# Large deviations of reflected weakly interacting particle systems

Ping Chen  School of Mathematical Sciences, University of Science and
Technology of China, Hefei, 230026, China<EMAIL_ADDRESS>,  Rong
Wei  School of Mathematics and Statistics, Anhui Normal University, Wuhu
241002, China<EMAIL_ADDRESS>and  Tusheng Zhang  Department of
Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United
Kingdom<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a large deviation principle for the empirical measures
of a system of weakly interacting diffusion with reflection. We adopt the weak
convergence approach. To make this approach work, we show that the sequence of
empirical measures of the controlled reflected system will converge to the
weak solution of an associated reflected McKean–Vlasov equation.

###### Key words and phrases:

large deviation; interacting particle systems; McKean–Vlasov equation;
Stochastic differential equation with reflection; weak convergence; sub-
martingale problem

###### 2020 Mathematics Subject Classification:

Primary 60F10, 60K35; secondary 60B10, 60H10, 34K50, 93E20.

##  1\. Introduction

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{D}$
an open domain in $\mathbb{R}^{d}$. The purpose of this paper is to establish
a large deviation principle for the empirical measures of weakly interacting
particle systems, which are given by reflected stochastic differential
equations (RSDEs) of the following form: for fixed $N\in\mathbb{N}$ and finite
interval $[0,T]$, $i\in\\{1,\ldots,N\\}$,

(1.1) $\displaystyle\left\\{\begin{aligned}
\mathrm{d}X^{i,N}(t)&=b(t,X^{i,N}(t),\mu^{N}(t))\mathrm{d}t+\sigma(t,X^{i,N}(t),\mu^{N}(t))\mathrm{d}W^{i}(t)\\\
&\ \ \ \ -\mathrm{d}K^{i,N}(t),\\\
|K^{i,N}|(t)&=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(X^{i,N}(s))\mathrm{d}|K^{i,N}|(s),\\\
K^{i,N}(t)&=\int^{t}_{0}\mathbf{n}(X^{i,N}(s))\mathrm{d}|K^{i,N}|(s),\\\
X^{i,N}(0)&=x^{i,N},\quad X^{i,N}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T]\end{aligned}\right.$

where $\mathbf{n}(x)$ is the outward unit normal vector at
$x\in\partial{\mathcal{D}}$, $\\{W^{i},1\leq i\leq N\\}$ are independent
standard $d_{1}$-dimensional Wiener processes and $K^{i,N}$ is a bounded
variation process with variation $|K^{i,N}|$ acting as a local time that
constrains the process $X^{i,N}$ to the domain $\overline{\mathcal{D}}$. Here,
$x^{i,N}\in\overline{\mathcal{D}}$ and

$\mu^{N}(t,\omega):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,N}(t,\omega)},\quad\omega\in\Omega,$

is the empirical measure of $(X^{1,N},\ldots,X^{N,N})$ at time $t\in[0,T]$.
For the precise conditions on $\mathcal{D}$, $b$ and $\sigma$, we refer the
readers to Section 2.

Large deviations of weakly interacting diffusion can characteristic the rate
of convergence of the particle system converging to a Mckean–Vlasov stochastic
differential equations. The pioneer work [8] for large deviation of weakly
interacting diffusion considers a system of uniformly non-degenerate diffusion
with interacting in the drift and establishes a large deviation principle for
the empirical measures using discretization arguments and exponential
estimates. There is now a number of papers on the large deviations of weakly
interacting diffusions for a variety of models, including multilevel large
deviations [9, 11], jump diffusions [16, 17], discrete-time system [7, 10] and
singular interaction [1, 14], interaction with common noise [3].

However, there is no study so far on the large deviation of empirical measures
of weakly interacting diffusion with reflection. The aim of this work is to
fill the gap. Our work is mainly inspired by the work [5]. We will use the
weak convergence methods, no longer requiring any time or space discretization
of the system and exponential probability estimates, along the same lines of
the approaches in [5] where the authors considered weakly interacting
diffusion with no reflection. As in [5], we allow the driving noise to be
degenerate and also allow both the drift and the diffusion coefficients to
depend on interactions.

The early work to study McKean–Vlasov stochastic differential equations (SDEs)
with reflecting boundaries goes back to [18], in which the pathwise well-
posedness and propagation of chaos are proved. McKean–Vlasov SDEs are SDEs
which are related to the mean-field interacting diffusions and where the
coefficients depend on the law of the solution. In a recent work [6], the
authors study a system, which is the combination of the interaction keeping
the average position prescribed, and the reflection at the boundaries. And
they show pathwise well-posedness for the McKean–Vlasov SDE and the
propagation of chaos. There is also some work on the small noise large
deviations of reflected McKean–Vlasov SDEs, see [2, 21].

With the weak convergence approach, we reduce the proof of the large deviation
principle to the study of asymptotic properties of certain controlled versions
of the original system with reflection. The main ingredients in our argument
are to characterize the weak limits of the controlled processes. The most
closed work to ours is [5], which considers large deviation property of weakly
interacting systems of stochastic differential equations. Although we both use
the weak convergence approach, there are major differences due to the
reflection on the boundary of the domain. Particularly, the method of
obtaining the Laplace upper bound in [5] no longer works for our model due to
the reflection. Instead, we fully exploit the close relationship between
stochastic differential equations with reflecting boundary and the so-called
submartingale problems. The remainder of this paper is organized as follows.
In Section 2, we describe the framework, present the main result of the paper,
and then outline how the result will be proved. In Section 3, we consider the
sub-martingale characterization of a weak solution of reflected stochastic
differential equations, which will be used in the proof. Finally, we complete
the proof of the main result by showing the Laplace upper and lower bounds in
Section 4.

We end this section with some standard notations.

