures induced on $C([0,T];\mathbb{R}^{d})$ by
$X^{\mathbf{x}_{\sigma},\sigma}$ and $X^{\mathbf{x},\sigma}$ respectively, are
exponentially equivalent (see [DZ10, Definition 4.2.10]) for any
$\\{\mathbf{x}_{\sigma}\\}_{\sigma>0}$ such that
$\mathbf{x}_{\sigma}\xrightarrow[\sigma\to 0]{d_{\mathfrak{X}}}\mathbf{x}$.

Indeed, if we define
$Z^{\sigma}_{t}=X^{\mathbf{x}_{\sigma},\sigma}_{t}-X^{\mathbf{x},\sigma}_{t}$,
where $X^{\mathbf{x}_{\sigma},\sigma}_{t}$ and $X^{\mathbf{x},\sigma}_{t}$ are
driven by the same Brownian motion, then

$\displaystyle|Z_{t}^{\sigma}|\leq|\mathrm{P}{\mathbf{x}_{\sigma}}-\mathrm{P}{\mathbf{x}}|+\int_{0}^{t}|\nabla
V(X_{s}^{\mathbf{x}_{\sigma},\sigma})-\nabla
V(X_{s}^{\mathbf{x},\sigma})|\differential{s}$
$\displaystyle+\int_{0}^{t}|\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}\nabla
F*\mathrm{M}{\mathbf{x}_{\sigma}}(X_{s}^{\mathbf{x}_{\sigma},\sigma})-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\nabla
F*\mathrm{M}{\mathbf{x}}(X_{s}^{\mathbf{x},\sigma})|\differential{s}$
$\displaystyle+\int_{0}^{t}\int_{0}^{s}\big{|}\frac{1}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}\nabla
F(X_{s}^{\mathbf{x}_{\sigma},\sigma}-X_{z}^{\mathbf{x}_{\sigma},\sigma})-\frac{1}{\mathrm{T}{\mathbf{x}}+s}\nabla
F(X_{s}^{\mathbf{x},\sigma}-X_{z}^{\mathbf{x},\sigma})\big{|}\differential{z}\differential{s}.$

In order to separate the effect of closeness of
$\mathrm{T}{\mathbf{x}_{\sigma}}$ to $\mathrm{T}{\mathbf{x}}$ from
$Z_{t}^{\sigma}$ in the last two integrals, we add and subtract
$\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}\nabla
F*\mathrm{M}{\mathbf{x}}(X_{s}^{\mathbf{x},\sigma})$ and
$\frac{1}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}\nabla
F(X_{s}^{\mathbf{x},\sigma}-X_{z}^{\mathbf{x},\sigma})$ in the corresponding
integrals. Since $\nabla V$ and $\nabla F$ are Lipschitz continuous, we get

$\displaystyle|Z_{t}^{\sigma}|$ $\displaystyle\leq(1+\text{Lip}_{\nabla
F})d_{\mathfrak{X}}(\mathbf{x}_{\sigma},\mathbf{x})+(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla F})\int_{0}^{t}|Z_{s}^{\sigma}|\differential{s}$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|\left(\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\right)\nabla
F(X_{s}^{\mathbf{x},\sigma}-z)\right|\mathrm{M}{\mathbf{x}}(\differential{z})\differential{s}$
$\displaystyle\quad+\text{Lip}_{\nabla
F}\int_{0}^{t}\frac{1}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}\int_{0}^{s}\big{(}|Z_{s}^{\sigma}|+|Z_{z}^{\sigma}|\big{)}\differential{z}\differential{s}$
$\displaystyle\leq(1+\text{Lip}_{\nabla
F})d_{\mathfrak{X}}(\mathbf{x}_{\sigma},\mathbf{x})+\left(\text{Lip}_{\nabla
V}+2\text{Lip}_{\nabla
F}+\frac{T}{\mathrm{T}{\mathbf{x}_{\sigma}}}\right)\int_{0}^{t}|Z_{s}^{\sigma}|\differential{s}$
$\displaystyle\quad+C_{\nabla
F}\int_{0}^{t}\left|\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\right|\differential{s},$

where we get the last inequality by applying assumption A-1.iii to
$\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|\left(\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\right)\nabla
F(X_{s}^{\mathbf{x},\sigma}-z)\right|\mathrm{M}{\mathbf{x}}(\differential{z})\differential{s}$.

Since the expression inside the integral is bounded, we can conclude that
$\int_{0}^{t}\left|\frac{\mathrm{T}{\mathbf{x}_{\sigma}}}{\mathrm{T}{\mathbf{x}_{\sigma}}+s}-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\right|\differential{s}=O(|\mathrm{T}{\mathbf{x}_{\sigma}}-\mathrm{T}{\mathbf{x}}|)=O(d_{\mathfrak{X}}(\mathbf{x}_{\sigma},\mathbf{x}))$.
Thus, by Grönwall’s inequality,

$\displaystyle|Z_{t}^{\sigma}|$ $\displaystyle\leq\left(1+\text{Lip}_{\nabla
F}+C_{\nabla F}\right)O(d_{\mathfrak{X}}(\mathbf{x}_{\sigma};\mathbf{x}))$
$\displaystyle\quad\times\exp{\left(\text{Lip}_{\nabla V}+2\text{Lip}_{\nabla
F}+\frac{T}{\mathrm{T}{\mathbf{x}_{\sigma}}}\right)T}.$

It means that $\mathbf{P}(|Z_{t}^{\sigma}|\geq\delta)=0$ for any $\delta$ if
we choose $\sigma$ to be small enough. That proves exponential equivalence of
$(\bar{\nu}^{\mathbf{x}_{\sigma},\sigma})_{\sigma>0}$ and
$(\bar{\nu}^{\mathbf{x},\sigma})_{\sigma>0}$, and, by contraction principle
(see [DZ10, Theorem 4.2.13]), the lemma itself.

∎

The following lemma elaborates on the idea of the LDP and provides a tool to
study the asymptotic behaviour ($\sigma\to 0$) of the process $X^{\sigma}$
with respect to its initial conditions.

######  Lemma 2.5.

For any compact subset $\mathcal{C}\subset\mathfrak{X}$ the following
inequalities hold:

  1. i)

For any closed $\Phi\subset C([0,T];\mathbb{R}^{d})$

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Phi)\leq-\inf_{\mathbf{x}\in\mathcal{C}}\inf_{\phi\in\Phi}I_{T}^{\mathbf{x}}(\phi).$

  2. ii)

Similarly, for any open $\Psi\subset C([0,T];\mathbb{R}^{d})$

$\liminf_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\inf_{\mathbf{x}\in\mathcal{C}}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Psi)\geq-\sup_{\mathbf{x}\in\mathcal{C}}\inf_{\phi\in\Psi}I_{T}^{\mathbf{x}}(\phi).$

###### Proof.

For the first inequality, for each fixed $\varepsilon>0$ let us define

$I^{\varepsilon}:=\min\left\\{\inf_{\mathbf{x}\in\mathcal{C}}\inf_{\phi\in\Phi}I_{T}^{\mathbf{x}}(\phi)-\varepsilon;\frac{1}{\varepsilon}\right\\}.$

By Lemma 2.4 for any $\mathbf{x}\in\mathcal{C}$ there exists small enough
$\sigma_{\mathbf{x}}>0$ such that for any $\sigma<\sigma_{\mathbf{x}}$

$\frac{\sigma^{2}}{2}\log\sup_{\mathbf{y}\in
B_{\sigma_{\mathbf{x}}}(\mathbf{x})}\mathbf{P}_{\mathbf{y}}(X^{\sigma}\in\Phi)\leq-I^{\varepsilon}.$

Since $\mathcal{C}$ is compact, we can cover it by finite amount of
$B_{\sigma_{\mathbf{x}_{i}}}(\mathbf{x}_{i})$ for some
$\mathbf{x}_{1},\dots,\mathbf{x}_{m}\in\mathcal{C}$. Then, for any
$\sigma<\min_{1\leq i\leq m}\sigma_{\mathbf{x}_{i}}$

$\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Phi)\leq-I^{\varepsilon},$

and that proves the first inequality. One can prove the second inequality the
same way. ∎

By definition, rate functions are lower semicontinuous, i.e. for any
$\mathbf{x}\in\mathfrak{X}$,
$\liminf_{\phi\to\phi_{0}}I^{\mathbf{x}}_{T}(\phi)\geq
I^{\mathbf{x}}_{T}(\phi_{0})$, or, equivalently, all level sets
$L_{\alpha}:=\\{\phi\in
C([0,T];\mathbb{R}^{d}):I^{\mathbf{x}}_{T}(\phi)\leq\alpha\\}$ are closed for
$0\leq\alpha<\infty$. An immediate consequence of this property is that infima
of $I^{\mathbf{x}}_{T}$ are achieved over compact sets. The following lemma
extends this lower semicontinuity property of our rate functions to the case
of converging initial conditions besides the argument of $I_{T}$. In other
words, $I_{T}$ as a function of two arguments
$(\mathbf{x},\phi)\in\mathfrak{X}\times C([0,T];\mathbb{R}^{d})$ still
possesses lower semicontinuity property.

######  Lemma 2.6.

For any $T>0$, $\mathbf{x}\in\mathfrak{X}$, $\phi\in C([0,T];\mathbb{R}^{d})$,
and any sequences $\\{\mathbf{x}_{n}\\}$, $\\{\phi_{n}\\}$, where
$\mathbf{x}_{n}\in\mathfrak{X}$, $\phi_{n}\in C([0,T];\mathbb{R}^{d})$, such
that $\mathbf{x}_{n}\xrightarrow[n\to\infty]{d_{\mathfrak{X}}}\mathbf{x}$ and
$\phi_{n}\xrightarrow[n\to\infty]{}\phi$ the following inequality holds:

$\liminf_{n\to\infty}I^{\mathbf{x}_{n}}_{T}(\phi_{n})\geq
I^{\mathbf{x}}_{T}(\phi).$

###### Proof.

Without loss of generality, let us assume that all $\phi_{n}\in
H_{1}^{\mathrm{P}{\mathbf{x}}}$ (those $\phi_{n}$ for which it is not true do
not influence the $\liminf$ since $I^{\mathbf{x}_{n}}_{T}(\phi_{n})=\infty$).
Then

$\displaystyle 4I_{T}^{\mathbf{x}_{n}}(\phi_{n})$
$\displaystyle=\int_{0}^{T}\big{|}\dot{\phi_{n}}(t)+\nabla V(\phi_{n}(t))$
(2.2)
$\displaystyle\quad\quad+\frac{\mathrm{T}{\mathbf{x}_{n}}}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$
$\displaystyle\quad\quad+\frac{1}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{0}^{t}\nabla
F(\phi_{n}(t)-\phi_{n}(u))\differential{u}\big{|}^{2}\differential{t}.$

Let us add and subtract two terms of the form
$A:=\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}}(\differential{u})$ and
$B:=\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\nabla
F(\phi_{n}(t)-\phi_{n}(u))\differential{u}$. Note that for any $c\in(0,1)$ and
for any $a,b\in\mathbb{R}^{d}$ the following inequality holds:
$|a+b|^{2}\geq(1-c)|a|^{2}+(1-\frac{1}{c})|b|^{2}$. Using two previous
statements, let us split (2.2) into two parts, one of which does not depend on
$\mathbf{x}_{n}$, but depends on $\mathbf{x}$ instead. We then have

$\displaystyle 4I_{T}^{\mathbf{x}_{n}}(\phi_{n})$
$\displaystyle\geq(1-c)4I_{T}^{\mathbf{x}}(\phi_{n})$
$\displaystyle\quad+\left(1-\frac{1}{c}\right)\int_{0}^{T}\Big{|}\frac{\mathrm{T}{\mathbf{x}_{n}}}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}_{n}}(\differential{u})-A$
$\displaystyle\quad+\frac{1}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{0}^{t}\nabla
F(\phi_{n}(t)-\phi_{n}(u))\differential{u}-B\Big{|}^{2}\differential{t}$
$\displaystyle=:(1-c)4I_{T}^{\mathbf{x}}(\phi_{n})+\left(1-\frac{1}{c}\right)\textbf{I}.$

Since $I_{T}^{\mathbf{x}}(\phi_{n})$ is a good rate function, by taking
$\liminf$ from both sides of the inequality above we get

$\liminf_{n\to\infty}I_{T}^{\mathbf{x}_{n}}(\phi_{n})\geq(1-c)I_{T}^{\mathbf{x}}(\phi)+\frac{1}{4}\left(1-\frac{1}{c}\right)\liminf_{n\to\infty}\textbf{I},$
(2.3)

for all $c\in(0,1)$. Therefore, in order to prove the lemma, it is enough to
show that $\liminf_{n\to\infty}\textbf{I}=0$. Then we could take the limit
$c\to 0$ from both sides of (2.3) and obtain the necessary result.

Let us consider I. We use inequality of the form $|a+b|^{2}\leq
2|a|^{2}+2|b|^{2}$ for I and get an upper bound of the form:

I $\displaystyle\leq
2\int_{0}^{T}\Big{|}\frac{\mathrm{T}{\mathbf{x}_{n}}}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$
$\displaystyle\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}}(\differential{u})\Big{|}^{2}\differential{t}$
$\displaystyle\quad+2\int_{0}^{T}\Big{|}\frac{1}{\mathrm{T}{\mathbf{x}_{n}}+t}\int_{0}^{t}\nabla
F(\phi_{n}(t)-\phi_{n}(u))\differential{u}$
$\displaystyle\qquad\qquad\qquad\quad\quad\quad\quad\quad\quad\quad\quad-\frac{1}{\mathrm{T}{\mathbf{x}}+t}\int_{0}^{t}\nabla
F(\phi_{n}(t)-\phi_{n}(u))\differential{u}\Big{|}^{2}\differential{t}$
$\displaystyle=:2\text{I}\text{I}_{1}+2\text{I}\text{I}_{2}.$

Let us apply dominated convergence theorem to both $\text{I}\text{I}_{1}$ and
$\text{I}\text{I}_{2}$. In order to do so, we prove that integrands are
uniformly bounded.

First, consider $\text{I}\text{I}_{1}$. Of course, fractions
$\frac{\mathrm{T}{\mathbf{x}_{n}}}{\mathrm{T}{\mathbf{x}_{n}}+t}$ and
$\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}$ are bounded by $1$.
As for $\int_{\mathbb{R}^{d}}|\nabla
F(\phi_{n}(t)-u)|\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$, we introduce
the following decomposition

$\displaystyle\int_{\mathbb{R}^{d}}$ $\displaystyle|\nabla
F(\phi_{n}(t)-u)|\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$ (2.4)
$\displaystyle=\int_{\mathbb{R}^{d}}|\nabla F(\phi_{n}(t)-u)-\nabla
F(\phi(t)-u)+\nabla F(\phi(t)-u)|\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$
$\displaystyle\leq\text{Lip}_{\nabla
F}|\phi_{n}(t)-\phi(t)|+\int_{\mathbb{R}^{d}}|\nabla
F(\phi(t)-u)|\mathrm{M}{\mathbf{x}_{n}}$ $\displaystyle\leq\text{Lip}_{\nabla
F}\|\phi_{n}-\phi\|_{\infty}+C_{\nabla
F}\left(\max_{t\in[0,T]}|\phi(t)|+\left(\int_{\mathbb{R}^{d}}|u|^{2}\mathrm{M}{\mathbf{x}_{n}}\right)^{1/2}\right),$

and since
$\int|u|^{2}\mathrm{M}{\mathbf{x}_{n}}\xrightarrow[n\to\infty]{}\int|u|^{2}\mathrm{M}{\mathbf{x}}$
(see [Vil09, Theorem 6.9]), the integral $\int_{\mathbb{R}^{d}}|\nabla
F(\phi_{n}(t)-u)|\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$ and, as a
consequence, $\int_{\mathbb{R}^{d}}|\nabla
F(\phi_{n}(t)-u)|\mathrm{M}{\mathbf{x}}(\differential{u})$ are uniformly (in
$t$) bounded.