Let $\overline{\mathcal{D}}$ be the closure of $\mathcal{D}$. Let
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ be a filterted
probability space where $(\mathcal{F}_{t})_{t\in[0,T]}$ satisfies the usual
conditions. Let $\mathcal{X}:=C([0,T];\overline{\mathcal{D}})$ and
$\mathcal{W}:=C([0,T];\mathbb{R}^{d_{1}})$, equipped with maximum norm, which
is denoted by $\|\cdot\|_{\infty}$. Let $|\cdot|$ be the Euclidean norm and
$\|\cdot\|$ stand for the Hilbert–Schmidt norm, i.e.,
$\|\sigma\|^{2}:=\sum_{i=1}^{d}{\sum_{j=1}^{d_{1}}{\sigma_{ij}^{2}}}$ for any
$d\times d_{1}$-matrix $\sigma=(\sigma_{ij})\in\mathbb{R}^{d\times d_{1}}$.
Given a Polish space $(E,d_{E})$, define $\mathcal{B}(E)$ as the Borel
$\sigma$-field, denote by $\mathcal{P}(E)$ the space of probability measures
on $\mathcal{B}(E)$, which is equipped with the topology of weak convergence.
A convenient metric on this space is the bounded Lipschitz metric, which is
denoted by

$\Pi_{E}(\mu,\nu):=\sup_{f\in\mbox{Lip}_{1}(E)}\int_{E}f\mathrm{d}(\mu-\nu),\
\ \mu,\nu\in\mathcal{P}(E),$

where $\mbox{Lip}_{1}(E):=\\{f\in C(E):\sup_{x\in E}|f(x)|\leq 1,\sup_{x\neq
y\in E}\frac{|f(x)-f(y)|}{d_{E}(x,y)}\leq 1\\}$. Let $C_{b}(E)$ be the set of
real-valued bounded continuous functions on $E$. Let
$C^{1,2}([0,T]\times\overline{\mathcal{D}})$ denote the set of real-valued
functions, whose elements are continuously differentiable once with respect to
the time variable and twice with respect to the space variable. For $f\in
C^{1,2}([0,T]\times\overline{\mathcal{D}})$, $\nabla_{x}f$ denotes the
gradient of $f$ with respect to the spatial variable $x$. Analogously, we can
define $C^{1,2,2}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$,
$C^{1,2,2}_{b}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$,
$C^{1,2,2}_{0}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$.

##  2\. Main result

Let $\mathcal{R}$ denote the set of all positive measures on
$\mathcal{B}(\mathbb{R}^{d_{1}}\times[0,T])$, say $r$, such that
$r(\mathbb{R}^{d_{1}}\times[0,t])=t$ for all $t\in[0,T]$, which will be the
space of all deterministic relaxed controls on
$\mathbb{R}^{d_{1}}\times[0,T]$. We also denote

$\displaystyle\mathcal{R}_{1}$ $\displaystyle:=$
$\displaystyle\Big{\\{}r\in\mathcal{R}:\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|r(\mathrm{d}y\times\mathrm{d}t)<\infty\Big{\\}},$

as the space of all deterministic relaxed controls on
$\mathbb{R}^{d_{1}}\times[0,T]$ with finite first moments. We equip
$\mathcal{R}$ and $\mathcal{R}_{1}$ with the topology of weak convergence of
measures and weak convergence of measures plus convergence of the first
moments respectively, which turns $\mathcal{R}$ and $\mathcal{R}_{1}$ into
Polish spaces. If $r\in\mathcal{R}$ and $B\in\mathcal{B}(\mathbb{R}^{d_{1}})$,
the mapping $[0,T]\ni t\mapsto r(B\times[0,t])$ will be absolutely continuous,
hence differentiable almost everywhere. Since
$\mathcal{B}(\mathbb{R}^{d_{1}})$ is countably generated, the time derivative
of $r$, denoted by $r_{t}$, exists almost everywhere and is a measurable
mapping from $[0,T]$ to $\mathcal{P}(\mathbb{R}^{d_{1}})$, such that
$r(\mathrm{d}y\times\mathrm{d}t)=r_{t}(\mathrm{d}y)\mathrm{d}t$.

Let
$b:[0,T]\times\overline{\mathcal{D}}\times\mathcal{P}(\overline{\mathcal{D}})\to\mathbb{R}^{d}$
and
$\sigma:[0,T]\times\overline{\mathcal{D}}\times\mathcal{P}(\overline{\mathcal{D}})\to\mathbb{R}^{d\times
d_{1}}$ be measurable mappings. Given a Borel measurable mapping
$\nu:[0,T]\to\mathcal{P}(\overline{\mathcal{D}})$ and an adapted
$\mathcal{R}_{1}$-valued random variable $\rho$. We will consider the
controlled RSDEs:

(2.1) $\displaystyle\left\\{\begin{aligned}
\mathrm{d}\bar{X}(t)&=b(t,\bar{X}(t),\nu(t))\mathrm{d}t+\int_{\mathbb{R}^{d_{1}}}\sigma(t,\bar{X}(t),\nu(t))y\rho_{t}(\mathrm{d}y)\mathrm{d}t\\\
&\ \ \ \ +\sigma(t,\bar{X}(t),\nu(t))\mathrm{d}W(t)-\mathrm{d}\bar{K}(t),\\\
|\bar{K}|(t)&=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\quad\bar{K}(t)=\int^{t}_{0}\mathbf{n}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\\\
\nu(0)&=\mbox{Law}(\bar{X}(0)),\quad\bar{X}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T],\\\ \end{aligned}\right.$

where $W$ is a $d_{1}$-dimensional $\mathcal{F}_{t}$-adapted standard Wiener
process. The above equation is the controlled analogue of the following
reflected McKean-Vlasov SDEs:

(2.2) $\displaystyle\left\\{\begin{aligned}
&\mathrm{d}X(t)=b(X(t),\mbox{Law}(X(t)))\mathrm{d}t+\sigma(X(t),\mbox{Law}(X(t)))\mathrm{d}W(t)-\mathrm{d}K(t),\\\
&|K(t)|=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(X(s))\mathrm{d}|K|(s),\quad
K(t)=\int^{t}_{0}\mathbf{n}(X(s))\mathrm{d}|K|(s),\\\
&\mbox{Law}(X(0))=\nu_{0},\quad X(t)\in\overline{\mathcal{D}},\quad
t\in[0,T],\end{aligned}\right.$

where $W$ is a $d_{1}$-dimensional $(\mathcal{F}_{t})$-adapted standard Wiener
process.

Recall the system given by (1.1): for fixed $N\in\mathbb{N}$ and finite
interval $[0,T]$, $i\in\\{1,\ldots,N\\}$,

$\displaystyle\left\\{\begin{aligned}
\mathrm{d}X^{i,N}(t)&=b(t,X^{i,N}(t),\mu^{N}(t))\mathrm{d}t+\sigma(t,X^{i,N}(t),\mu^{N}(t))\mathrm{d}W^{i}(t)\\\
&\ \ \ \ -\mathrm{d}K^{i,N}(t),\\\
|K^{i,N}|(t)&=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(X^{i,N}(s))\mathrm{d}|K^{i,N}|(s),\\\
K^{i,N}(t)&=\int^{t}_{0}\mathbf{n}(X^{i,N}(s))\mathrm{d}|K^{i,N}|(s),\\\
X^{i,N}(0)&=x^{i,N},\quad X^{i,N}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T].\end{aligned}\right.$

Define

$\mu^{N}(\cdot,\omega):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,N}(\cdot,\omega)},\quad\omega\in\Omega$

as the empirical measure of $(X^{1,N},\ldots,X^{N,N})$ over the path space
$\mathcal{X}:=C([0,T];\overline{\mathcal{D}})$. For any $t\in[0,T]$, $\mu^{N}$
and $\mu^{N}(t)$ have relationship as

$\mu^{N}(t)=\mu^{N}\circ\pi^{-1}_{t},$

where $\pi_{t}:\mathcal{X}\to\overline{\mathcal{D}}$ is a projection map at
time $t$. It is known that under mild conditions on the coefficients the
empirical measures $\mu^{N}$ converges to the law of the solution to reflected
McKean-Vlasov SDE (2.2).