For $\text{I}\text{I}_{2}$, we also use lipschitzness of $\nabla F$ when
needed, convergence of $\phi_{n}$ in uniform norm topology towards $\phi$, as
well as bounds of the form $|\phi(t)|\leq\max_{t\in[0,T]}|\phi(t)|$. That
easily gives us uniform (in $t$) boundedness of the integrand of the integral
$\text{I}\text{I}_{2}$.

Thus, to calculate $\lim_{n\to\infty}\textbf{I}$, we can use the dominated
convergence theorem and pass the limit inside both of the integrals
$\text{I}\text{I}_{1}$ and $\text{I}\text{I}_{2}$.

Since $\nabla F$ is continuous and since the uniform boundedness in $t$ and in
$u$ on finite time interval $[0,T]$ of $\nabla F(\phi_{n}(t)-\phi_{n}(u))$ can
be easily established, $\lim_{n\to\infty}\text{I}\text{I}_{2}$ is clearly
equal to $0$. Limits of components of $\text{I}\text{I}_{1}$ are also obvious,
except for $\int_{\mathbb{R}^{d}}|\nabla
F(\phi_{n}(t)-u)|\mathrm{M}{\mathbf{x}_{n}}(\differential{u})$ that needs some
attention. Let us show that expression $\int_{\mathbb{R}^{d}}|\nabla
F(\phi(t)-u)|\mathrm{M}{\mathbf{x}}(\differential{u})$ is actually its limit
with $n\to\infty$. Similarly to computations in (2.4), we get

$\displaystyle\Big{|}\int_{\mathbb{R}^{d}}\nabla
F(\phi_{n}(t)-u)\mathrm{M}{\mathbf{x}_{n}}(\differential{u})-\int_{\mathbb{R}^{d}}\nabla
F(\phi(t)-u)\mathrm{M}{\mathbf{x}}(\differential{u})\Big{|}$
$\displaystyle\leq\Big{|}\int_{\mathbb{R}^{d}}\nabla
F(\phi(t)-u)\mathrm{M}{\mathbf{x}_{n}}(\differential{u})-\int_{\mathbb{R}^{d}}\nabla
F(\phi(t)-u)\mathrm{M}{\mathbf{x}}(\differential{u})\Big{|}$
$\displaystyle\qquad+\text{Lip}_{\nabla F}\|\phi_{n}-\phi\|_{\infty}.$

As was pointed out before, convergence of measures in Wasserstein distance
gives convergence of respective integrals, since $\nabla F$ is Lipschitz
continuous [Vil09, Theorem 6.9].

This is the last remark needed to observe that
$\lim_{n\to\infty}\textbf{I}=0$. Thus, the lemma is proved by (2.3). ∎

As was pointed out before, lower semicontinuity guarantees that infima of a
function are achieved over compact sets. We summarise this property by the
following corollary.

######  Corollary 2.7.

For any $T>0$, any compact set $\Phi\subset C([0,T];\mathbb{R}^{d})$, and any
compact set $\mathcal{C}\subset\mathfrak{X}$ there exist $\phi^{*}\in\Phi$ and
$\mathbf{x}^{*}\in\mathcal{C}$ such that

$\inf_{\mathbf{x}\in\mathcal{C}}\inf_{\phi\in\Phi}I^{\mathbf{x}}_{T}(\phi)=I^{\mathbf{x}^{*}}_{T}(\phi^{*}).$

###  2.3 Compactness results

In this paper compact subsets of $\mathfrak{X}\times C([0,T];\mathbb{R}^{d})$
of particular form are considered. Let us present two results about
compactness of some sets that will be used later in the proof.

######  Lemma 2.8.

For any $T>0$ and any compact subsets $C_{1},C_{2}\subset\mathbb{R}^{d}$ the
following set

$\mathcal{C}:=\\{\mathbf{x}\in\mathfrak{X}:T\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{P}{\mathbf{x}}\in
C_{1},\;\mathrm{M}{\mathbf{x}}\in\mathcal{P}_{2}(C_{2})\\}.$

is a compact subset of $\mathfrak{X}$.

###### Proof.

Projections of $\mathcal{C}$ on first two axes are obviously compact subsets
in $[0,\infty]$ and $\mathbb{R}^{d}$. For compactness of the projection on the
third axis, note that by Prokhorov’s theorem this set is compact in weak
topology which is metrizable by Wasserstein-2 distance (see [Vil09, Theorem
6.9]). ∎

######  Lemma 2.9.

Let $\Phi^{\mathbf{x}}:=\\{\phi\in
C([0,T];\mathbb{R}^{d}):I^{\mathbf{x}}_{T}(\phi)\leq 1\\}$, where
$\mathbf{x}\in\mathcal{C}$, $T>0$, and $\mathcal{C}$ is a compact subset of
$\mathfrak{X}$. Then

$\Phi=\bigcup_{\mathbf{x}\in\mathcal{C}}\Phi^{\mathbf{x}}$

is a totally bounded subset of $C([0,T];\mathbb{R}^{d})$.

###### Proof.

By definition, the set $\Phi$ is totally bounded if for any $\varepsilon>0$
there exists a finite cover of $\Phi$ with open balls of radius $\varepsilon$.
The strategy of the proof is the following. We prove that for any
$\varepsilon>0$ and any $\mathbf{x}\in\mathcal{C}$ there exists
$\delta_{\mathbf{x}}>0$ small enough such that
$\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})}\Phi^{\mathbf{y}}$
is totally bounded. Since $\mathcal{C}$ is itself a compact subset of
$\mathfrak{X}$ and
$\\{\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})\\}_{\mathbf{x}\in\mathcal{C}}$
is its cover by open sets, we can extract finite subcover
$\\{\mathbb{B}_{\delta_{\mathbf{x}_{i}}}(\mathbf{x}_{i})\\}_{1}^{n}$ and thus
prove the lemma, since
$\Phi\subset\bigcup_{i=1,\dots,n}\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}_{i}}}(\mathbf{x}_{i})}\Phi^{\mathbf{y}}$
is covered by the finite amount of open balls of radius $\varepsilon$.

$\mathcal{C}$$\Phi$$\mathbf{x}$$\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})$$\Phi^{\mathbf{x}}$$\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})}\Phi^{\mathbf{y}}$

As a result, all we have to prove is that for any $\varepsilon>0$ and any
$\mathbf{x}\in\mathcal{C}$ there exists $\delta_{\mathbf{x}}>0$ small enough
such that
$\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})}\Phi^{\mathbf{y}}$
is a totally bounded subset of $C([0,T];\mathbb{R}^{d})$. First of all, for
any function
$\phi\in\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})}\Phi^{\mathbf{y}}$
if $\delta_{\mathbf{x}}$ is small enough, the following integral is bounded by
a positive constant $C_{1}$:

$\int_{0}^{T}|\dot{\phi}(s)|^{2}\differential{s}\leq C_{1}.$

Indeed, by adding and subtracting $\nabla
V(\phi(s))+\frac{\mathrm{T}{\mathbf{y}}}{\mathrm{T}{\mathbf{y}}+s}\nabla
F*\mathrm{M}{\mathbf{y}}(\phi(s))+\\\
\frac{1}{\mathrm{T}{\mathbf{y}}+s}\int_{0}^{s}\nabla
F(\phi(s)-\phi(z))\differential{z}$ inside the absolute value, we get:

$\displaystyle\int_{0}^{t}|\dot{\phi}(s)|^{2}\differential{s}$
$\displaystyle\leq 4\int_{0}^{t}\Big{|}\dot{\phi}(s)+\nabla
V(\phi(s))+\frac{\mathrm{T}{\mathbf{y}}}{\mathrm{T}{\mathbf{y}}+s}\nabla
F*\mathrm{M}{\mathbf{y}}(\phi(s))$
$\displaystyle\quad+\frac{1}{\mathrm{T}{\mathbf{y}}+s}\int_{0}^{s}\nabla
F(\phi(s)-\phi(z))\differential{z}\Big{|}^{2}\differential{s}+4\int_{0}^{t}|\nabla
V(\phi(s))|^{2}\differential{s}$
$\displaystyle\quad+4\int_{0}^{t}\left|\frac{\mathrm{T}{\mathbf{y}}}{\mathrm{T}{\mathbf{y}}+s}\nabla
F*\mathrm{M}{\mathbf{y}}(\phi(s))\right|^{2}\differential{s}$
$\displaystyle\quad+4\int_{0}^{t}\left|\frac{1}{\mathrm{T}{\mathbf{y}}+s}\int_{0}^{s}\nabla
F(\phi(s)-\phi(z))\differential{z}\right|^{2}\differential{s}.$

Since $I_{T}^{\mathbf{y}}(\phi)\leq 1$ for respective
$\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(\mathbf{x})$ and using
Lipschitz continuity of $\nabla V$ and $\nabla F$ (Assumption A-1.ii), we get

$\displaystyle\int_{0}^{t}|\dot{\phi}(s)|^{2}\differential{s}$
$\displaystyle\leq 4+4\text{Lip}^{2}_{\nabla
V}\\!\int_{0}^{t}|\phi(s)|^{2}\differential{s}+4\text{Lip}_{\nabla
F}^{2}\int_{0}^{t}\\!\\!\int_{\mathbb{R}^{d}}|\phi(s)-z|^{2}\mathrm{M}{\mathbf{y}}(\differential{z})\differential{s}$
$\displaystyle\quad+\frac{4t\text{Lip}^{2}_{\nabla
F}}{\mathrm{T}{\mathbf{y}}^{2}}\int_{0}^{t}\\!\\!\int_{0}^{s}|\phi(s)-\phi(z)|^{2}\differential{z}\differential{s}$
$\displaystyle\leq 4+4\left(\text{Lip}^{2}_{\nabla V}+2\text{Lip}^{2}_{\nabla
F}+\frac{4T^{2}\text{Lip}_{\nabla
F}^{2}}{\mathrm{T}{\mathbf{y}}^{2}}\right)\int_{0}^{t}|\phi(s)|^{2}\differential{s}$
$\displaystyle\quad+8\text{Lip}_{\nabla
F}^{2}T\int_{\mathbb{R}^{d}}|z|^{2}\mathrm{M}{\mathbf{y}}(\differential{z}).$

Since all $\mathbf{y}$ belong to a ball of radius $\delta_{\mathbf{x}}$ of
$\mathbf{x}$, there exists $C_{\mathbf{x}}>0$ such that the following
inequalities hold: $|\mathrm{T}{\mathbf{y}}-\mathrm{T}{\mathbf{x}}|\leq
C_{\mathbf{x}}$,
$\left|\int|z|^{2}\mathrm{M}{\mathbf{y}}(\differential{z})-\int|z|^{2}\mathrm{M}{\mathbf{x}}(\differential{z})\right|\leq
C_{\mathbf{x}}$ and $|\mathrm{P}{\mathbf{y}}-\mathrm{P}{\mathbf{x}}|\leq
C_{\mathbf{x}}$. In addition, by Hölder’s inequality, we can bound
$\int_{0}^{t}|\phi(s)|^{2}\differential{s}\leq
T\int_{0}^{t}\int_{0}^{s}|\dot{\phi}(z)|^{2}\differential{z}\differential{s}$.
As a result, there exist constants $C_{2},C_{3}>0$ such that

$\int_{0}^{t}|\dot{\phi}(s)|^{2}\differential{s}\leq
C_{2}+C_{3}\int_{0}^{t}\\!\\!\int_{0}^{s}|\dot{\phi}(z)|^{2}\differential{z}\differential{s}.$

So, by Grönwall’s inequality, we get

$\int_{0}^{T}|\dot{\phi}(s)|^{2}\differential{s}\leq C_{2}e^{C_{3}T}=:C_{1}.$
(2.5)

Finally, we use the Arzelà–Ascoli theorem to prove that the set
$\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(x)}\Phi^{\mathbf{y}}$
is totally bounded. We use the bound (2.5) and get the following inequalities
showing uniform equicontinuity and pointwise boundedness of functions that
belong to
$\bigcup_{\mathbf{y}\in\mathbb{B}_{\delta_{\mathbf{x}}}(x)}\Phi^{\mathbf{y}}$.

$|\phi(t_{2})-\phi(t_{1})|^{2}\leq(t_{2}-t_{1})\int_{t_{1}}^{t_{2}}\left|\dot{\phi}(s)\right|^{2}\differential{s}\leq
C_{1}(t_{2}-t_{1}).$

This completes the proof.

∎

##  3 Exit-time

First, we present the following assumptions on domain $G$ and geometry of the
potentials that drive the system inside the domain.

######  Assumptions A-2.

  1. i)

(domain $G$) $G\subset\mathbb{R}^{d}$ is an open, bounded, connected set such
that $\partial G=\partial\overline{G}$. The boundary $\partial G$ is a smooth
$(d-1)$-dimensional hypersurface.

  2. ii)

(point of attraction $a$) There exists $a\in G$ such that
$\\{X^{x_{0},0}_{t}\\}_{t\geq 0}\subset G$ and
$X^{x_{0},0}_{t}\xrightarrow[t\to\infty]{}a$.

  3. iii)

(stability of $\overline{G}$ under the effective potential) Let $\phi$ be
defined as $\phi_{t}^{x}=x-\int_{0}^{t}\nabla
V(\phi_{s})\differential{s}-\int_{0}^{t}\nabla F(\phi_{s}-a)\differential{s}$.
For any $x\in\overline{G}$, $\\{\phi^{x}_{t}\\}_{t>0}\subset G$ and
$\phi^{x}_{t}\xrightarrow[t\to\infty]{}a$. Moreover, $\nabla V(a)=\nabla
F(0)=0$.

  4. iv)

(strong attraction around $a$) There exist $\Delta_{\mu},\Delta_{x}>0$ small
enough and a constant $K>0$, such that for any
$\mu\in\mathbb{B}_{\Delta_{\mu}}(\delta_{a})$ and for any $x\in
B_{\Delta_{x}}(a)$, $\langle\nabla V(x)+\nabla F*\mu(x);x-a\rangle\geq
K|x-a|^{2}$.

Assumption A-2.i defines domain $G$ and its regularity properties. Assumption
A-2.ii defines point $a$ as a unique attractor of the deterministic process
$X^{0}$ starting at fixed $x_{0}\in G$. Since, as was proved in Theorem 2.2,
the process $X^{\sigma}$ satisfies LDP for any finite time interval $[0;T]$,
we expect it to be close to its deterministic limit and also to converge
towards $a$ with high probability. As a consequence, we also expect
$\mu_{t}^{\sigma}$ to converge towards $\delta_{a}$ in a finite time (this is
shown in Lemma 3.1). Assumption A-2.iii ensures that for
$\mathbf{x}\in\mathfrak{X}$ such that $\mathrm{M}{\mathbf{x}}$ is close to
$\delta_{a}$, starting from any $\mathrm{P}{\mathbf{x}}\in G$, deterministic
process $X^{\mathbf{x},0}$ will be still attracted to $a$. Assumption A-2.iv
elaborates on this idea and suggests that the attracting forces around $a$
will be stronger than interaction forces. These facts combined produce Lemma
3.2. These effects also suggest that $\mu^{\sigma}$ for small $\sigma$ should
stay close to $\delta_{a}$ with high probability at least until exit-time,
since inside $G$ there will always be a force that pushes $X^{\sigma}$ towards
$a$. That is shown in Lemma 3.5.