Let $\mathcal{Z}:=\mathcal{X}\times\mathcal{R}_{1}\times\mathcal{W}$. For any
$z\in\mathcal{Z}$, we write it as $(\phi,r,w)$ with the understanding that
$\phi\in\mathcal{X}$, $r\in\mathcal{R}_{1}$, and $w\in\mathcal{W}$. If the
triple $(\bar{X},\rho,W)$ defined on some filtered probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t}),\tilde{\mathbb{P}})$
solves equation (2.1) for some measurable
$\nu:[0,T]\to\mathcal{P}(\overline{\mathcal{D}})$, then the distribution of
$(\bar{X},\rho,W)$ under $\tilde{\mathbb{P}}$ is an element of
$\mathcal{P}(\mathcal{Z})$ and is called a weak solution of equation (2.1).
For any $\Theta\in\mathcal{P}(\mathcal{Z})$, define
$\nu_{\Theta}:[0,T]\to\mathcal{P}(\overline{\mathcal{D}})$ as

(2.3)
$\nu_{\Theta}(t)(B):=\Theta\big{(}\\{(\phi,\rho,w)\in\mathcal{Z}:\phi(t)\in
B\\}\big{)},\quad B\in\mathcal{B}(\overline{\mathcal{D}}),\quad t\in[0,T],$

which is the distribution under $\Theta$ of the first component of $Z$ at time
$t$. Note that if $\Theta$ is a weak solution of equation (2.1) with
$\nu_{\Theta}(t)=\nu(t),\ \forall t\in[0,T]$, then $\Theta$ is a weak solution
of the controlled RSDEs

(2.4) $\displaystyle\left\\{\begin{aligned}
&\mathrm{d}\bar{X}(t)=b(t,\bar{X}(t),\nu_{\Theta}(t))\mathrm{d}t+\int_{\mathbb{R}^{d_{1}}}\sigma(t,\bar{X}(t),\nu_{\Theta}(t))y\rho_{t}(\mathrm{d}y))\mathrm{d}t\\\
&\quad\quad\quad\quad+\sigma(t,\bar{X}(t),\nu_{\Theta}(t)))\mathrm{d}W(t)-\mathrm{d}\bar{K}(t),\\\
&|\bar{K}|(t)=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\quad\bar{K}(t)=\int^{t}_{0}\mathbf{n}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\\\
&\nu_{\Theta}(t)=\mbox{Law}(\bar{X}(t)),\quad\bar{X}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T].\\\ \end{aligned}\right.$

For a probability measure $\Theta\in\mathcal{P}(\mathcal{Z})$, let
$\Theta_{\mathcal{X}}$, $\Theta_{\mathcal{R}}$ and
$\Theta_{\mathcal{\mathcal{W}}}$ denote the first, second and the third
marginal measures, respectively. Let $\mathcal{P}_{\infty}$ be the set of all
probability measures $\Theta\in\mathcal{P}(\mathcal{Z})$ such that

  * (i)

$\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}}(\mathrm{d}r)<\infty.$

  * (ii)

$\Theta$ is a weak solution to equation (2.4).

  * (iii)

$\nu_{\Theta}(0)=\nu_{0}$, where
$\nu_{0}\in\mathcal{P}(\overline{\mathcal{D}})$ is the initial distribution.

Define the map
$\bar{v}:\mathcal{Z}\to\mathcal{Z}^{0}:=\overline{\mathcal{D}}\times\mathcal{R}_{1}\times\mathcal{W}$
as $\bar{v}(\phi,r,w)=(\phi(0),r,w)$.

######  Definition 2.1 (Weak Uniqueness).

We say that the solution of equation (2.4) is weakly unique if for any
$\Theta$ and $\tilde{\Theta}\in\mathcal{P}_{\infty}$ such that
$\Theta\circ\bar{v}^{-1}=\tilde{\Theta}\circ\bar{v}^{-1}$, we have
$\Theta=\tilde{\Theta}$.

Now we are going to state the precise assumption and the main result.

######  Definition 2.2 (Rate Function [12, 13]).

A function $I:\mathcal{P}(\mathcal{X})\to[0,\infty]$ is called a rate function
on $\mathcal{P}(\mathcal{X})$, if for any $C<\infty$, the level set
$\\{\theta\in\mathcal{P}(\mathcal{X}):I(\theta)\leq C\\}$ is compact.

######  Definition 2.3 (Large Deviation Principle [12, 13]).

Let $I$ be a rate function on $\mathcal{P}(\mathcal{X})$. The sequence
$\\{\mu^{N},N\in\mathbb{N}\\}$ is said to satisfy large deviation principle on
$\mathcal{P}(\mathcal{X})$ with the rate function $I$ if, for all Borel subset
$\Gamma$ of $\mathcal{P}(\mathcal{X})$,

$-\inf_{\theta\in\Gamma^{o}}I(\theta)\leq\liminf_{N\to\infty}\frac{1}{N}\log\mathbb{P}\\{\mu^{N}\in\Gamma\\}\leq\limsup_{N\to\infty}\frac{1}{N}\log\mathbb{P}\\{\mu^{N}\in\Gamma\\}\leq-\inf_{\theta\in\overline{\Gamma}}I(\theta),$

where $\Gamma^{o}$ is the interior of $\Gamma$.

######  Definition 2.4 (Laplace Principle [13]).

Let $I$ be a rate function on $\mathcal{P}(\mathcal{X})$. The sequence
$\\{\mu^{N},N\in\mathbb{N}\\}$ is said to satisfy the Laplace principle on
$\mathcal{P}(\mathcal{X})$ with rate function $I$ if for all bounded
continuous function $F:\mathcal{P}(\mathcal{X})\to\mathbb{R}$,

$\lim_{N\to\infty}-\frac{1}{N}\log\mathbb{E}\big{\\{}\exp[-N\cdot
F(\mu^{N})]\big{\\}}=\inf_{\theta\in\mathcal{P}(\mathcal{X})}\\{F(\theta)+I(\theta)\\}.$

It is known that the Laplace principle holds in the above setting if and only
if $\\{\mu^{N},N\in\mathbb{N}\\}$ satisfies a large deviation principle with
rate function $I$ (see [13, Section 1.2]).