$x$$V$$a$$G$

$x$$F$$0$$R$$R$

Figure 1: Examples of possible $V$, $F$, and $G$ in dimension $d=1$. Despite
Assumptions A-1.ii and A-1.iii are not satisfied globally, they are satisfied
inside $G$ and for $R:=\sup\\{|z-y|:y,z\in G\\}$, which is enough for exit-
time result.

Now, let us remind the main theorem before proving it. Under the sets of
Assumptions A-1 and A-2 the following theorem holds:

######  Theorem 1.1.

Let the process $X^{\sigma}$ be the unique strong solution of the system (1.1)
with initial condition $x_{0}\in G$. Let
$\tau_{G}^{\sigma}:=\inf\\{t:X_{t}^{\sigma}\notin G\\}$ denote the first time
when $X^{\sigma}$ exits the domain $G$. Let $H:=\inf_{x\in\partial
G}\\{V(x)+F(x-a)-V(a)\\}$ be the height of the effective potential. Then, the
following two results hold:

  1. i)

Kramers’ law: for any $\delta>0$

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}\left(e^{\frac{2(H-\delta)}{\sigma^{2}}}<\tau_{G}^{\sigma}<e^{\frac{2(H+\delta)}{\sigma^{2}}}\right)=1;$
(3.1)

  2. ii)

Exit-location: for any closed set $N\subset\partial G$ such that $\inf_{z\in
N}\\{V(z)+F(z-a)-V(a)\\}>H$ the following limit holds:

$\lim_{\sigma\to 0}\mathbf{P}_{x_{0}}(X^{\sigma}_{\tau_{G}^{\sigma}}\in N)=0.$
(3.2)

###  3.1 Auxiliary results

####  3.1.1 Initial descent to the point of attraction

The first result that we will use for proving the main theorem describes
convergence of the solution to the system (1.1) with vanishing noise towards
point $a$ in a constant time. Moreover, we show that its occupation measure
also converges towards $\delta_{a}$ at some constant time with high
probability for small $\sigma$. As a matter of fact, as will be shown in Lemma
3.5, with $\sigma\to 0$, not only $\mu^{\sigma}$ converges towards
$\delta_{a}$ in some constant time, but it also stays in the neighbourhood of
$\delta_{a}$ at least until exit-time of $X^{\sigma}$ from the domain $G$. We
call this “effect of stabilization of the occupation measure”.

Consider the following lemma:

######  Lemma 3.1.

For any $\rho>0$ small enough there exists big enough time
$T_{\text{st}}^{\rho}>0$ such that

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}\left(X^{\sigma}_{T_{\text{st}}^{\rho}}\notin
B_{\rho}(a)\right)=0$ (3.3)

and

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}\left(\mu^{\sigma}_{T_{\text{st}}^{\rho}}\notin\mathbb{B}_{\rho}(\delta_{a})\right)=0.$
(3.4)

####  3.1.2 Convergence of deterministic process towards $a$

The following lemma claims that the deterministic process $X^{\mathbf{x},0}$
driven by (1.8) with some suitable initial conditions converges towards point
$a$ at most at some constant time (that depends on how close should the
process get to the point $a$). This result is shown for any starting point
$\mathrm{P}\mathbf{x}\in G$, any $\mathrm{M}\mathbf{x}$ close enough to
$\delta_{a}$
($\mathrm{M}\mathbf{x}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$), and
any $\mathrm{T}{\mathbf{x}}$ big enough ($\mathrm{T}{\mathbf{x}}\geq
T_{\text{st}}^{\rho}$). We also show that the occupation measure $\mu^{0}$
will not move far from $\delta_{a}$ in the process of convergence.

Let $X^{\mathbf{x},0}$ be the solution of (1.8) and define

$\mathcal{C}_{1}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in G\\}.$

######  Lemma 3.2.

For any $\rho>0$ small enough there exists $T^{\rho}_{1}>0$ such that for any
$\varepsilon,\vartheta>0$ small enough and any $\mathbf{x}\in\mathcal{C}_{1}$
the following estimations hold:

  1. i)

$X^{\mathbf{x},0}_{s}\in G$ and
$\mu_{s}^{\mathbf{x},0}\in\mathbb{B}_{(1+2\varepsilon)\rho}(\delta_{a})$ for
any $0\leq s<T_{1}^{\rho}$.

  2. ii)

$X^{\mathbf{x},0}_{s}\in B_{(1-\vartheta)\rho}(a)$ and
$\mu_{s}^{\mathbf{x},0}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$ for
any $s\geq T_{1}^{\rho}$.

####  3.1.3 Attraction of stochastic process towards $a$

The following lemma claims that there exists time $T_{1}^{\rho}>0$ such that
at most at that time with high probability the following event happens.
Starting at any point $\mathrm{P}{\mathbf{x}}$ inside $G$ with some suitable
initial conditions $\mathrm{M}{\mathbf{x}}$ and $\mathrm{T}{\mathbf{x}}$,
stochastic process $X^{\mathbf{x},\sigma}$ either comes sufficiently close to
point $a$ or leaves domain $G$ entirely.

Define the first time when the process either comes close enough to point $a$
or leaves the domain $G$ as the stopping time
$\tau_{0}^{\mathbf{x}}:=\inf\\{t\geq 0:X^{\mathbf{x},\sigma}_{t}\in
B_{\rho}(a)\cup\partial G\\}$. Let
$\mathcal{C}_{1}:=\\{\mathbf{x}\in\mathfrak{X}:T^{\rho}_{\text{st}}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in G\\}$. Then the following lemma holds.

######  Lemma 3.3.

For any $\rho,\varepsilon>0$ small enough there exist time $T_{1}^{\rho}>0$
big enough such that

$\lim_{\sigma\to
0}\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{0}>T_{1}^{\rho})=0.$
(3.5)

####  3.1.4 Behaviour in the annulus between $B_{\rho}(a)$ and $\partial G$

The following lemma claims that the probability that, with $t$ big enough, the
process $X^{\mathbf{x},\sigma}$ stays in between $B_{\rho}(a)$ and
$\mathbb{R}^{d}\setminus G$ without touching any of those sets, decays
exponentially with $\sigma\to 0$. Moreover, with $t\to+\infty$ the rate of
this decay tends to $-\infty$.

Introduce the following stopping time $\gamma:=\inf\\{t\geq
T_{\text{st}}^{\rho}:\mu_{t}^{\mathbf{x},\sigma}\notin\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\\}$,
where $\mu^{\sigma}$ is part of the solution to equation (1.1). In other
words, $\gamma$ is a random time that indicates when does the occupation
measure $\mu^{\sigma}$ move significantly far away from $\delta_{a}$ after
first descending to it by Lemma 3.1. We also remind that
$\tau_{0}^{\mathbf{x}}:=\inf\\{t\geq 0:\;X_{t}^{\sigma}\in
B_{\rho}(a)\cup\partial G\\}$, and
$\mathcal{C}_{1}=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a});\text{and
}\mathrm{P}{\mathbf{x}}\in G\\}$.

######  Lemma 3.4.

For any $\rho,\varepsilon>0$ small enough the following inequality holds

$\lim_{t\to\infty}\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(t<\tau_{0}<\gamma)=-\infty.$
(3.6)

####  3.1.5 Stabilization of the occupation measure

Given that $\gamma:=\inf\\{t\geq
T_{\text{st}}^{\rho}:\mu^{\sigma}_{t}\notin\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\\}$,
the following lemma establishes control of the occupation measure
$\mu^{\sigma}$ until exit-time $\tau_{G}^{\sigma}$. We remind that
$H=\inf_{z\in\partial G}\\{V(z)+F(z-a)-V(a)\\}$ and present the following
lemma.

######  Lemma 3.5.

For any $\rho,\varepsilon>0$ the following limit holds:

$\lim_{\sigma\to
0}\mathbf{P}_{x_{0}}\\!\\!\left(\gamma\leq\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}\right)=0.$

####  3.1.6 Exit before nearing $a$

We remind that $H=\inf_{z\in\partial G}\\{V(z)+F(z-a)-V(a)\\}$ and state that
the following asymptotic upper bound for the probability of exiting the domain
$G$ before approaching even smaller neighbourhood of $a$ holds. Let
$\mathcal{C}_{2}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\}$. Consider the
following lemma:

######  Lemma 3.6.

For any $\varepsilon>0$ small enough

$\lim_{\rho\to 0}\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{0}=\tau_{G}^{\sigma},\tau_{0}<\gamma)\leq-H.$

####  3.1.7 Control of dynamics for small time intervals

The following result claims that for any $\epsilon>0$ and $c>0$ there exists
small enough time $T(\epsilon,c)$ such that during this time the probability
that the diffusion (1.7) drifts farther away than $\epsilon$ from any starting
point $\mathrm{T}{\mathbf{x}}\geq T_{\text{st}}^{\rho}$,
$\mathrm{P}{\mathbf{x}}\in G$ and
$\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{\rho}(\delta_{a})$ decreases
exponentially with the given rate $c$.

######  Lemma 3.7.

For any $\epsilon>0$ and $c>0$ there exists time $T(\epsilon,c)>0$ such that
for any $\rho,\varepsilon>0$ small enough

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}\left(\sup_{t\in[0,T(\epsilon,c)]}|X^{\sigma}_{t}-\mathrm{P}{\mathbf{x}}|\geq\epsilon\right)\leq-c,$

where
$\mathcal{C}_{1}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in G\\}$.

####  3.1.8 Uniform lower bound for probability of exit from $G$

The following lemma provides a uniform lower bound for the probability of an
exit from domain $G$ starting from a position that is close to $a$, given that
the empirical measure of the process does not move far away from $\delta_{a}$.

######  Lemma 3.8.

For any $\eta>0$ there exists time $T_{0}>0$ such that for any $\rho>0$ small
enough there exists $T_{\text{st}}^{\rho}>0$ big enough such that

$\liminf_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\inf_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},\tau_{G}^{\sigma}<\gamma)>-(H+\eta),$

where
$\mathcal{C}_{2}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\}$.

###  3.2 Proof of the main theorem

####  3.2.1 Kramers’ law

Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most
of its time near $a$ with $\sigma$ small enough. In order to have more
information about this behaviour, we introduce the following stopping times

$\displaystyle\tau_{1}$ $\displaystyle:=\inf\\{t\geq
T_{\text{st}}^{\rho}:\;X_{t}^{\sigma}\in B_{\rho}(a)\cup\partial G\\},$ (3.7)
$\displaystyle\theta_{m}$
$\displaystyle:=\inf\\{t\geq\tau_{m}:X_{t}^{\sigma}\in
S_{(1+\varepsilon)\rho}(a)\\},$ $\displaystyle\tau_{m+1}$
$\displaystyle:=\inf\\{t\geq\theta_{m}:X_{t}^{\sigma}\in
B_{\rho}(a)\cup\partial G\\};$

for $m\in\mathbb{N}$ with the convention that $\theta_{m+1}=\infty$ if
$\tau_{m}=\tau_{G}^{\sigma}$. We hereby can consider separately the intervals
where $X^{\sigma}$ is close to point $a$ and those where it is not the case,
i.e. the intervals of the type $[\tau_{k};\theta_{k}]$ and
$[\theta_{k};\tau_{k+1}]$. Besides, parameter $\rho$ controls the desired
closeness of $X^{\sigma}$ to $a$. By $\tau_{0}^{\mathbf{x}}$ and
$\theta_{0}^{\mathbf{x}}$ we denote the corresponding stopping times
regardless of the number of exits that we had before, i.e.

$\displaystyle\tau_{0}^{\mathbf{x}}$ $\displaystyle:=\inf\\{t\geq
0:X^{\mathbf{x},\sigma}_{t}\in B_{\rho}(a)\cup\partial G\\};$
$\displaystyle\theta_{0}^{\mathbf{x}}$
$\displaystyle:=\inf\\{t:\;X_{t}^{\mathbf{x},\sigma}\in
S_{(1+\varepsilon)\rho}(a)\\}.$

$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{\rho}(a)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{(1+\varepsilon)\rho}(a)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}G$$\tau_{1}$$\theta_{1}$$\tau_{2}$$\theta_{2}$$\tau_{3}$$\theta_{3}$$t$
Figure 2: Illustration of the definitions of $\tau_{k}$ and $\theta_{k}$ for
$d=2$.

Lower bound. Take the time $T_{0}:=T(\varepsilon\rho,H)$, where
$T(\epsilon,c)$ is defined by Lemma 3.7, i.e., it is small enough time such
that the probability that the process $X^{\sigma}$ moves further away than
$\varepsilon\rho$ from its starting point within time $T_{0}$ decays
exponentially with rate at least $H$.