Let us make the following assumptions with respect to the domain, coefficients
$b$, $\sigma$, and the family $\\{x^{i,N}\\}\subset\overline{\mathcal{D}}$ of
initial conditions:

######  Assumption 2.1.

  * (A1)

$\mathcal{D}$ is a bounded, convex, smooth domain in $\mathbb{R}^{d}$.

  * (A2)

There exists $\nu_{0}\in\mathcal{P}(\overline{\mathcal{D}})$ such that for all
$\nu_{0}$-integrable $f:\overline{\mathcal{D}}\rightarrow\mathbb{R}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}f(x^{i,N})=\int_{\overline{\mathcal{D}}}f(x)\mathrm{d}\nu_{0}(x).$

  * (A3)

Let
$b:[0,T]\times\overline{\mathcal{D}}\times\mathcal{P}(\overline{\mathcal{D}})\to\mathbb{R}^{d}$
and
$\sigma:[0,T]\times\overline{\mathcal{D}}\times\mathcal{P}(\overline{\mathcal{D}})\to\mathbb{R}^{d\times
d_{1}}$ be measurable and there exist constants $L$ and $K\in(0,\infty)$ such
that for each $t\in[0,T],x,y\in\overline{\mathcal{D}}$ and
$\mu,\nu\in\mathcal{P}(\overline{\mathcal{D}})$,

$|b(t,x,\mu)|+\|\sigma(t,x,\mu)\|\leq L$

and

$|b(t,x,\mu)-b(t,y,\nu)|+\|\sigma(t,x,\mu)-\sigma(t,y,\nu)\|\leq
K(|x-y|+\Pi_{\overline{\mathcal{D}}}(\mu,\nu)).$

  * (A4)

Weak uniqueness of solution holds for Eq.(2.4).

######  Remark 2.5.

  * (R1)

Under Assumption 2.1(A3), it follows by standard arguments that for each $N$,
equation (1.1) has a unique solution. Moreover, according to Theorem 3.2 in
[2], equation (2.2) admits a unique solution.

  * (R2)

The assumption (A4) is satisfied if the diffusion coefficients depended only
on state variables like $\sigma(t,x,\mu)=\sigma(t,x)$, or only on
distributions such as $\sigma(t,x,\mu)=\sigma(t,\mu)$ (cf. [3, Lemma 3.1]).

  * (R3)

Since the domain $\mathcal{D}$ is convex, we have for any $y\in\mathcal{D}$
and $x\in\partial\mathcal{D}$,

$\langle\mathbf{n}(x),y-x\rangle\leq 0.$

The main result is stated as follows.

######  Theorem 2.6.

Let Assumption 2.1 hold. Then the family of empirical measures
$\\{\mu^{N},N\in\mathbb{N}\\}$ of the solutions of the interacting reflected
system (1.1) satisfies the Laplace Principle with rate function

$I(\theta)=\inf_{\Theta\in\mathcal{P}_{\infty}:\Theta_{\mathcal{X}}=\theta}\frac{1}{2}\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}}(\mathrm{d}r)$

where $\theta\in\mathcal{P}(\mathcal{X})$ and $\inf\emptyset:=\infty$ by
convention.

To explain the steps of the proof, we need the controlled system of RSDEs
(1.1). For $N\in\mathbb{N}$, let $\mathcal{H}_{N}$ be the space of all
$(\mathcal{F}_{t})$-progressively measurable functions
$h:[0,T]\times\Omega\to\mathbb{R}^{N\times d_{1}}$ such that

$\mathbb{E}\left[\int^{T}_{0}|h(t)|^{2}\mathrm{d}t\right]<\infty,$

where $h$ is written as $h=(h_{1},\ldots,h_{N})$, and $h_{i}$ is its $i$-th
entry, which is $d_{1}$-dimensional.

Given $h\in\mathcal{H}_{N}$, we consider the following controlled system of
RSDEs

(2.5) $\displaystyle\left\\{\begin{aligned}
&\mathrm{d}\bar{X}^{i,N}(t)=b(t,\bar{X}^{i,N}(t),\bar{\mu}^{N}(t))\mathrm{d}t+\sigma(t,\bar{X}^{i,N}(t),\bar{\mu}^{N}(t))h_{i}(t)\mathrm{d}t\\\
&\qquad\quad\quad\quad+\sigma(t,\bar{X}^{i,N}(t),\bar{\mu}^{N}(t))\mathrm{d}W^{i}(t)-\mathrm{d}\bar{K}^{i,N}(t),\\\
&|\bar{K}^{i,N}|(t)=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(\bar{X}^{i,N}(s))\mathrm{d}|\bar{K}^{i,N}|(s),\\\
&\bar{K}^{i,N}(t)=\int^{t}_{0}\mathbf{n}(\bar{X}^{i,N}(s))\mathrm{d}|\bar{K}^{i,N}|(s),\\\
&\bar{X}^{i,N}(0)=x^{i,N},\quad\bar{X}^{i,N}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T],\end{aligned}\right.$

where $\bar{\mu}^{N}(t)$ is empirical measure of
$(\bar{X}^{1,N}(t),\ldots,\bar{X}^{N,N}(t))$ defined by

$\bar{\mu}^{N}(t,\omega):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\bar{X}^{i,N}(t,\omega)},\quad\omega\in\Omega.$

We also use $\bar{\mu}^{N}$ to denote the process version of
$\bar{\mu}^{N}(t)$.

The proof of Theorem 2.6 is based on following variational representation
given by Theorem 3.6 in [4], i.e. for any $F\in
C_{b}(\mathcal{P}(\mathcal{X}))$

$-\frac{1}{N}\log\mathbb{E}\left\\{\exp[-N\cdot
F(\mu^{N})]\right\\}=\inf_{h^{N}\in\mathcal{H}_{N}}\Big{\\{}\frac{1}{2}\mathbb{E}\big{[}\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}|^{2}\mathrm{d}t\big{]}+\mathbb{E}\left[F(\bar{\mu}^{N})\right]\Big{\\}},$

where $\bar{\mu}^{N}$ is the empirical measure of the solution to equation
(LABEL:II-eq2.3) for $h^{N}\in\mathcal{H}_{N}$. According to the arguments in
[5, section 3] ,the Laplace principle (hence the large deviation principle)
can be established through the following two steps.