Consider the following event $\\{\tau_{G}^{\sigma}\leq kT_{0}\\}$. We can
split this event into two parts accordingly to whether
$\\{\gamma\leq\tau_{G}^{\sigma}\\}$ holds or not. Define
$\mathcal{C}_{2}=\\{\mathbf{x}\in\mathfrak{X}:T^{\rho}_{\text{st}}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\}$. Note that if
the event $\\{\tau_{G}^{\sigma}\leq kT_{0},\gamma>\tau_{G}^{\sigma}\\}$ takes
place, then either first $k$ of disjoint events
$\\{\tau_{i}=\tau_{G}^{\sigma},\gamma>\tau_{G}^{\sigma}\\}$ occur, or at least
one period $\theta_{i+1}-\tau_{i}$ was less than $T_{0}$, i.e.,

$\displaystyle\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq kT_{0})$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+\sum_{i=1}^{k}\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}=\tau_{i},\gamma>\tau_{G}^{\sigma})$
$\displaystyle\quad+\mathbf{P}_{x_{0}}(\min_{1\leq i\leq
k-1}\\{\theta_{i}-\tau_{i-1}\\}<T_{0},\gamma>\tau_{G}^{\sigma})$
$\displaystyle\quad+\mathbf{P}_{x_{0}}(\gamma\leq\tau_{G}^{\sigma}\leq
kT_{0}).$

That gives us:

$\displaystyle\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq kT_{0})$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+k\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}=\tau_{0},\gamma>\tau_{G}^{\sigma})$
$\displaystyle\quad+(k-1)\sup_{\begin{subarray}{c}T^{\rho}_{\text{st}}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\\\
\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a});\\\
\mathrm{P}{\mathbf{x}}\in
B_{\rho}(a)\end{subarray}}\\!\\!\\!\mathbf{P}_{\mathbf{x}}(\theta_{0}<T_{0},\gamma>\tau_{G}^{\sigma})$
$\displaystyle\quad+\mathbf{P}_{x_{0}}(\gamma\leq\tau_{G}^{\sigma}\leq
kT_{0}).$

By Lemmas 3.6 and 3.7, we can choose $\sigma$ small enough such that the
respective
$\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{G}=\tau_{0},\gamma>\tau_{G})$
and
$\sup_{\mathbf{x}}\mathbf{P}_{\mathbf{x}}(\theta_{0}<T_{0},\gamma>\tau_{G})$
are less or equal than $\exp{-\frac{2(H-\delta/2)}{\sigma^{2}}}$. We also use
Lemma 3.1 in order to deal with the probability of exit before stabilization
time $T_{\text{st}}^{\rho}$. Therefore, we establish the following upper
bound:

$\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq kT_{0})\leq
o_{\sigma}+ke^{\frac{-2(H-\delta/2)}{\sigma^{2}}}+(k-1)e^{\frac{-2(H-\delta/2)}{\sigma^{2}}}+\mathbf{P}_{x_{0}}(\gamma\leq\tau_{G}^{\sigma}\wedge
kT_{0}),$

where $o_{\sigma}$ is an infinitesimal w.r.t. $\sigma$. Finally, choose
$k=\left\lfloor\frac{1}{T_{0}}\exp{\frac{2(H-\delta)}{\sigma^{2}}}\right\rfloor$,
where $\delta>0$ is small enough. By Lemma 3.5 and simple calculations we
establish the following result:

$\lim_{\sigma\to 0}\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
e^{\frac{2(H-\delta)}{\sigma^{2}}})=0.$ (3.8)

Upper bound. Fix $\delta>0$, let $\eta=\delta/2$ and $\rho>0$, $T_{0}>0$ be
defined by Lemma 3.8. Let us define event
$A:=\left\\{\gamma>\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}\right\\}$.
Since $\\{\gamma>\tau_{G}^{\sigma}\\}\subset A$, the result of the Lemma 3.8
takes the form:

$\liminf_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\inf_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},A)>-(H+\eta).$

By Lemma 3.3, there exists $T_{1}^{\rho}>0$ such that

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{0}>T_{1}^{\rho},A)\leq\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{0}>T_{1}^{\rho})<0,$

where
$\mathcal{C}_{1}=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\;\mathrm{M}{\mathbf{x}}\in
B_{(1+\varepsilon)\rho}(\delta_{a})\text{ and }\mathrm{P}{\mathbf{x}}\in G\\}$
and $\tau^{\mathbf{x}}_{0}:=\inf\\{t\geq 0:\;X_{t}^{\mathbf{x},\sigma}\in
B_{\rho}(a)\cup\partial G\\}$. Let $T:=T_{0}+T_{1}^{\rho}$. Then, for $\sigma$
small enough we can provide the following bound.

$\displaystyle q$
$\displaystyle:=\inf_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T,A)\geq\inf_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{0}\leq
T_{1}^{\rho},A)\inf_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},A)$ (3.9) $\displaystyle\geq\exp{-\frac{2(H+\eta)}{\sigma^{2}}}.$

This lower bound will help us to calculate the probability of event
$\\{\tau_{G}^{\sigma}>kT,A\\}$ for any $k\in\mathbb{N}$ in the following way.
Consider the following inequalities:

$\displaystyle\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}>(k+1)T,A)$
$\displaystyle=(1-\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq(k+1)T,A|\tau_{G}^{\sigma}>kT,A))$
$\displaystyle\quad\times\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}>kT,A)$
$\displaystyle\leq(1-q)\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}>kT,A).$

Therefore, by induction in $k$, we get:

$\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}>kT,A)\leq(1-q)^{k}.$
(3.10)

Note that
$\tau_{G}^{\sigma}\mathbbo{1}_{A}\leq\Big{(}T+\sum_{k=1}^{\infty}T\mathbbo{1}_{\\{\tau_{G}^{\sigma}>kT)\\}}\Big{)}\mathbbo{1}_{A}$.
Therefore, by (3.10) and (3.9), we can express the following expectation as:

$\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbb{E}_{\mathbf{x}}\big{(}\tau_{G}^{\sigma}\mathbbo{1}_{A}\big{)}\leq
T\sum_{k=0}^{\infty}(1-q)^{k}=\frac{T}{q}\leq
T\exp{\frac{2(H+\eta)}{\sigma^{2}}}.$

By Markov’s inequality, we get:

$\displaystyle\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}\left(\tau_{G}^{\sigma}>\exp{\frac{2(H+\delta)}{\sigma^{2}}},A\right)$
$\displaystyle=\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}\left(\tau_{G}^{\sigma}\mathbbo{1}_{A}>\exp{\frac{2(H+\delta)}{\sigma^{2}}}\right)$
$\displaystyle\leq\exp{-\frac{2(H+\delta)}{\sigma^{2}}}\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbb{E}_{\mathbf{x}}\big{(}\tau_{G}^{\sigma}\mathbbo{1}_{A}\big{)}$
$\displaystyle\leq Te^{-\delta/\sigma^{2}}\xrightarrow[\sigma\to 0]{}0.$

We remark that by Lemmas 3.1, 3.5 and derivations above, we have

$\displaystyle\mathbf{P}_{x_{0}}\left(\tau_{G}^{\sigma}\geq\exp{\frac{2(H+\delta)}{\sigma^{2}}}\right)$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+\mathbf{P}_{x_{0}}\left(A^{c}\right)$
$\displaystyle\quad+\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}\left(\tau_{G}^{\sigma}\geq\exp{\frac{2(H+\delta)}{\sigma^{2}}},A\right)$
$\displaystyle\xrightarrow[\sigma\to 0]{}0.$

This concludes the proof.

####  3.2.2 Exit-location

Before proving the exit-location result, we provide some observations about
the geometry of the domain $G$ with respect to the effective potential. Let
$W_{a}(x)=V(x)+F(x-a)-V(a)$ be the effective potential. Note that the level
sets $L_{\alpha}:=\\{x\in\mathbb{R}^{d}:W_{a}(x)=\alpha\\}$ are smooth
hypersurfaces, since both $V$ and $F$ belong to
$C^{2}(\mathbb{R}^{d};\mathbb{R})$. Here, solely for simpler notation, we
denote by $L_{\alpha}$ only the parts of the level sets that intersect with
$\overline{G}$. By Assumption A-2.iii, for $\alpha=H$ there is in fact only
one “part” of the hypersurface intersecting the set $\overline{G}$. Namely, it
can be shown that $L_{H}$ is the boundary of the set
$L^{-}_{H}:=\\{x\in\mathbb{R}^{d}:W_{a}(x)\leq H\\}$ that itself belongs to
$\overline{G}$ and is a bounded connected set (otherwise the Assumption
A-2.iii is violated).

Note that there exists a constant $C_{H}>0$ such that for any point on the
surface $x\in L_{H}$: $\langle n(x);-\nabla W_{a}(x)\rangle\geq C_{H}$, where
$n(x)$ is the inner unit normal vector to the surface $L_{H}$ at the
respective point. Indeed, if it is not the case, then either by the continuity
argument there exists $x^{*}\in L_{H}$ such that $n(x^{*})=0$, which violets
Assumption A-2.iii, or all the vectors $-\nabla W_{a}(x)$ pin outside of the
set $L^{-}_{H}$, which also contradicts the assumption above. It means that if
we plug interior $\mathsf{Int}(L^{-}_{H})$ and the set $L_{H}$ in Assumptions
A-2 instead of $G$ and $\partial G$, all the conditions will be satisfied.
Then, by the proof of Kramers’ law above, for
$\tau^{\mathbf{x},\sigma}_{\mathsf{Int}(L^{-}_{H})}:=\inf\\{t\geq
0:X^{\mathbf{x},\sigma}_{t}\notin\mathsf{Int}(L^{-}_{H})\\}$, we have for any
$\delta>0$:

$\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(e^{\frac{2(H-\delta)}{\sigma^{2}}}<\tau^{\sigma}_{\mathsf{Int}(L^{-}_{H})}<e^{\frac{2(H+\delta)}{\sigma^{2}}}\right),$

where
$\mathcal{C}_{2}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\}$.

$a$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}L_{H}$$G$
Figure 3: Domain $G$ with level sets of $W_{a}=V+F(\cdot-a)-V(a)$.
$L_{H}=\\{x\in\mathbb{R}^{d}:W_{a}(x)=H\\}$ is the smallest level set that
touches the boundary $\partial G$

Since $W_{a}$ is continuous, there exists a small enlargement of $L_{H}$ such
that the property $\langle n(x);-\nabla W_{a}(x)\rangle\geq\text{Const}>0$ is
still satisfied. Namely, there exists $\bar{\eta}>0$ small enough such that
for any $\eta<\bar{\eta}$ there exists a constant $C_{H+\eta}>0$ such that
$\langle n(x);-\nabla W_{a}(x)\rangle\geq C_{H+\eta}$. By analogy with the
previous statement, it means that Assumptions A-2 are satisfied for $\eta>0$
small enough and thus, for any $\delta>0$,

$\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(e^{\frac{2(H+\eta-\delta)}{\sigma^{2}}}<\tau^{\sigma}_{\mathsf{Int}(L^{-}_{H+\eta})}<e^{\frac{2(H+\eta+\delta)}{\sigma^{2}}}\right).$

Now we are ready to prove the exit-location result. Take the set
$N\subset\partial G$ described in the theorem. By definition, $\inf_{z\in
N}\\{V(z)+F(z-a)-V(a)\\}>H$. That in particular means that we can choose
$\eta<\bar{\eta}$ such that $H+\eta$ is smaller then $\inf_{z\in
N}\\{V(z)+F(z-a)-V(a)\\}$. It means that $N\cap\mathsf{Cl}(L_{H+\eta}^{-})$ is
an empty set.

The fact that $X_{\tau_{G}^{\sigma}}^{\sigma}\in N$ means that, after
converging first towards $a$, we left the domain $L^{-}_{H+\eta}$ before time
$\tau_{G}^{\sigma}$, which is the exit-time from $G$. The following
inequalities show that it is unlikely. Consider:

$\displaystyle\mathbf{P}_{x_{0}}(X^{\sigma}_{\tau_{G}^{\sigma}}\in N)$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau^{\sigma}_{L^{-}_{H+\eta}}\leq\tau_{G}^{\sigma}).$

Taking $\delta=\eta/3$, it gives us:

$\displaystyle\mathbf{P}_{x_{0}}(X^{\sigma}_{\tau_{G}^{\sigma}}\in N)$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(\tau_{G}^{\sigma}\geq
e^{\frac{2(H+\delta)}{\sigma^{2}}}\right)$
$\displaystyle\quad+\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(\tau^{\sigma}_{L^{-}_{H+\eta}}\leq\tau_{G}^{\sigma}\leq
e^{\frac{2(H+\delta)}{\sigma^{2}}}\right)$
$\displaystyle\leq\mathbf{P}_{x_{0}}(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho})+\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(\tau_{G}^{\sigma}\geq
e^{\frac{2(H+\delta)}{\sigma^{2}}}\right)$
$\displaystyle\quad+\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\left(\tau^{\sigma}_{L^{-}_{H+\eta}}\leq
e^{\frac{2(H+\eta-\delta)}{\sigma^{2}}}\right)$
$\displaystyle\xrightarrow[\sigma\to 0]{}0,$

by Lemma 3.5 and Kramers’ law for $G$ and $L_{H+\eta}^{-}$ proved above.

###  3.3 Proofs of auxiliary lemmas

####  3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction

Let us fix $l:=\inf_{t>0,z\in\partial G}|X^{0}_{t}-z|$, the distance between
the deterministic path starting at $x_{0}$ ($X^{0}$) and the frontier of the
domain $G$. Let us also define as $T_{\text{st}}^{\rho}$ the first time when
$\mu^{0}_{t}\in\mathbb{B}_{\rho/2}(\delta_{a})$ as well as $X_{t}^{0}\in
B_{\rho/2}(a)$, where
$\mu^{0}_{t}:=\frac{1}{t}\int_{0}^{t}\delta_{X^{0}_{s}}\differential{s}$ is
the empirical measure of the deterministic process. This time obviously
exists, since, by Assumption A-2.ii, $X_{t}^{0}\xrightarrow[t\to\infty]{}a$.
Let us decrease $l=l_{\rho}$ if necessary (it will now depend on $\rho$) such
that $l_{\rho}<\frac{\rho}{2}$. Define $\Phi:=\\{\phi\in
C([0;T_{\text{st}}^{\rho}]):\|X^{0}-\phi\|_{\infty}\geq l_{\rho}\\}$. Then, by
Theorem 2.2,

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\mathbf{P}_{x_{0}}(X^{\sigma}_{T_{\text{st}}^{\rho}}\notin
B_{\rho}(a))\leq-\inf_{\phi\in\Phi}I_{T_{\text{st}}^{\rho}}^{x_{0}}(\phi)$

and

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\mathbf{P}_{x_{0}}(\mu^{\sigma}_{T_{\text{st}}^{\rho}}\notin\mathbb{B}_{\rho}(\delta_{a}))\leq-\inf_{\phi\in\Phi}I_{T_{\text{st}}^{\rho}}^{x_{0}}(\phi).$

Since $I_{T_{\text{st}}^{\rho}}^{x_{0}}$ has only one minimum which is given
by $X^{0}$, $I_{T_{\text{st}}^{\rho}}^{x_{0}}(X^{0})=0$, and as the distance
between $X^{0}$ and the set $\Phi$ is strictly positive, we conclude that
$\inf_{\phi\in\Phi}I_{T_{\text{st}}^{\rho}}^{x_{0}}(\phi)>0$. That completes
the proof.

####  3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards
$a$

We separate the dynamics of the system into the following three parts (see
Figure 4). First, we show that in time $\hat{T}_{1}>0$ for any
$\mathrm{P}{\mathbf{x}}$ the process converges inside $B_{\Delta_{x}}(a)$,
whereas its occupation measure $\mu$ does not move far from $\delta_{a}$ and
stays inside $\mathbb{B}_{(1+2\varepsilon)\rho}(\delta_{a})$ for some small
$\varepsilon>0$, that we can decrease by increasing $t_{0}$. Second, we show
that inside the set $B_{\Delta_{x}}(a)$, the attraction force of the potential
$V+F*\delta_{a}$ becomes so strong that, in some time $\hat{T}_{2}>0$, it
drags $X^{0}$ inside small ball $B_{(1-\vartheta)\rho}(a)$ for some
$0<\vartheta<1$, whereas its occupation measure, as before, does not move far
from $\delta_{a}$ and still stays inside
$\mathbb{B}_{(1+2\varepsilon)\rho}(\delta_{a})$. Third, we show that for small
enough $\rho>0$ and after hitting $B_{(1-\vartheta)\rho}(a)$, $X^{0}$ stays
inside this ball for a long time $\hat{T}_{3}$ that is enough to attract the
occupation measure $\mu$ back inside
$\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$. Thus, the required time is
represented through the following sum
$T_{1}^{\rho}:=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}$.

$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}G$$\mathrm{P}{\mathbf{x}}$$\hat{T}_{1}$$\hat{T}_{2}$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{\Delta_{x}}(a)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}B_{(1-\vartheta)\rho}(a)$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a$
Figure 4: Dynamics of $X^{\mathbf{x},0}$

Convergence inside $B_{\Delta_{x}}(a)$. By Assumption A-2.iii, for any
$x\in\overline{G}$, the flow $\phi^{x}$, generated by the effective potential
$V+F(\cdot-a)$, converges towards $a$. Since, by Assumption A-1.ii, both
$\nabla V$ and $\nabla F$ are Lipschitz continuous and, by Assumption A-2.i,
$\overline{G}$ is a compact set, there exists a uniform upper bound
$\hat{T}_{1}$ for the time in which $\phi^{x}$ converges inside
$B_{\Delta_{x}/2}(a)$.