  * Step 1:

We establish the Laplace principle upper bound by showing that for any
sequence $\\{h^{N},N\in\mathbb{N}\\}$ with $h^{N}\in\mathcal{H}_{N}$,

(2.6)
$\displaystyle\begin{split}&\liminf_{N\mapsto\infty}\Big{\\{}\frac{1}{2}\mathbb{E}\Big{[}\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}|^{2}\mathrm{d}t\Big{]}+\mathbb{E}\big{[}F(\bar{\mu}^{N})\big{]}\Big{\\}}\\\
&\geq\inf_{\Theta\in\mathcal{P}_{\infty}}\Big{\\{}\frac{1}{2}\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}_{1}}(\mathrm{d}r)+F(\Theta_{\mathcal{X}})\Big{\\}}.\end{split}$

This will be done in Section 4.1.

  * Step 2:

We verify the Laplace principle lower bound in Section 4.2, by showing that
for any $\Theta\in\mathcal{P}_{\infty}$ there exists a sequence
$\\{h^{N},N\in\mathbb{N}\\}$ with $h^{N}\in\mathcal{H}_{N}$ such that

(2.7)
$\displaystyle\begin{split}&\limsup_{N\to\infty}\Big{\\{}\frac{1}{2}\mathbb{E}\big{[}\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}|^{2}\mathrm{d}t\big{]}+\mathbb{E}[F(\bar{\mu}^{N})]\Big{\\}}\\\
&\leq\frac{1}{2}\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}_{1}}(\mathrm{d}r)+F(\Theta_{\mathcal{X}}).\end{split}$

##  3\. Sub-martingale Problem

As a preparation for the proof of the main result, in this section, we will
introduce some sub-martingale problems and provide the relations to the weak
solution of RSDEs (2.4).

To begin with, we introduce the definition of the sub-martingale problem
described in [19]. Let
$\bar{b}:[0,T]\times\overline{\mathcal{D}}\to\mathbb{R}^{d}$ be a bounded
measurable function and
$\bar{\sigma}:[0,T]\times\overline{\mathcal{D}}\to\mathbb{R}^{d\times d_{1}}$
a bounded continuous function.

Let $\tilde{X}$ be the canonical process on $\mathcal{X}$. Set
$\mathcal{F}^{0}_{t}=\sigma(\tilde{X}(s),\ s\leq t)$ and define operator

$\mathcal{L}_{t}:=\frac{1}{2}\sum^{d}_{i,j=1}(\bar{\sigma}\bar{\sigma}^{T})_{ij}(t,x)\frac{\partial^{2}}{\partial
x_{i}\partial x_{j}}+\sum^{d}_{i=1}\bar{b}_{i}(t,x)\frac{\partial}{\partial
x_{i}}.$

######  Definition 3.1 (Sub-martingale Problem).

We say that a probability measure $\hat{\mathbb{P}}$ on
$(\mathcal{X},\mathcal{B}(\mathcal{X}))$ solves a sub-martingale problem for
coefficients $\bar{b}$, $\bar{\sigma}$ and $\mathbf{n}(x)$ if

$f(t,\tilde{X}(t))-f(0,\tilde{X}(0))-\int^{t}_{0}(f_{s}+\mathcal{L}_{s}f)(s,\tilde{X}(s))\mathrm{d}s$

is a $\hat{\mathbb{P}}$-sub-martingale with respect to the canonical
filtration $(\mathcal{F}^{0}_{t})$ for any $f\in
C^{1,2}_{0}([0,T]\times\overline{\mathcal{D}})$ satisfying

$\langle\nabla_{x}f(t,x),\mathbf{n}(x)\rangle\leq
0\quad\mbox{on}\quad[0,T]\times\partial\mathcal{D}.$

Consider the following RSDEs

(3.1) $\displaystyle\left\\{\begin{aligned}
&\mathrm{d}Y(t)=\bar{b}(t,Y(t))\mathrm{d}t+\bar{\sigma}(t,Y(t))\mathrm{d}W(t)-\mathrm{d}L(t),\\\
&|L(t)|=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(Y(s))\mathrm{d}|L|(s),\quad
L(t)=\int^{t}_{0}\mathbf{n}(Y(s))\mathrm{d}|L|(s),\\\
&Y(t)\in\overline{\mathcal{D}},\quad t\in[0,T],\end{aligned}\right.$

where $W$ is a $d_{1}$-dimensional $\mathcal{F}_{t}$-adapted Wiener process.

For any $f\in C^{1,2}([0,T]\times\overline{\mathcal{D}})$, we define a real-
valued process $(M_{f}(t))_{t\in[0,T]}$ on probability space
$(\mathcal{X},\mathcal{B}(\mathcal{X}),\hat{\mathbb{P}})$ by:

$\displaystyle\begin{split}M_{f}(t,\tilde{X})&:=f(t,\tilde{X}(t))-f(0,\tilde{X}(0))-\int^{t}_{0}(f_{s}+\mathcal{L}_{s}f)(s,\tilde{X}(s))\mathrm{d}s.\\\
\end{split}$

The next result provides the relationship between a sub-martingale problem and
the weak solutions of RSDEs (3.1).

######  Lemma 3.2.

Assume that the measure $\hat{\mathbb{P}}\in\mathcal{P}(\mathcal{X})$, then
$\hat{\mathbb{P}}$ is a weak solution of equation (3.1) if and only if
$\hat{\mathbb{P}}$ solves a sub-martingale problem for coefficients $\bar{b}$,
$\bar{\sigma}$ and $\mathbf{n}$, or equivalently, $M_{f}$ is a
$\hat{\mathbb{P}}$-sub-martingale with respect to the canonical filtration
$(\mathcal{F}^{0}_{t})$ for any $f\in
C^{1,2}([0,T]\times{\overline{\mathcal{D}}})$ satisfying
$\langle\nabla_{x}f(t,x),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}$.

###### Proof.

The “$\Rightarrow$” part is obvious by Itô’s formula.

Now we show the “$\Leftarrow$” part. Assume $\hat{\mathbb{P}}$ is a solution
to the sub-martingale problem. By Theorem 2.4 in [19], we know that there
exists a unique, continuous, non-decreasing, adapted function
$L:[0,T]\times\mathcal{X}\mapsto[0,\infty)$ such that $L(0)=0$,
$L(t)=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(\tilde{X}(s))\mathrm{d}L(s)$
and for any $f\in C^{1,2}([0,T]\times{\overline{\mathcal{D}}})$

$\displaystyle\tilde{M}_{f}(t,\tilde{X}):=M_{f}(t,\tilde{X})+\int^{t}_{0}\langle\nabla_{x}f(s,\tilde{X}(s)),\mathbf{n}(\tilde{X}(s))\rangle\mathrm{d}L(s)$

is a $\hat{\mathbb{P}}$-martingale.