Let us use this fact to establish the time of convergence of $X^{0}$ inside.
Define $\eta_{t}:=|X^{\mathbf{x},0}_{t}-\phi^{\mathrm{P}{\mathbf{x}}}_{t}|$.
Applying Assumption A-1.ii, we get

$\displaystyle\eta_{t}$ $\displaystyle\leq\text{Lip}_{\nabla
V}\int_{0}^{t}\eta_{s}\differential{s}+\text{Lip}_{\nabla
F}\int_{0}^{t}\int_{\mathbb{R}^{d}}|(X^{\mathbf{x},0}_{s}-z)-(\phi^{\mathrm{P}{\mathbf{x}}}_{s}-a)|\mu^{\mathbf{x},0}_{s}(\differential{z})\differential{s}$
$\displaystyle\leq\text{Lip}_{\nabla
V}\int_{0}^{t}\eta_{s}\differential{s}+\text{Lip}_{\nabla
F}\left(\int_{0}^{t}\eta_{s}\differential{s}+\int_{0}^{t}\mathbb{W}_{2}(\mu_{s}^{\mathbf{x},0};\delta_{a})\differential{s}\right).$

Then, by Grönwall’s inequality:

$\eta_{t}\leq\int_{0}^{t}\mathbb{W}_{2}(\mu_{s}^{\mathbf{x},0};\delta_{a})\differential{s}\cdot\exp{(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla F})t}.$ (3.11)

Since $\mathbf{x}\in\mathcal{C}_{1}$,
$\mathbb{W}_{2}(\mu_{0}^{\mathbf{x},0};\delta_{a})\leq(1+\varepsilon)\rho$. We
can show, by (3.11), that if we take $\rho$ to be small enough, then
$\mathbb{W}_{2}(\mu_{t}^{\mathbf{x},0};\delta_{a})\leq(1+1.5\varepsilon)\rho$
at least for $t\leq\hat{T}_{1}$. Indeed, if we introduce a generic time
$T^{\prime}:=\inf\\{t\geq
0:\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{t};\delta_{a})>(1+1.5\varepsilon)\rho\\}$,
we can express, using (3.11), for $t\leq T^{\prime}\wedge\hat{T}_{1}$:

$\eta_{t}\leq(1+1.5\varepsilon)\rho\hat{T}_{1}e^{(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla F})\hat{T}_{1}}.$ (3.12)

Since $\phi_{t}$ belongs to $\overline{G}$ for any $t\geq 0$, by inequality
above, there exists a constant $C>0$ such that $|X_{t}^{\mathbf{x},0}-a|\leq
C$ for any $t\leq T^{\prime}\wedge\hat{T}_{1}$. At the same time, by the
definition of $\mu^{\mathbf{x},0}$,

$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{t};\delta_{a})\leq\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+t}\mathbb{W}_{2}(\mathrm{M}{\mathbf{x}};\delta_{a})+\frac{t}{\mathrm{T}{\mathbf{x}}+t}\mathbb{W}_{2}\left(\frac{1}{t}\int_{0}^{t}\delta_{X_{s}^{\mathbf{x},0}}\differential{s};\delta_{a}\right).$

Therefore, for any $t\leq T^{\prime}\wedge\hat{T}_{1}$,

$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{t};\delta_{a})\leq(1+\varepsilon)\rho+\frac{\hat{T}_{1}}{T_{\text{st}}^{\rho}+\hat{T}_{1}}C.$

Note that, without loss of generality, we can increase $T_{\text{st}}^{\rho}$
if necessary such that
$\frac{\hat{T}_{1}}{T_{\text{st}}^{\rho}+\hat{T}_{1}}C\leq 0.5\varepsilon\rho$
for any $\rho>0$. Thus, for any $t\leq T^{\prime}\wedge\hat{T}_{1}$:

$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{t};\delta_{a})\leq(1+1.5\varepsilon)\rho.$

That includes the time $t=T^{\prime}\wedge\hat{T}_{1}$. At the same time, by
the definition of $T^{\prime}$,
$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{T^{\prime}};\delta_{a})>(1+1.5\varepsilon)\rho$
that means that $T^{\prime}\wedge\hat{T}_{1}=\hat{T}_{1}$.

Now we can conclude that equation (3.12) holds for any $t\leq\hat{T}_{1}$.
Thus, we can choose $\rho>0$ small enough such that $\eta_{t}<\Delta_{x}/2$
for any $t\leq\hat{T}_{1}$. That proves the uniform (in initial point
$\mathrm{P}{\mathbf{x}}\in G$) convergence inside $B_{\Delta_{x}}(a)$.

Convergence inside $B_{(1-\vartheta)\rho}(a)$. By Assumption A-2.iv, in the
set $B_{\Delta_{x}}(a)$ attraction forces towards the point $a$ prevail over
the interaction forces. First of all, we decrease $\rho$ to be small enough
such that $(1+2\varepsilon)\rho<\Delta_{\mu}$, where $\Delta_{\mu}$ is defined
in A-2.iv. As was shown before,
$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}};\delta_{a})\leq(1+1.5\varepsilon)\rho$
and thus
$\mu^{\mathbf{x},0}_{\hat{T}_{1}}\in\mathbb{B}_{\Delta_{\mu}}(\delta_{a})$. We
can show that moreover
$\mu^{\mathbf{x},0}_{\hat{T}_{1}+t}\in\mathbb{B}_{(1+2\varepsilon)\rho}(\delta_{a})$
long enough such that in $X^{\mathbf{x},0}$ converges inside
$B_{(1-\vartheta)\rho}(a)$.

Let $T^{\prime\prime}:=\inf\\{t\geq
0:\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}+t};\delta_{a})>(1+2\varepsilon)\rho,\text{
or }|X_{\hat{T}_{1}+t}^{\mathbf{x},0}-a|>\Delta_{x}\\}$ and consider
$\xi_{t}:=|X_{\hat{T}_{1}+t}^{\mathbf{x},0}-a|^{2}$. By Assumption A-2.iv, for
any $t\leq T^{\prime\prime}$, its derivative is bounded by:

$\dot{\xi}_{t}\leq-2K\xi_{t}.$

That guarantees exponentially fast convergence towards 0:

$\xi_{t}\leq\Delta_{x}e^{-2Kt}.$

Thus, all we need to prove is that $\hat{T}_{2}:=\inf\\{t\geq
0:\Delta_{x}\exp{-2Kt}\leq(1-\vartheta)\rho\\}$ is less or equal then
$T^{\prime\prime}$. We can prove that fact by contradiction. For any $t\leq
T^{\prime\prime}\wedge\hat{T}_{2}$:

$\displaystyle\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}+t};\delta_{a})$
$\displaystyle\leq\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}};\delta_{a})+\frac{t}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+t}\mathbb{W}_{2}\left(\frac{1}{t}\int_{0}^{t}\delta_{X_{\hat{T}_{1}+s}^{\mathbf{x},0}}\differential{s};\delta_{a}\right)$
$\displaystyle\leq(1+1.5\varepsilon)\rho+\frac{\hat{T}_{2}}{T_{\text{st}}^{\rho}+\hat{T}_{1}+\hat{T}_{2}}\Delta_{x}.$

For any $\rho>0$ we can choose, without loss of generality,
$T_{\text{st}}^{\rho}$ to be big enough such that
$\frac{\hat{T}_{2}}{T_{\text{st}}^{\rho}+\hat{T}_{1}+\hat{T}_{2}}\Delta_{x}\leq
0.5\varepsilon\rho$. That gives, for any $t\leq
T^{\prime\prime}\wedge\hat{T}_{2}$:

$\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}+t};\delta_{a})\leq(1+2\varepsilon)\rho.$

Thus, as before, $\hat{T}_{2}<T^{\prime\prime}$ or else we get a contradiction
between the definition of $T^{\prime\prime}$ and the inequality above along
with the fact that $\xi$ decreases for any $t\leq
T^{\prime\prime}\wedge\hat{T}_{2}$.

Return of the occupation measure back inside
$\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$. The last time period can be
found easily. Note that since $X^{\mathbf{x},0}$ belongs to
$B_{(1-\vartheta)\rho}(a)$ at time $\hat{T}_{1}+\hat{T}_{2}$, as well as
$\mu^{\mathbf{x},0}$ belongs to
$\mathbb{B}_{(1+2\varepsilon)\rho}(\delta_{a})$, then by Assumption A-2.iv,
$X^{\mathbf{x},0}_{t}$ will not leave $B_{(1-\vartheta)\rho}(a)$ for any
$t\geq\hat{T}_{1}+\hat{T}_{2}$. Using this fact and estimations that we had on
$\mu^{\mathbf{x},0}$ for time intervals $[0;\hat{T}_{1}]$ and
$[\hat{T}_{1};\hat{T}_{2}]$, we can provide the following bound:

$\displaystyle\mathbb{W}_{2}(\mu^{\mathbf{x},0}_{\hat{T}_{1}+\hat{T}_{2}+t};\delta_{a})$
$\displaystyle\leq\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}(1+\varepsilon)\rho+\frac{\hat{T}_{1}}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}(1+1.5\varepsilon)\rho$
$\displaystyle\quad+\frac{\hat{T}_{2}}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}(1+2\varepsilon)\rho+\frac{t}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}(1-\vartheta)\rho$
$\displaystyle=\rho+\frac{\mathrm{T}{\mathbf{x}}\varepsilon+1.5\hat{T}_{1}\varepsilon+2\hat{T}_{2}\varepsilon-t\vartheta}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}\rho$
$\displaystyle\leq\rho+\frac{(\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t)+\hat{T}_{2}-t\left(1+\frac{\vartheta}{\varepsilon}\right)}{\mathrm{T}{\mathbf{x}}+\hat{T}_{1}+\hat{T}_{2}+t}\varepsilon\rho$
$\displaystyle\leq(1+\varepsilon)\rho+\frac{\hat{T}_{2}+0.5\hat{T}_{1}-t\left(1+\frac{\vartheta}{\varepsilon}\right)}{T_{\text{st}}^{\rho}+\hat{T}_{1}+\hat{T}_{2}+t}\varepsilon\rho.$

We just need to choose $\hat{T}_{3}$ big enough such that
$\Big{(}\hat{T}_{2}+0.5\hat{T}_{1}-\hat{T}_{3}\left(1+\frac{\vartheta}{\varepsilon}\right)\Big{)}<0$.

We complete the prove by choosing
$T_{1}^{\rho}=\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}$.

####  3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards $a$

According to Lemma 3.2, there exists an upper bound for time of convergence of
deterministic process $X^{\mathbf{x},0}$ inside $B_{(1-\vartheta)\rho}$ for
some $0<\vartheta<1$ and for any $\mathrm{T}{\mathbf{x}}\geq
T_{\text{st}}^{\rho},\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\text{,
and }\mathrm{P}{\mathbf{x}}\in G$. Denote this time as $T_{1}^{\rho}$.

Let us define

$\Psi^{x_{0}}:=\\{\psi\in
C([0,T_{1}^{\rho}];\mathbb{R}^{d}),\psi(0)=x_{0},\psi(s)\in\mathsf{Cl}(G\setminus
B_{\rho}(a))\;\forall s\leq T_{1}^{\rho}\\}.$

The following inclusion of the events holds: $\\{\tau_{0}^{\mathbf{x}}\geq
T_{1}^{\rho}\\}\subset\\{X^{\mathbf{x},\sigma}\in\Psi^{\mathrm{P}{\mathbf{x}}}\\}$
(the definition of $\tau_{0}^{\mathbf{x}}$ was presented on page 3.2.1). Let
$\Psi=\bigcup_{x_{0}\in G}\Psi^{x_{0}}$. Note that $\Psi$ is a closed set and
the following enlargement of $\mathcal{C}_{1}$:
$\mathcal{C}_{1}^{\prime}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\mathrm{P}{\mathbf{x}}\in\overline{G},\text{
and},\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\\}$
is a compact set by Lemma 2.8. By Lemma 2.5 and the inclusion of the events,
we get

$\displaystyle\limsup_{\sigma\to 0}\frac{\sigma^{2}}{2}$
$\displaystyle\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(\tau_{0}>T_{1}^{\rho})$
$\displaystyle\leq\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Psi)\leq-\inf_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\inf_{\phi\in\Psi}I_{T_{1}^{\rho}}^{\mathbf{x}}(\phi).$

What is left to prove is that

$\inf_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\inf_{\phi\in\Psi}I_{T_{1}^{\rho}}^{\mathbf{x}}(\phi)>0.$
(3.13)

We can not apply Corollary 2.7 directly since $\Psi$ is not necessarily
compact. Let
$\Psi^{\mathbf{x}}_{1}:=\\{\psi\in\Psi:I_{T_{1}^{\rho}}^{\mathbf{x}}(\psi)\leq
1\\}$. Of course, if all $\Psi^{\mathbf{x}}_{1}$ are empty, then expression in
(3.13) is indeed strictly greater than 0. If it is not the case, then the sets
$\\{\Psi^{\mathbf{x}}_{1}\\}_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}$ satisfy
conditions of Lemma 2.9, and, therefore, union
$\Psi_{1}:=\bigcup_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\Psi^{\mathbf{x}}_{1}$
is precompact, and $\Psi_{2}=\overline{\Psi_{1}}$ is a compact set. Thus, by
Corollary 2.7, there exists some $\mathbf{x}^{*}\in\mathcal{C}_{1}^{\prime}$,
$\phi^{*}\in\Psi_{2}$ such that

$\inf_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\inf_{\phi\in\Psi_{2}}I_{T_{1}^{\rho}}^{\mathbf{x}}(\phi)=I_{T_{1}^{\rho}}^{\mathbf{x}^{*}}(\phi^{*}).$

By Lemma 3.2, for any $\mathbf{x}\in\mathcal{C}_{1}^{\prime}$, the
corresponding deterministic trajectories converge inside
$B_{(1-\vartheta)\rho}(a)$ in time $T_{1}^{\rho}$ for some $\vartheta$. Thus,
for any $\psi\in\Psi$: $\|\psi-X^{\mathbf{x},0}\|_{\infty}>\vartheta$ for any
$\mathbf{x}\in\mathcal{C}_{1}^{\prime}$. The same bound obviously holds for
$\Psi_{2}$ since it is a closure of some subset of $\Psi$. Since the
deterministic trajectories are the unique functions in
$C([0,T_{1}^{\rho}];\mathbb{R}^{d})$ for which
$I^{\mathbf{x}}_{T_{1}^{\rho}}(X^{\mathbf{x},0})=0$, we conclude that

$\inf_{\mathbf{x}\in\mathcal{C}_{1}^{\prime}}\inf_{\phi\in\Psi}I_{T_{1}^{\rho}}^{\mathbf{x}}(\phi)>I_{T_{1}^{\rho}}^{\mathbf{x}^{*}}(\phi^{*})>0.$

And this concludes the proof of the lemma.

####  3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between $B_{\rho}(a)$
and $\partial G$

The idea of the proof is to show that, since $T_{1}^{\rho}$ represents the
time in which the noiseless process converges inside $B_{\rho}(a)$, after each
time interval of length $T_{1}^{\rho}$ it should be more and more unlikely
that diffusion $X^{\sigma}$ did not follow the deterministic path even once.

We introduce the following set of functions whose path stay inside the annulus
$\mathsf{Cl}(G\setminus B_{\rho}(a))$:

$\displaystyle\Psi_{t}$ $\displaystyle:=\\{\psi\in
C([0,t];\mathbb{R}^{d}):\psi_{s}\in\mathsf{Cl}(G\setminus B_{\rho}(a))$
$\displaystyle\qquad\qquad\text{ and
}\frac{1}{s}\int_{0}^{s}\delta_{\psi_{u}}\differential{u}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\;\forall
s\in[0,t]\\}.$

Obviously, the following inclusion of events takes place
$\\{\tau_{0}^{\mathbf{x}}>t,\gamma>\tau_{0}^{\mathbf{x}}\\}\subset\\{X^{\mathbf{x},\sigma}\in\Psi_{t}\\}$.
By Lemma 2.5 for any $t<\infty$:

$\lim_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Psi_{t})\leq-\inf_{\psi\in\Psi_{t}}I_{t}(\psi),$

where
$I_{t}(\psi):=\inf_{\mathbf{x}\in\mathcal{C}_{1}}I_{t}^{\mathbf{x}}(\psi)$.
That means that it is enough to show that
$\lim_{t\to\infty}\inf_{\psi\in\Psi_{t}}I_{t}(\psi)=\infty$. Let us assume
that it is not true and there exists $M<\infty$ such that for any $n$ there
exists some function $\psi^{n}\in\Psi_{nT_{1}^{\rho}}$ such that
$I_{nT_{1}^{\rho}}(\psi^{n})\leq M$, where $T_{1}^{\rho}$ is defined by Lemma
3.3. Then, we can separate the path of $\psi^{n}$ into $n$ parts and establish
the following lower bound:

$M\geq
I_{nT_{1}^{\rho}}(\psi^{n})\geq\sum_{k=0}^{n-1}I^{\mathbf{x}_{k}^{n}}_{T_{1}^{\rho}}(\psi^{n,k})\geq
n\min_{k\leq n}I_{T_{1}^{\rho}}^{\mathbf{x}_{k}^{n}}(\psi^{n,k}),$ (3.14)

where $\mathbf{x}^{n}_{k}\in\mathfrak{X}$ are such that:

$\displaystyle\mathrm{T}{\mathbf{x}_{k}^{n}}$
$\displaystyle=\mathrm{T}{\mathbf{x}_{0}}+kT_{1}^{\rho},$
$\displaystyle\mathrm{P}{\mathbf{x}_{k}^{n}}$
$\displaystyle=\psi^{n,k-1}(kT_{1}^{\rho}),$
$\displaystyle\mathrm{M}{\mathbf{x}_{k}^{n}}$
$\displaystyle=\frac{\mathrm{T}{\mathbf{x}_{0}}}{\mathrm{T}{\mathbf{x}_{0}}+kT_{1}^{\rho}}\mathrm{M}{\mathbf{x}_{0}}+\frac{1}{\mathrm{T}{\mathbf{x}_{0}}+kT_{1}^{\rho}}\int_{0}^{kT_{1}^{\rho}}\delta_{\psi_{s}^{n}}\differential{s};$

and $\psi^{n,k}\in C([0,T_{1}^{\rho}];\mathbb{R}^{d})$ are the corresponding
peaces of $\psi^{n}$ of length $T_{1}^{\rho}$, i.e.,
$\psi^{n,k}(s)=\psi^{n}(kT_{1}^{\rho}+s)$ for $s\in[0,T_{1}^{\rho}]$.

For equation (3.14) to hold for any $n$ it is required that there is a
sequence of functions $\\{\psi^{n,k_{n}}\\}_{n=1}^{\infty}$ such that
$I^{\mathbf{x}^{n}_{k_{n}}}_{T_{1}^{\rho}}(\psi^{n,k_{n}})\to 0$. Thus, after
some $n_{0}$ all of these functions belong to the set

$\Phi=\bigcup_{\mathbf{x}\in\mathcal{C}_{1}}\Phi^{\mathbf{x}},$

where $\Phi^{\mathbf{x}}:=\\{f\in C([0,T_{1}^{\rho}]):I^{\mathbf{x}}(f)\leq
1\\}$. It also means that $\inf_{\phi\in\Phi}I_{T_{1}^{\rho}}(\phi)=0$. We use
the same logic as in the proof of Lemma 3.3 and Lemma 3.9. If all the sets
$\\{\Phi_{\mathbf{x}}\\}_{\mathbf{x}\in\mathcal{C}}$ are empty, then we get
the contradiction with the fact that infimum of rate functions over this set
should be equal to 0. In the other case, we can apply Lemma 2.9 to conclude
that $\Phi$ is precompact set, which makes $\Phi_{1}:=\overline{\Phi}$ to be a
compact set. By Corollary 2.7, there exist $x^{*}\in\mathcal{C}_{1}$ and
$\phi^{*}\in\Phi_{1}$ such that

$\inf_{\phi\in\Phi_{1}}\inf_{\mathbf{x}\in\mathcal{C}_{1}}I^{\mathbf{x}}_{T_{1}^{\rho}}(\phi)=I^{\mathbf{x}^{*}}_{T_{1}^{\rho}}(\phi^{*}).$

By Lemma 3.1, all deterministic processes $X^{\mathbf{x},0}$ converge inside
$B_{(1-\vartheta)\rho}(a)$ for some $\vartheta>0$ in time $T_{1}^{\rho}$.
Thus, for any $\phi\in\Psi_{T_{1}^{\rho}}$ (and as a consequence in
$\Phi_{1}$): $\|\phi-X^{\mathbf{x},0}\|_{\infty}\geq\vartheta$. That means
that $I^{\mathbf{x}^{*}}_{T_{1}^{\rho}}(\phi^{*})$ is strictly positive, which
contradicts existence of $M<\infty$ in (3.14) and, thus, proves the lemma.

####  3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure

The proof of this result requires an additional Lemma 3.9. This lemma claims
that for any choice of constant $T_{2}>0$ we can make $\sigma$ small enough
such that with high probability the stochastic process $X^{\mathbf{x},\sigma}$
spends inside small neighbourhood of the point of attraction $a$ at least time
$T_{2}$.

We remind that
$\theta^{\mathbf{x}}_{0}:=\inf\\{t:\;X_{t}^{\mathbf{x},\sigma}\in
S_{(1+\varepsilon)\rho}(a)\\}$ is the time that the process starting in some
point inside $B_{\rho}(a)$ spends inside the ball
$B_{(1+\varepsilon)\rho}(a)$. Let
$\mathcal{C}_{3}:=\\{\mathbf{x}\in\mathfrak{X}:T^{\rho}_{\text{st}}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a});\mathrm{P}{\mathbf{x}}\in
B_{\rho}(a)\\}$. The following lemma holds.

######  Lemma 3.9.

For any $\rho,\varepsilon>0$ small enough and for any constant $T_{2}>0$ the
following limit holds

$\lim_{\sigma\to
0}\sup_{\mathbf{x}\in\mathcal{C}_{3}}\mathbf{P}_{\mathbf{x}}(\theta_{0}<T_{2})=0.$
(3.15)

###### Proof.

The proof of the following lemma follow the same logic as the one of Lemma
3.3.

By Lemma 3.2, the deterministic process $X^{\mathbf{x},0}$ starting with any
initial conditions $\mathbf{x}\in\mathfrak{X}$:
$T^{\rho}_{\text{st}}\leq\mathrm{T}{\mathbf{x}}\leq\infty$,
$\mathrm{P}{\mathbf{x}}\in B_{\rho}(a)$, and
$\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$ stays
inside $B_{\rho}(a)$ for any $t\geq 0$.

Fix $T_{2}$ and define $\Psi^{y}:=\\{\psi\in
C([0,T_{2}];\mathbb{R}^{d}):\psi(0)\in
y,|\psi(s)-a|\geq(1+\varepsilon)\rho,\text{ for some }s\leq T_{2}\\}$.
Obviously, the following inclusion holds: $\\{\theta^{\mathbf{x}}_{0}\geq
T_{2}\\}\subset\\{X^{\mathbf{x},\sigma}\in\Psi^{\mathrm{P}{\mathbf{x}}}\\}$.
Let $\Psi:=\mathsf{Cl}(\bigcup_{y\in B_{\rho}(a)}\Psi^{y})$. Note that, by
Lemma 2.8, the set $\mathcal{C}_{3}$ is compact. Then, by Lemma 2.5, we get

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\mathbf{x}\in\mathcal{C}_{3}}\mathbf{P}_{\mathbf{x}}(\theta_{0}>T_{2})\leq-\inf_{\mathbf{x}\in\mathcal{C}_{3}}\inf_{\phi\in\Psi}I_{T_{2}}^{\mathbf{x}}(\phi).$

We have to show now that

$\inf_{\mathbf{x}\in\mathcal{C}_{3}}\inf_{\phi\in\Psi}I_{T_{2}}^{\mathbf{x}}(\phi)>0.$

In order to apply Corollary 2.7, consider
$\Psi_{1}^{\mathbf{x}}:=\\{\psi\in\Psi:I_{T_{2}}^{\mathbf{x}}(\psi)\leq 1\\}$.
If all the sets are empty then the inequality above is trivially satisfied. If
not, the family of sets
$\\{\Psi_{1}^{\mathbf{x}}\\}_{\mathbf{x}\in\mathcal{C}_{3}}$ satisfies
conditions of Lemma 2.9. Thus, the union
$\bigcup_{\mathbf{x}\in\mathcal{C}_{3}}\Psi_{1}^{\mathbf{x}}$ is precompact
and $\Psi_{2}:=\overline{\Psi_{1}}$ is a compact set. Thus, by Corollary 2.7,
there exist some $\mathbf{x}^{*}\in\mathcal{C}_{3}$ and $\phi^{*}\in\Psi_{2}$
such that

$\inf_{\mathbf{x}\in\mathcal{C}_{3}}\inf_{\phi\in\Psi}I_{T_{2}}^{\mathbf{x}}(\phi)=I_{T_{2}}^{\mathbf{x}^{*}}(\phi^{*}).$

As was pointed out before, $X^{\mathbf{x},0}$ starting with initial conditions
$\mathbf{x}\in\mathcal{C}_{3}$ stays inside $B_{\rho}(a)$ for any $0\leq t\leq
T_{2}$. By definition of $\Psi$, any function $X^{\mathbf{x},0}$ has positive
distance of at least $\varepsilon\rho$ to $\Psi$. Same holds for $\Psi_{2}$.
Since $X^{\mathbf{x},0}$ are unique minimizers of $I_{T_{2}}^{\mathbf{x}}$ for
corresponding initial conditions $\mathbf{x}$ and
$I_{T_{2}}^{\mathbf{x}}(X^{\mathbf{x},0})=0$,

$I_{T_{2}}^{\mathbf{x}^{*}}(\phi^{*})>0.$

This completes the proof. ∎

Now we are ready to prove Lemma 3.5 itself.

###### Proof of Lemma 3.5.

Dynamics of $\mu_{t}^{\sigma}$. We separate the path of $\mu_{t}^{\sigma}$
into two parts: when it belongs to the ball of radius $\rho$ and when big
excursions occur. Let
$\mathbf{x}\in\mathcal{C}_{4}:=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty,\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{\rho}(\delta_{a}),\text{
and }\mathrm{P}{\mathbf{x}}\in B_{\rho}(a)\\}$. Let
$\mu_{0}=\mathrm{M}{\mathbf{x}}$ and $t_{0}=\mathrm{T}{\mathbf{x}}$. Consider
the following equations for $t\in[\theta_{m};\tau_{m+1}]$ (the definitions of
$\theta_{k}$ and $\tau_{k}$ were presented on page 3.2.1).

$\displaystyle\mathbb{W}_{2}(\mu_{t}^{\sigma};\delta_{a})$
$\displaystyle\leq\frac{t_{0}}{t_{0}+t}\mathbb{W}_{2}(\mu_{0};\delta_{a})+\frac{t}{t_{0}+t}\mathbb{W}_{2}\left(\frac{1}{t}\int_{0}^{t}\delta_{X_{s}}\differential{s};\delta_{a}\right)$
$\displaystyle=\frac{t_{0}}{t_{0}+t}\mathbb{W}_{2}(\mu_{0};\delta_{a})$
$\displaystyle\quad+\sum_{k=1}^{m-1}\frac{\tau_{k+1}-\theta_{k}}{t_{0}+t}\mathbb{W}_{2}\left(\frac{1}{\tau_{k+1}-\theta_{k}}\int_{\theta_{k}}^{\tau_{k+1}}\delta_{X_{s}}\differential{s};\delta_{a}\right)$
$\displaystyle\quad+\sum_{k=1}^{m}\frac{\theta_{k}-\tau_{k}}{t_{0}+t}\mathbb{W}_{2}\left(\frac{1}{\theta_{k}-\tau_{k}}\int_{\tau_{k}}^{\theta_{k}}\delta_{X_{s}}\differential{s}\\!;\delta_{a}\right)$
$\displaystyle\quad+\frac{t-\theta_{m}}{t_{0}+t}\mathbb{W}_{2}\left(\frac{1}{t-\theta_{m}}\int_{\theta_{m}}^{t}\delta_{X_{s}}\differential{s}\\!;\delta_{a}\right).$

That gives us:

$\mathbb{W}_{2}(\mu_{t}^{\sigma};\delta_{a})\leq\frac{t_{0}}{t_{0}+t}\rho+\left(\sum_{k=1}^{m-1}\frac{\tau_{k+1}-\theta_{k}}{t_{0}+t}+\frac{t-\theta_{m}}{t_{0}+t}\right)\\!\\!R+\left(\sum_{k=1}^{m}\frac{\theta_{k}-\tau_{k}}{t_{0}+t}\right)\\!(1+\varepsilon)\rho,$

where $R:=\sup_{z\in\partial G}|z-a|$ is the maximal distance between the
point $a$ and the frontier of $G$. Let us define
$\mathcal{T}_{\text{out}}(m):=\sum_{k=1}^{m}(\tau_{k+1}-\theta_{k})$ that is
the total amount of time spent significantly outside of the ball $B_{\rho}(a)$
after $m$ full exits, as well as
$\mathcal{T}_{\text{in}}(m):=\sum_{k=1}^{m}(\theta_{k}-\tau_{k})$. Taking into
account this notation for $t\in[\theta_{m};\tau_{m+1}]$ we express

$\displaystyle\mathbb{W}_{2}(\mu_{t}^{\sigma};\delta_{a})$
$\displaystyle\leq\frac{t_{0}}{t_{0}+t}\rho+\frac{\mathcal{T}_{\text{out}}(m)}{t_{0}+t}R+\frac{t-\mathcal{T}_{\text{out}}(m-1)}{t_{0}+t}(1+\varepsilon)\rho$
(3.16)
$\displaystyle\leq\rho+\frac{\mathcal{T}_{\text{out}}(m)}{t_{0}+t}R\leq\rho+\frac{\mathcal{T}_{\text{out}}(m)}{\mathcal{T}_{\text{in}}(m)}R.$

Here we emphasize that $\mu_{t}^{\sigma}$, $\tau_{k}$, $\theta_{k}$ and, as a
consequence, $\mathcal{T}_{\text{out}}(m)$ depend also on elementary event
$\omega$. Now, in order to prove that
$\gamma>\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}$ with high
probability, it suffices to show that $\mathcal{T}_{\text{out}}(m)$
constitutes such a small part of $t$ that it will not be able to move the
occupation measure $\mu_{t}^{\sigma}$ significantly away from $\delta_{a}$.