In particular, choose $f^{i}(t,x)=x_{i}$ to obtain that

$\displaystyle\begin{split}\tilde{M}_{f^{i}}(t,\tilde{X})&:=\tilde{X}_{i}(t)-\tilde{X}_{i}(0)-\int^{t}_{0}\bar{b}_{i}(s,\tilde{X}(s))\mathrm{d}s\\\
&\ \ \ \ +\int^{t}_{0}n_{i}(\tilde{X}(s))\mathrm{d}L(s)\\\ \end{split}$

is a $\hat{\mathbb{P}}$-martingale.
Similarly, for each $i,j\in\\{1,2,\ldots,d\\}$, letting
$f^{i,j}(t,x)=x_{i}x_{j}$, we see that

(3.2)
$\displaystyle\begin{split}\tilde{M}_{f^{i,j}}(t,\tilde{X})&:=\tilde{X}_{i}(t)\tilde{X}_{j}(t)-\tilde{X}_{i}(0)\tilde{X}_{j}(0)-\int^{t}_{0}\bar{b}_{i}(s,\tilde{X}(s))\tilde{X}_{j}(s)\mathrm{d}s\\\
&\ \ \ \
-\int^{t}_{0}\bar{b}_{j}(s,\tilde{X}(s))\tilde{X}_{i}(s)\mathrm{d}s+\int^{t}_{0}\tilde{X}_{i}(s)n_{j}(\tilde{X}(s))\mathrm{d}L(s)\\\
&\ \ \ \
+\int^{t}_{0}\tilde{X}_{j}(s)n_{i}(\tilde{X}(s))\mathrm{d}L(s)-\int^{t}_{0}(\bar{\sigma}\bar{\sigma}^{T})_{ij}(s,\tilde{X}(s))\mathrm{d}s\end{split}$

is also a $\hat{\mathbb{P}}$-martingale. Applying Itô’s formula to
$\tilde{X}_{i}(t)\tilde{X}_{j}(t)$ and comparing with (3.2), we deduce that

$\langle\tilde{M}_{f^{i}},\tilde{M}_{f^{j}}\rangle(t)=\int^{t}_{0}\sum^{d_{1}}_{k=1}\bar{\sigma}_{ik}\bar{\sigma}_{kj}(s,\tilde{X}(s))\mathrm{d}s.$

Then, according to Theorem II.7.1’ in [15], there exists a $d_{1}$-dimensional
$(\tilde{\mathcal{F}}_{t})$-Wiener process $\tilde{W}=(\tilde{W}(t))$ on an
extension of
$(\tilde{\Omega},\tilde{\mathcal{F}},(\tilde{\mathcal{F}}_{t}),\tilde{\mathbb{P}})$
such that

$\tilde{M}_{f^{i}}(t)=\sum^{d_{1}}_{i=1}\int^{t}_{0}\bar{\sigma}_{ik}(s,\tilde{X}(s))\mathrm{d}\tilde{W}^{k}(s),\quad
i=1,2,\ldots,d.$

Therefore, $(\tilde{X},\tilde{W},L)$ is a weak solution to the reflected
stochastic differential equation (3.1). ∎

In the remaining part of this section, we will show that the measure
$\Theta\in\mathcal{P}(\mathcal{Z})$ is a weak solution of equation (2.4) if
and only if it solves a sub-martingale problem.

Recall that $\mathcal{Z}=\mathcal{X}\times\mathcal{R}_{1}\times\mathcal{W}$
and $\nu_{\Theta}$ is given by (2.3). Let $(\bar{X},\rho,W)$ be the canonical
process on $\mathcal{Z}$, namely

$\bar{X}(t,(\phi,r,w)):=\phi(t),\
\rho(t,(\phi,r,w)):=r_{|\mathcal{B}(\mathbb{R}^{d_{1}}\times[0,t])},\
W(t,(\phi,r,w)):=w(t),$

and $(\mathcal{G}_{t})$ the canonical filtration in $\mathcal{B}(\mathcal{Z})$
defined as

$\mathcal{G}_{t}:=\sigma((\bar{X}(s),\rho(s),W(s)):0\leq s\leq t),\quad
t\in[0,T].$

Let $\Theta\in\mathcal{P}(\mathcal{Z})$. Given $f\in
C^{1,2,2}_{0}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$,
we define a real-valued process $(M^{\Theta}_{f}(t))_{t\in[0,T]}$ on
probability space $(\mathcal{Z},\mathcal{B}(\mathcal{Z}),\Theta)$ as

(3.3) $\displaystyle M^{\Theta}_{f}(t,(\phi,r,w))$ $\displaystyle:=$
$\displaystyle f(t,\phi(t),w(t))-f(0,\phi(0),w(0))-\int^{t}_{0}\frac{\partial
f}{\partial s}(s,\phi(s),w(s))\mathrm{d}s$
$\displaystyle-\int^{t}_{0}\int_{\mathbb{R}^{d_{1}}}\mathcal{A}^{\Theta}_{s}(f)(s,\phi(s),y,w(s))r_{s}(\mathrm{d}y)\mathrm{d}s,$

where for any $s\in[0,T]$, $x\in{\overline{\mathcal{D}}}$,
$y,z\in\mathbb{R}^{d_{1}}$, $\mathcal{A}^{\Theta}_{s}(f)$ is defined as

(3.4)
$\displaystyle\begin{split}\mathcal{A}^{\Theta}_{s}(f)(s,x,y,z)&:=\langle
b(s,x,\nu_{\Theta}(s))+\sigma(s,x,\nu_{\Theta}(s))y,\nabla_{x}f(s,x,z)\rangle\\\
&\quad+\frac{1}{2}\sum^{d}_{i,j=1}(\sigma\sigma^{T})_{ij}(s,x,\nu_{\Theta}(s))\frac{\partial^{2}f}{\partial{x_{i}}\partial{x_{j}}}(s,x,z)\\\
&\quad+\sum^{d}_{i=1}\sum^{d_{1}}_{j=1}\sigma_{ij}(s,x,\nu_{\Theta}(s))\frac{\partial^{2}f}{\partial{x_{i}}\partial{z_{j}}}(s,x,z)\\\
&\quad+\frac{1}{2}\sum^{d_{1}}_{i=1}\frac{\partial^{2}f}{\partial{z_{i}}\partial{z_{i}}}(s,x,z).\end{split}$

The last result of this section is stated as follows.