Control of $\mathcal{T}_{\text{in}}$. By Lemma 3.4 if we choose $\sigma$ small
enough, there exists $T_{1}^{\rho}>0$ big enough such that

$\sup_{\mathbf{x}\in\mathcal{C}_{4}}\mathbf{P}(\tau_{0}>T_{1}^{\rho},\gamma>\tau_{0})<\exp{-\frac{8(H+1)}{\sigma^{2}}}.$
(3.17)

Define $T_{2}^{\rho}>0$ as a number that is big enough such that:

$\frac{T_{1}^{\rho}}{T_{2}^{\rho}}R<\varepsilon\rho.$ (3.18)

We recall that Lemma 3.9 establishes the following asymptotic behaviour for
small $\sigma$:

$\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\theta_{0}<T_{2}^{\rho})=o_{\sigma},$
(3.19)

where $o_{\sigma}$ is an infinitesimal w.r.t. $\sigma$. Thus, we can get the
following lower bound for time spent inside the ball
$B_{(1+\varepsilon)\rho}(a)$. For any $\mathbf{x}\in\mathcal{C}_{4}$,

$\displaystyle\mathbf{P}_{\mathbf{x}}$
$\displaystyle(\mathcal{T}_{\text{in}}(m)<\frac{m}{2}T_{2}^{\rho},\gamma\geq\tau_{m+1})$
(3.20) $\displaystyle\leq\mathbf{P}_{\mathbf{x}}(\\#\\{i\leq
m:\theta_{i}-\tau_{i}<T_{2}^{\rho}(\sigma)\\}>\frac{m}{2},\gamma\geq\tau_{m+1})$
$\displaystyle\leq\sum_{k=\lceil\frac{m}{2}\rceil}^{m}\sum_{(i_{1},\dots,i_{k})}\\!\\!\mathbf{P}_{\mathbf{x}}\\!\left(\bigcap_{j}\\{\theta_{i_{j}}-\tau_{i_{j}}<T_{2}^{\rho}(\sigma)\\},\gamma\geq\tau_{m+1}\right),$

where the summation is taken with respect to all tuples of the form
$(i_{1},\leavevmode\nobreak\ \dots,i_{k})$ for $i_{1}<\dots<i_{k}$. Note that
we can roughly estimate the number of such tuples to be less than $2^{m}$. We
also emphasize that each respective probability can be expressed as

$\displaystyle\mathbf{P}_{\mathbf{x}}$
$\displaystyle\left(\bigcap_{j=1}^{k}\\{\theta_{i_{j}}-\tau_{i_{j}}<T_{2}^{\rho}\\},\gamma\geq\tau_{m+1}\right)$
$\displaystyle=\prod_{j=1}^{k}\mathbf{P}_{\mathbf{x}}\left(\theta_{i_{j}}-\tau_{i_{j}}<T_{2}^{\rho},\gamma\geq\tau_{m+1}\Big{|}\bigcap_{z\leq
j}\\{\theta_{i_{m}}-\tau_{i_{z}}<T_{2}^{\rho}\\}\right)$
$\displaystyle\leq\left(\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\big{(}\theta_{0}<T_{2}^{\rho}\big{)}\right)^{k}.$

By equation (3.19) we conclude the following bound for probability (3.20). For
any $m\in\mathbb{N}$ and $\mathbf{x}\in\mathcal{C}_{4}$,

$\displaystyle\mathbf{P}_{\mathbf{x}}$
$\displaystyle(\mathcal{T}_{\text{in}}(m)<\frac{m}{2}T_{2}^{\rho},\gamma\geq\tau_{m+1})$
(3.21)
$\displaystyle\leq\sum_{k=\lceil\frac{m}{2}\rceil}^{m}2^{m}\left(\sup_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}\big{(}\theta_{0}<T_{2}^{\rho}\big{)}\right)^{k}\leq
2^{m}\frac{o_{\sigma}^{\lceil\frac{m}{2}\rceil}(o_{\sigma}^{\lceil\frac{m}{2}\rceil}-1)}{o_{\sigma}-1}\leq\frac{o_{\sigma}^{\lceil\frac{m}{2}\rceil}}{1-o_{\sigma}}.$

Control of the number of excursion. Inequality (3.21) also provides us with
the following upper bound for $\tau_{m}$. Let
$m^{*}:=\lceil\frac{2}{T_{2}^{\rho}}\rceil\exp{\frac{2(H+1)}{\sigma^{2}}}$ and
consider

$\displaystyle\mathbf{P}_{\mathbf{x}}\left(\tau_{m^{*}}<\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\leq\mathbf{P}_{\mathbf{x}}\left(\mathcal{T}_{\text{in}}(m^{*})<\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
(3.22)
$\displaystyle\leq\frac{o_{\sigma}^{\lceil\frac{m^{*}}{2}\rceil}}{1-o_{\sigma}}.$

Note that $m^{*}$ tends to infinity with $\sigma\to 0$. It means that the
probability in (3.22) tends to zero, which provides us with an asymptotic
upper bound for $\tau_{m}$ knowing that $m$ is large enough. Inequality (3.22)
in particular means that the probability that there were more than $m^{*}$
(that grows exponentially fast with $\sigma$) excursions before time
$\exp{\frac{2(H+1)}{\sigma^{2}}}$ is very small.

Control of $\mathcal{T}_{\text{out}}$. For time spent significantly outside of
$B_{\rho}(a)$ – $\mathcal{T}_{\text{out}}(m)$ – we provide the following
simple bound. For any $\mathbf{x}\in\mathcal{C}_{1}$,

$\displaystyle\mathbf{P}_{\mathbf{x}}\big{(}\mathcal{T}_{\text{out}}(m)>mT_{1}^{\rho},\gamma\geq\tau_{m}\big{)}$
$\displaystyle\leq\mathbf{P}_{\mathbf{x}}\big{(}\\{\exists i\leq
m:\tau_{i}-\theta_{i}>T_{1}^{\rho}\\},\gamma\geq\tau_{m}\big{)}$ (3.23)
$\displaystyle\leq
m\sup_{\mathbf{x}\in\mathcal{C}_{1}}\mathbf{P}_{\mathbf{x}}\big{(}\tau_{0}>T_{1}^{\rho}\big{)}$
$\displaystyle\leq m\exp{-\frac{8(H+1)}{\sigma^{2}}},$

where we obtain the last inequality by (3.17).

Control of $\gamma$. Note that, by definition of $\gamma$,
$\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho$ a.s. Consider
the following inequalities. For any $\mathbf{x}\in\mathcal{C}_{1}$,

$\displaystyle\mathbf{P}_{\mathbf{x}}$
$\displaystyle\left(\gamma\leq\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle=\mathbf{P}_{\mathbf{x}}\left(\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\gamma\leq\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\leq\sum_{m=1}^{\infty}\mathbf{P}_{\mathbf{x}}\left(\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\tau_{m}\leq\gamma\leq\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\leq\sum_{m=1}^{m^{*}}\mathbf{P}_{\mathbf{x}}\left(\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\tau_{m}\leq\gamma\leq\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\quad\quad+\sum_{m=m^{*}}^{\infty}\mathbf{P}_{\mathbf{x}}\left(\tau_{m}\leq\exp{\frac{2(H+1)}{\sigma^{2}}}\right)=:A+B.$

Let us first deal with the term $B$. By (3.22), we have

$\displaystyle
B\leq\sum_{m=m^{*}}^{\infty}\frac{o_{\sigma}^{\lceil\frac{m}{2}\rceil}}{o_{\sigma}-1}\leq\frac{o_{\sigma}^{\lceil\frac{m^{*}}{2}\rceil}}{(o_{\sigma}-1)^{2}}\xrightarrow[\sigma\to
0]{}0.$

Consider now $A$. Separate each probability inside the sum the following way

$\displaystyle\mathbf{P}_{\mathbf{x}}\left(\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\tau_{m}\leq\gamma\leq\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\leq\sum_{k=m}^{m^{*}}\mathbf{P}_{\mathbf{x}}\Big{(}\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\mathcal{T}_{\text{out}}(k)<kT_{1}^{\rho},$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\mathcal{T}_{\text{in}}(k)>\frac{k}{2}T_{2}^{\rho},\gamma\in[\theta_{k};\tau_{k+1}]\Big{)}$
$\displaystyle\quad\quad+\sum_{k=m}^{m^{*}}\mathbf{P}_{\mathbf{x}}(\mathcal{T}_{\text{out}}(k)\geq
kT_{1}^{\rho},\gamma\geq\tau_{k})+\sum_{k=m}^{m^{*}}\mathbf{P}_{\mathbf{x}}(\mathcal{T}_{\text{in}}(k)\leq\frac{k}{2}T_{2}^{\rho},\gamma\geq\tau_{k}).$

Note that all the probabilities of the following form
$\mathbf{P}_{\mathbf{x}}(\mathbb{W}_{2}(\mu_{\gamma};\delta_{a})\geq(1+\varepsilon)\rho,\mathcal{T}_{\text{out}}(k)<kT_{1}^{\rho},\mathcal{T}_{\text{in}}(k)>\frac{k}{2}T_{2}^{\rho},\gamma\in[\theta_{k};\tau_{k+1}])$
are equal to $0$ by (3.16) and . For the latter two sums we can use (3.21) and
(3.23). Finally, we can conclude that the twice summation of infinitesimals
above gives us

$\displaystyle
A\leq\frac{o_{\sigma}}{(1-o_{\sigma})^{3}}+(1+m^{*})^{3}\exp{-\frac{8(H+1)}{\sigma^{2}}}.$

We remind that, by definition,
$m^{*}=\lceil\frac{2}{T_{2}^{\rho}}\rceil\exp{\frac{2(H+1)}{\sigma^{2}}}$. It
means that $A$ is also bounded by some function that tends to 0. Combining the
bounds above for $A$ and $B$ as well as Lemma 3.1 (decrease $\sigma$ if
necessary such that $\exp{\frac{2(H+1)}{\sigma^{2}}}>T_{\text{st}}^{\rho}$),
we finally get

$\displaystyle\mathbf{P}_{x_{0}}\left(\gamma\leq\tau_{G}^{\sigma}\right)$
$\displaystyle\leq\mathbf{P}_{x_{0}}\left(\tau_{G}^{\sigma}\leq
T_{\text{st}}^{\rho}\right)+\sup_{\mathbf{x}\in\mathcal{C}_{4}}\mathbf{P}_{\mathbf{x}}\left(\gamma\leq\tau_{G}^{\sigma}\wedge\exp{\frac{2(H+1)}{\sigma^{2}}}\right)$
$\displaystyle\xrightarrow[\sigma\to 0]{}0.$

∎

####  3.3.6 Proof of Lemma 3.6: Exit before nearing $a$

First, we show that it is possible to establish a lower bound for

$\inf_{z\in\partial G}Q^{\rho}(x,z):=\frac{1}{2}\inf_{z\in\partial
G}\inf_{t>0}\inf_{\phi}I_{t}^{\mathbf{x}}(\phi),$

where $\mathbf{x}\in\mathcal{C}_{1}$ and
$\mathcal{C}_{1}=\\{\mathbf{x}\in\mathfrak{X}:T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty;\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a});\mathrm{P}{\mathbf{x}}\in
G\\}$, and infimum is taken over functions $\phi\in C([0,t];\mathbb{R}^{d})$
such that $\phi_{t}=z$ and
$\mu^{\phi}_{s}:=\frac{1}{s}\int_{0}^{s}\delta_{\phi_{z}}\differential{z}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$
for any $s\leq t$. Moreover, we show that this lower bound approaches $H$ with
$\rho\to 0$. Indeed, for any $f\in C([0,T];\mathbb{R}^{d})$, such that
$\mu^{f}_{t}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$ for any $t\leq
T$, and $f_{0}\in G$, $f_{T}\in\partial G$:

$\displaystyle\frac{1}{4}\int_{0}^{T}$ $\displaystyle|\dot{f}_{t}+\nabla
V(f_{t})+\nabla F*\delta_{a}(f_{t})|^{2}\differential{t}$
$\displaystyle=\frac{1}{4}\int_{0}^{T}|\dot{f}_{t}+\nabla V(f_{t})+\nabla
F*\mu^{f}_{t}(f_{t})|^{2}\differential{t}$
$\displaystyle\quad+\frac{1}{4}\int_{0}^{T}|\nabla F*\mu^{f}_{t}(f_{t})-\nabla
F*\delta_{a}(f_{t})|^{2}\differential{t}$
$\displaystyle\quad+\frac{1}{2}\int_{0}^{T}\big{\langle}\dot{f}_{t}+\nabla
V(f_{t})+\nabla F*\delta_{a}(f_{t});\nabla F*\mu^{f}_{t}(f_{t})-\nabla
F*\delta_{a}(f_{t})\big{\rangle}\differential{t}$
$\displaystyle\leq\frac{1}{4}\int_{0}^{T}|\dot{f}_{t}+\nabla V(f_{t})+\nabla
F*\mu^{f}_{t}(f_{t})|^{2}\differential{t}$
$\displaystyle\quad+\frac{\text{Lip}_{\nabla
F}}{4}T(1+2\varepsilon)^{2}\rho^{2}$
$\displaystyle\quad+\frac{1}{2}\sqrt{\int_{0}^{T}|\dot{f}_{t}+\nabla
V(f_{t})+\nabla
F*\mu^{f}_{t}(f_{t})|^{2}\differential{t}}\cdot\text{Lip}_{\nabla
F}(1+\varepsilon)\rho.$

We can first take the infimum of both sides of the inequality above over all
$f$ such that $\mu^{f}_{t}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$ and
$f_{0}\in G$, $f_{T}\in\partial G$, then over all $T>0$. Note that $H$ will be
less than the infimum in the left-hand side. Thus, we conclude that

$\displaystyle H$ $\displaystyle\leq\inf_{z\in\partial
G}Q^{\rho}(a,z)+\text{Lip}_{\nabla
F}(1+\varepsilon)\rho\sqrt{\inf_{z\in\partial G}Q^{\rho}(a,z)}$
$\displaystyle\leq Q^{\rho}(a,y)+\inf_{z\in\partial
G}Q^{\rho}(y,z)+\text{Lip}_{\nabla
F}(1+\varepsilon)\rho\sqrt{\inf_{z\in\partial G}Q^{\rho}(a,z)},$

or, equivalently,

$\inf_{z\in\partial G}Q^{\rho}(y,z)\geq H-C(\rho),$ (3.24)

where $C(\rho)>0$ is some function of $\rho$, that tends to $0$, whenever
$\rho\to 0$.