######  Theorem 3.3.

Let measure $\Theta\in\mathcal{P}(\mathcal{Z})$ satisfy
$\Theta(\\{(\phi,r,w)\in\mathcal{Z}:w(0)=0\\})=1$. Then $\Theta$ is a weak
solution of equation (2.4) if and only if $M^{\Theta}_{f}$ is a sub-martingale
under $\Theta$ with respect to the canonical filtration $(\mathcal{G}_{t})$
for all $f\in
C^{1,2,2}_{0}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$
with $\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$.

Before proving Theorem 3.3, we state the following Lemmas whose proofs are
similar to those of Lemma 2.2 and Theorem 2.4 in [19].

######  Lemma 3.4.

Suppose that $M^{\Theta}_{f}$ is a $\Theta$-sub-martingale for any $f\in
C^{1,2,2}_{0}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$
satisfying $\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$. Then, for each
$f\in
C^{1,2,2}_{b}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$
with $\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$,
$M^{\Theta}_{f}(t)$ is a $\Theta$-local-sub-martingale.

###### Proof.

Assume that $f\in
C^{1,2,2}_{b}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$
with $\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$. For each $n\geq
1$, choose $\eta_{n}\in C_{0}^{\infty}(\mathbb{R}^{d_{1}})$ such that
$0\leq\eta_{n}\leq 1$, $\eta_{n}=1$ on $\\{z\in\mathbb{R}^{d_{1}}:|z|\leq
n\\}$ and all derivatives of $\eta_{n}$ up to the second order are uniformly
bounded on $n$. Let

$f_{n}=\eta_{n}\cdot f.$

Then $f_{n}\in
C^{1,2,2}_{0}([0,T]\times{\overline{\mathcal{D}}}\times\mathbb{R}^{d_{1}})$
and $\langle\nabla_{x}f_{n}(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$. So
$M^{\Theta}_{f_{n}}(t)$ is a $\Theta$-sub-martingale. For each
$M\in\mathbb{N}$, define a stopping time

$\tau_{M}((\phi,r,w)):=\inf\\{t\in[0,T]:\int_{\mathbb{R}^{d_{1}}\times[0,t]}|y|r(\mathrm{d}y\times\mathrm{d}s)\geq
M\\}.$

Then $M^{\Theta}_{f_{n}}(t\wedge\tau_{M})$ is a $\Theta$-sub-martingale.
Obviously, $M^{\Theta}_{f_{n}}(t\wedge\tau_{M})\rightarrow
M^{\Theta}_{f}(t\wedge\tau_{M})$ boundedly and so
$M^{\Theta}_{f}(t\wedge\tau_{M})$ is a $\Theta$-sub-martingale. It follows
that $M^{\Theta}_{f}(t)$ is a $\Theta$-local-sub-martingale. ∎

######  Lemma 3.5.

There exists a continuous, non-decreasing and adapted function
$\xi:[0,T]\times\mathcal{Z}\mapsto[0,\infty)$ such that $\xi(0)=0$,

$\xi(t)=\int_{0}^{t}\mathbf{1}_{\partial{D}}(\phi(s))\mathrm{d}\xi(s),$

and for all $f\in
C^{1,2,2}_{b}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$,

$M^{\Theta}_{f}+\int_{0}^{t}\langle\nabla_{x}f(s,\phi(s),w(s)),\mathbf{n}(\phi(s))\rangle\mathrm{d}\xi(s)$

is a $\Theta$-local-martingale.

The proof of this lemma is a minor modification of the proof of Theorem 2.4 in
[19]. We omit the details.

Now we come back to the proof of Theorem 3.3

###### Proof.

The “$\Rightarrow$” part is obvious by Itô’s formula.

Now we show the “$\Leftarrow$” part. From Lemma 3.5, we know that there exists
a continuous, non-decreasing and adapted function
$\xi:[0,T]\times\mathcal{Z}\mapsto[0,\infty)$ such that $\xi(0)=0$,
$\xi(t)=\int_{0}^{t}\mathbf{1}_{\overline{\mathcal{D}}}(\phi(s))\mathrm{d}\xi(s)$,
and for all $f\in
C^{1,2,2}_{b}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$,

$\displaystyle\tilde{M}^{\Theta}_{f}(t,(\phi,r,w)):=M^{\Theta}_{f}(t,(\phi,r,w))+\int^{t}_{0}\langle\nabla_{x}f(s,\phi(s),w(s)),\mathbf{n}(\phi(s))\rangle\mathrm{d}\xi(s)$

is a $\Theta$-local-martingale.

In particular, for each $i\in\\{1,2,\ldots,d\\}$, choose $f^{i}(t,x,z)=x_{i}$
to obtain that

$\displaystyle\begin{split}\tilde{M}^{\Theta}_{f^{i}}(t,(\phi,r,w))&:=\phi_{i}(t)-\phi_{i}(0)-\int_{0}^{t}b_{i}(s,\phi(s),\nu_{\Theta}(s))\mathrm{d}s\\\
&\quad-\int_{0}^{t}\int_{\mathbb{R}^{d_{1}}}\sum^{d_{1}}_{k=1}\sigma_{ik}(s,\phi(s),\nu_{\Theta}(s))y_{k}r_{s}(\mathrm{d}y)\mathrm{d}s\\\
&\quad+\int_{0}^{t}n_{i}(\phi(s))\mathrm{d}\xi(s)\\\ \end{split}$

is a $\Theta$-local-martingale.

For each $i,j\in\\{1,2,\ldots,d\\}$, choosing $f^{i,j}(t,x,z)=x_{i}x_{j}$, we
see that

(3.5)
$\displaystyle\begin{split}\tilde{M}^{\Theta}_{f^{i,j}}(t,(\phi,r,w))&:=\phi_{i}(t)\phi_{j}(t)-\phi_{i}(0)\phi_{j}(0)\\\
&\quad-\int_{0}^{t}b_{i}(s,\phi(s),\nu_{\Theta}(s))\phi_{j}(s)\mathrm{d}s-\int_{0}^{t}b_{j}(s,\phi(s),\nu_{\Theta}(s))\phi_{i}(s)\mathrm{d}s\\\
&\quad-\int_{0}^{t}\int_{\mathbb{R}^{d_{1}}}\sum^{d_{1}}_{k=1}\sigma_{ik}(s,\phi(s),\nu_{\Theta}(s))y_{k}\phi_{j}(s)r_{s}(\mathrm{d}y)\mathrm{d}s\\\
&\quad-\int_{0}^{t}\int_{\mathbb{R}^{d_{1}}}\sum^{d_{1}}_{k=1}\sigma_{jk}(s,\phi(s),\nu_{\Theta}(s))y_{k}\phi_{i}(s)r_{s}(\mathrm{d}y)\mathrm{d}s\\\
&\quad-\int_{0}^{t}\sum^{d_{1}}_{k=1}\sigma_{ik}\sigma_{kj}(s,\phi(s),\nu_{\Theta}(s))\mathrm{d}s+\int_{0}^{t}\phi_{i}(s)n_{j}(\phi(s))\mathrm{d}\xi(s)\\\
&\quad+\int_{0}^{t}\phi_{j}(s)n_{i}(\phi(s))\mathrm{d}\xi(s)\\\ \end{split}$

is also a $\Theta$-local-martingale. Applying Itô’s formula to
$\bar{X}_{i}(t)\bar{X}_{j}(t)$ and comparing with (3.5), we deduce that