By Lemma 3.4, there exists time $T>0$ big enough such that,

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup\mathbf{P}_{\mathbf{x}}(\tau_{0}>T,\gamma>\tau_{0})<-H,$

where supremum is taken over $\mathrm{P}{\mathbf{x}}\in
S_{(1+\varepsilon)\rho}(a)$. Consider the following set:

$\Phi:=\\{\phi\in C([0,T]):\exists t\in[0,T]:\phi_{t}\in\partial G,\text{ and
}\mu^{\phi}_{s}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\;\forall
s\in[0,T]\\}.$

By the definition of $Q^{\rho}(x,z)$ and proved above facts,

$\inf_{\begin{subarray}{c}T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty\\\
\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\\
\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\\\
\end{subarray}}\inf_{\phi\in\Phi}I_{T}^{\mathbf{x}}(\phi)\geq\inf_{\begin{subarray}{c}y\in
S_{(1+\varepsilon)\rho}(a)\\\ z\in\partial G\end{subarray}}Q^{\rho}(y,z)\geq
H-C(\rho).$

By Lemma 2.5,

$\displaystyle\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\begin{subarray}{c}T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty\\\
\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\\
\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\end{subarray}}\sup_{\phi\in\Phi}\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Phi)$
$\displaystyle\quad\quad\leq-\inf_{\begin{subarray}{c}T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty\\\
\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\\
\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\end{subarray}}\inf_{\phi\in\Phi}I_{T}^{\mathbf{x}}(\phi)\leq-H+C(\rho).$

And, since
$\mathbf{P}_{\mathbf{x}}(\tau_{0}=\tau_{G}^{\sigma},\tau_{0}<\gamma)\leq\mathbf{P}_{\mathbf{x}}(\tau_{0}<T,\tau_{0}<\gamma)+\mathbf{P}_{\mathbf{x}}(X^{\sigma}\in\Phi)$,
we get:

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\sup_{\begin{subarray}{c}T_{\text{st}}^{\rho}\leq\mathrm{T}{\mathbf{x}}\leq\infty\\\
\mathrm{P}{\mathbf{x}}\in S_{(1+\varepsilon)\rho}(a)\\\
\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})\end{subarray}}\sup_{\phi\in\Phi}\mathbf{P}_{\mathbf{x}}(\tau_{0}=\tau_{G}^{\sigma},\tau_{0}<\gamma)\leq-H+C(\rho),$

which proves the lemma by taking $\rho\to 0$.

####  3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals

Let us investigate the dynamics of $|X^{\sigma}-\mathrm{P}{\mathbf{x}}|$. We
plug in the expression describing $X^{\sigma}_{t}$ from (1.7), use
Lipschitzness of $\nabla V$ and $\nabla F$ (Assumption A-1.ii) as well as
$\nabla V(a)=0$, $\nabla F(0)=0$ (Assumption A-2.iii), and express for any
$t$:

$\displaystyle|X_{t}^{\sigma}-\mathrm{P}{\mathbf{x}}|$
$\displaystyle\leq\sigma|W_{t}|+\text{Lip}_{\nabla
V}\int_{0}^{t}|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|\differential{s}+\text{Lip}_{\nabla
V}|\mathrm{P}{\mathbf{x}}-a|t$
$\displaystyle\quad+\int_{0}^{t}\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\text{Lip}_{\nabla
F}\Big{(}|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|+\int|z-\mathrm{P}{\mathbf{x}}|\mathrm{M}{\mathbf{x}}(\differential{z})\Big{)}\differential{s}$
$\displaystyle\quad+\int_{0}^{t}\frac{1}{\mathrm{T}{\mathbf{x}}+s}\text{Lip}_{\nabla
F}\int_{0}^{s}\Big{(}|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|+|X_{u}^{\sigma}-\mathrm{P}{\mathbf{x}}|\Big{)}\differential{u}\differential{s}$

Use the Jensen’s inequality, integrate the second part of the last integral
over $s$, and get:

$\displaystyle|X_{t}^{\sigma}-\mathrm{P}{\mathbf{x}}|$
$\displaystyle\leq\sigma|W_{t}|+\text{Lip}_{\nabla
V}\int_{0}^{t}|X_{s}-\mathrm{P}{\mathbf{x}}|\differential{s}+\text{Lip}_{\nabla
V}|\mathrm{P}{\mathbf{x}}-a|t$ $\displaystyle\quad+\text{Lip}_{\nabla
F}\int_{0}^{t}|X^{\sigma}_{s}-\mathrm{P}{\mathbf{x}}|\differential{s}+\text{Lip}_{\nabla
F}t\Big{(}\mathbb{W}_{2}(\mathrm{M}{\mathbf{x}};\delta_{a})+|\mathrm{P}{\mathbf{x}}-a|\Big{)}$
$\displaystyle\quad+\text{Lip}_{\nabla
F}\Big{(}\int_{0}^{t}|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|\differential{s}+\int_{0}^{t}\log(\frac{\mathrm{T}{\mathbf{x}}+t}{\mathrm{T}{\mathbf{x}}+s})|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|\differential{s}\Big{)}$

Finally, introducing $R:=\sup_{z\in\partial G}|z-a|$ and using the fact that
$\mathbf{x}$ is assumed, in the lemma, to belong to $\mathcal{C}_{4}$, we get
the bound:

$\displaystyle|X_{t}^{\sigma}-\mathrm{P}{\mathbf{x}}|$
$\displaystyle\leq\sigma|W_{t}|+\Big{(}(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla F})R+\text{Lip}_{\nabla F}(1+\varepsilon)\rho\Big{)}t$
$\displaystyle\quad+\left(\text{Lip}_{\nabla V}+2\text{Lip}_{\nabla
F}+\log(\frac{\mathrm{T}{\mathbf{x}}+t}{\mathrm{T}{\mathbf{x}}})\right)\int_{0}^{t}|X_{s}^{\sigma}-\mathrm{P}{\mathbf{x}}|\differential{s}.$

We apply Grönwall’s inequality and get:

$\displaystyle|X_{t}^{\sigma}-$
$\displaystyle\mathrm{P}{\mathbf{x}}|\leq\Big{(}\sigma|W_{t}|+\Big{(}(\text{Lip}_{\nabla
V}+\text{Lip}_{\nabla F})R+\text{Lip}_{\nabla F}\rho\Big{)}t\Big{)}$ (3.25)
$\displaystyle\quad\times\exp{(\text{Lip}_{\nabla V}+2\text{Lip}_{\nabla
F})t+\mathrm{T}{\mathbf{x}}\frac{\mathrm{T}{\mathbf{x}}+t}{\mathrm{T}{\mathbf{x}}}\log(\frac{\mathrm{T}{\mathbf{x}}+t}{\mathrm{T}{\mathbf{x}}})-t}.$

Thus, it follows that, for the absolute value of the Brownian motion itself,

$\sigma|W_{t}|\geq-C_{1}t+C_{2}(t)|X_{t}^{\sigma}-\mathrm{P}{\mathbf{x}}|,$

where, for simplicity of further derivations, we introduced a positive
constant $C_{1}:=(\text{Lip}_{\nabla V}+\text{Lip}_{\nabla
F})R+\text{Lip}_{\nabla F}\rho$ and a function $C_{2}(t)$ that is one over the
exponent that appears in the equation (3.25). For our purposes, it is not the
form of $C_{2}$ itself that is important, but rather its following properties:
$C_{2}(t)\geq 0$ for any $t\geq 0$ and $C_{2}(t)\xrightarrow[t\to 0]{}1$.

The infimum of $C_{2}$ over any time interval $[0,T]$, for $T<1$, is either
equal to $C_{2}(T)$, if $\text{Lip}_{\nabla V}+2\text{Lip}_{\nabla F}\geq 1$,
or can be bounded by $C_{2}(T)\exp\\{\text{Lip}_{\nabla V}+2\text{Lip}_{\nabla
F}\\\ -1\\}$ otherwise. This observation can be expressed in the following
form:

$\inf_{t\in[0,T]}C_{2}(t)\geq\min\\{C_{2}(T);C_{2}(T)\exp{\text{Lip}_{\nabla
V}+2\text{Lip}_{\nabla F}-1}\\}.$

Moreover, it is easy to check that $\lim_{T\to 0}C_{2}^{2}(T)/T=\infty$.

Taking into account these remarks, we can now use the Schilder theorem [DZ10,
Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence,
for some fixed $\epsilon$ and $0<T<1$:

$\displaystyle\mathbf{P}_{\mathbf{x}}(\sup_{t\in[0,T]}|X^{\sigma}_{t}-\mathrm{P}{\mathbf{x}}|\geq\epsilon)$
$\displaystyle\leq\mathbf{P}_{\mathbf{x}}(\sigma\sup_{t\in[0,T]}|W_{t}|\geq-
C_{1}t+C_{2}(t)\epsilon)$ $\displaystyle\leq
4d\exp{-\frac{(-C_{1}t+\inf_{t\in[0,T]}C_{2}(t)\epsilon)^{2}}{4dT}\cdot\frac{2}{\sigma^{2}}},$

where $d$ is the dimension of the space.

Note that for any $\epsilon>0$ the following limit holds:

$\frac{\big{(}-C_{1}T+\inf_{t\in[0,T]}C_{2}(t)\epsilon\big{)}^{2}}{T}\xrightarrow[T\to
0]{}\infty.$

Thus, for any given in advance $c$ and $\epsilon$, we can choose
$T=T(\epsilon,c)$ to be small enough, such that the rate
$-\big{(}-C_{1}T(\epsilon,c)+\inf_{t\in[0,T(\epsilon,c)]}C_{2}(t)\epsilon\big{)}^{2}/\big{(}4dT(\epsilon,c)\big{)}$
will be less or equal to $-c$, which completes the proof.

####  3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit
from $G$

In order to prove the lemma, we need to find for some $\delta>0$ small enough,
for some $T_{0}>0$ and for any $\mathbf{x}\in\mathcal{C}_{2}$ a function
$\psi^{\mathbf{x}}\in C([0,T_{0}];\mathbb{R}^{d})$ such that
$\psi^{\mathbf{x}}(0)=\mathrm{P}{\mathbf{x}}$, $\inf_{z\in
G}|\psi^{\mathbf{x}}(s)-z|>\delta$ for some $s\leq T_{0}$ and
$I^{\mathbf{x}}_{T_{0}}(\psi^{\mathbf{x}})<H+\eta$. Moreover, this function
should posses an empirical measure that is close to $\delta_{a}$ at any point
of time, i.e.
$\mathbb{W}_{2}\left(\frac{1}{t}\int_{0}^{t}\delta_{\psi^{\mathbf{x}}_{s}}\differential{s};\delta_{a}\right)\leq\rho$.
Given such a function, by a simple inclusion of events, we can get the
following bound for probability of leaving domain $G$ before time $T_{0}$:

$\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},\gamma>\tau_{G}^{\sigma})\geq\mathbf{P}_{\mathbf{x}}(X^{\mathbf{x},\sigma}\in\Psi),$

where $\Psi:=\bigcup_{\mathbf{x}\in\mathcal{C}_{2}}\Psi^{\mathbf{x}}$ and
$\Psi^{\mathbf{x}}:=\\{\phi\in
C([0;T_{0}];\mathbb{R}^{d}):\|\phi-\psi^{\mathbf{x}}\|_{\infty}<\delta\\}$. Of
course, we should take $\delta$ small enough such that any $\phi\in\Psi$ has
an occupation measure that satisfies
$\mathbb{W}_{2}\left(\frac{1}{t}\int_{0}^{t}\delta_{\phi_{s}}\differential{s};\delta_{a}\right)$.
Note that $\Psi$ is an open subset of $C([0,T_{0}];\mathbb{R}^{d})$, as a
union of open sets. Therefore, we can use Theorem 2.2 and get

$\limsup_{\sigma\to
0}\frac{\sigma^{2}}{2}\log\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},\gamma>\tau_{G}^{\sigma})\geq-\inf_{\phi\in\Phi}I^{\mathbf{x}}_{T_{0}}(\phi)\geq-I^{\mathbf{x}}_{T_{0}}(\psi^{\mathbf{x}})>-(H+\eta).$

The same lower bound obviously holds for infimum
$\inf_{\mathbf{x}\in\mathcal{C}_{2}}\mathbf{P}_{\mathbf{x}}(\tau_{G}^{\sigma}\leq
T_{0},\gamma>\tau_{G}^{\sigma})$, which is what the lemma claims. Thus, we
only need to find a $\psi^{\mathbf{x}}$ with the properties given above.

Construction of $\psi^{\mathbf{x}}$. The function $\psi^{\mathbf{x}}$ will be
represented as a consecutive gluing of four functions (see Figure 5):
$\psi_{0}^{\mathbf{x}}\in C([0;1];\mathbb{R}^{d})$, $\psi_{a}\in
C([0;T_{a}];\mathbb{R}^{d})$, $\psi_{1}\in C([0;T_{1}];\mathbb{R}^{d})$, and
$\psi_{2}\in C([0;1];\mathbb{R}^{d})$, for some $T_{a},T_{1}$ that will be
defined below and will determine $T_{0}$ as $T_{0}:=T_{a}+T_{1}+2$.

$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}G$$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}a$$\mathrm{P}{\mathbf{x}}$$\psi_{0}^{\mathbf{x}}$$\psi_{1}$$\psi_{2}$$x_{\text{out}}$
Figure 5: Construction of $\psi^{\mathbf{x}}$

1\. The first function $\psi_{0}^{\mathbf{x}}$ will bring us from point
$\mathrm{P}{\mathbf{x}}$ to point $a$ in at most time $1$ with small rate
function. Define

$\psi^{\mathbf{x}}_{0}(s)=\begin{cases}\mathrm{P}{\mathbf{x}}+\frac{a-\mathrm{P}{\mathbf{x}}}{|a-\mathrm{P}{\mathbf{x}}|}s,\text{
for }0\leq s\leq|a-\mathrm{P}{\mathbf{x}}|,\\\ a,\text{ for
}|a-\mathrm{P}{\mathbf{x}}|\leq s\leq 1.\end{cases}$

We can establish the following bound for its rate function. Let
$\mathbf{x}_{0}=(\infty,\delta_{a},\mathrm{P}{\mathbf{x}})$ and
$\mu^{\psi_{0}^{\mathbf{x}}}_{s}:=\frac{\mathrm{T}{\mathbf{x}}}{\mathrm{T}{\mathbf{x}}+s}\mathrm{M}{\mathbf{x}}+\frac{1}{\mathrm{T}{\mathbf{x}}+s}\int_{0}^{s}\delta_{\psi_{0}^{\mathbf{x}}(s)}\differential{s}$.
We want to separate the interaction . Then:

$\displaystyle I_{1}^{\mathbf{x}}(\psi_{0}^{\mathbf{x}})$ $\displaystyle\leq
2I_{1}^{\mathbf{x}_{0}}(\psi_{0}^{\mathbf{x}})+\frac{1}{2}\int_{0}^{1}\Big{|}\nabla
F*\mu^{\psi_{0}^{\mathbf{x}}}_{s}(\psi_{0}^{\mathbf{x}}(s))-\nabla
F(\psi_{0}^{\mathbf{x}}(s)-a)\Big{|}^{2}\differential{s}.$

Note that, since $\psi^{\mathbf{x}}_{0}$ never leaves $B_{\rho}(a)$ and since
$\mathrm{M}{\mathbf{x}}\in\mathbb{B}_{(1+\varepsilon)\rho}(\delta_{a})$, then
$\mathbb{W}_{2}(\mu^{\psi_{0}^{\mathbf{x}}}_{s};\delta_{a})\leq(1+\varepsilon)\rho$
for any $0\leq s\leq 1$. We use Lipschitz continuity of $\nabla F$ (Assumption
A-1.ii) and get

$\displaystyle I_{1}^{\mathbf{x}}(\psi_{0}^{\mathbf{x}})$ $\displaystyle\leq