(3.6)
$\langle\tilde{M}^{\Theta}_{f^{i}},\tilde{M}^{\Theta}_{f^{j}}\rangle(t)=\int^{t}_{0}\sum^{d_{1}}_{k=1}\sigma_{ik}\sigma_{kj}(s,\phi(s),\nu_{\Theta}(s))\mathrm{d}s.$

For each $i\in\\{1,2,\ldots,d_{1}\\}$, choose $g^{i}\in
C_{0}^{1,2,2}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$ such
that $g^{i}(t,x,z)=z_{i}$ on
$[0,T]\times\overline{\mathcal{D}}\times\\{z\in\mathbb{R}^{d_{1}}:|z|\leq
n\\}$ and for each $n\in\mathbb{N}$, define the stopping time
$\tau_{n}=\inf\\{t\in[0,T]:|w(t)|>n\\}$ to obtain that

$\tilde{M}^{\Theta}_{g^{i}}(t\wedge\tau_{n},(\phi,r,w)):=w_{i}(t\wedge\tau_{n})-w_{i}(0)$

is a $\Theta$-martingale. It follows that
$\\{\tilde{M}^{\Theta}_{g^{i}}(t),t\geq 0\\}$ is a $\Theta$-local-martingale.

Similarly, for each $i,j\in\\{1,2,\ldots,d_{1}\\}$, choosing
$g^{i,j}(t,x,z)\in
C_{0}^{1,2,2}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$ such
that $g^{i,j}(t,x,z)=z_{i}z_{j}$ on
$[0,T]\times\overline{\mathcal{D}}\times\\{z\in\mathbb{R}^{d_{1}}:|z|\leq
n\\}$, we have

(3.7)
$\tilde{M}^{\Theta}_{g^{i,j}}(t,(\phi,r,w)):=w_{i}(t)w_{j}(t)-w_{i}(0)w_{j}(0)-\frac{1}{2}\delta_{ij}t$

is a $\Theta$-local-martingale. Applying Itô’s formula to $W_{i}(t)W_{j}(t)$
and comparing with (3.7), we deduce that

(3.8)
$\langle\tilde{M}^{\Theta}_{g^{i}},\tilde{M}^{\Theta}_{g^{j}}\rangle(t)=\langle
W_{i},W_{j}\rangle(t)=\delta_{ij}t.$

Therefore, $W(t)$ is a $(\mathcal{G}_{t})$-Wiener process on
$(\mathcal{Z},\mathcal{B}(\mathcal{Z}),(\mathcal{G}_{t}),\Theta)$.

For each $i\in\\{1,2,\ldots,d\\}$, $j\in\\{1,2,\ldots,d_{1}\\}$, by choosing
$h^{i,j}(t,x,z)\in
C_{0}^{1,2,2}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$ such
that $h^{i,j}(t,x,z)=x_{i}z_{j}$ on
$[0,T]\times\overline{\mathcal{D}}\times\\{z\in\mathbb{R}^{d_{1}}:|z|\leq
n\\}$, using a similar argument as (3.8), we see that

(3.9)
$\langle\tilde{M}^{\Theta}_{f^{i}},\tilde{M}^{\Theta}_{g^{j}}\rangle(t)=\int_{0}^{t}\sigma_{ij}(s,\phi(s),\nu_{\Theta}(s))\mathrm{d}s.$

Then, by (3.6), (3.8) and (3.9) and using the arguments in the proof of
Theorem II.7.1’ in [15], we obtain that

$M^{\Theta}_{f^{i}}(t,(\phi,r,w))=\sum_{k=1}^{d_{1}}\int_{0}^{t}\sigma_{ik}(s,\phi(s),\nu_{\Theta}(s))\mathrm{d}W_{k}(s),i=1,2,\ldots,d.$

Therefore, the canonical processes $(\bar{X},\rho,W)$ is a solution of
equation (2.4).

∎

##  4\. Laplace Principle

In this section, we will prove the upper and the lower bound of the Laplace
principle. We start by presenting an auxiliary lemma, which will be used in
the proofs of the Laplace principle. Recall the controlled RSDEs (LABEL:II-
eq2.3), the empirical measure $\bar{\mu}^{N}$ and the path space
$\mathcal{X}:=C([0,T];\overline{\mathcal{D}})$.

######  Lemma 4.1.

Suppose Assumption 2.1 holds. Let $\\{h^{N},N\in\mathbb{N}\\}$ be a sequence
of elements in $\mathcal{H}_{N}$ that satisfies

(4.1)
$\sup_{N\in\mathbb{N}}\mathbb{E}\left[\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}(t)|^{2}\mathrm{d}t\right]<\infty.$

Then the family of the laws of the $\mathcal{P}(\mathcal{X})$-valued random
variables $\bar{\mu}^{N}$ is tight.

###### Proof.

For positive constants $\alpha$ and $M$, we define a mapping $G_{\alpha}$ by

$\mathcal{X}\ni f\mapsto G_{\alpha}(f):=\sup_{0\leq s<t\leq
T}\frac{|f(t)-f(s)|}{|t-s|^{\alpha}}$

and set

$H_{M}:=\Big{\\{}\mu\in\mathcal{P}\big{(}\mathcal{X}\big{)}:\mu(G_{\alpha}(f))\leq
M\Big{\\}}.$

We claim that $H_{M}$ is tight (relatively compact) in
$\mathcal{P}\big{(}\mathcal{X}\big{)}$. Indeed, for each
$\tilde{M}\in(0,+\infty)$, since the domain $\mathcal{D}$ is bounded, we see
that the set
$B_{\tilde{M}}:=\big{\\{}f\in\mathcal{X},G_{\alpha}(f)\leq\tilde{M}\big{\\}}$
is relatively compact in $\mathcal{X}$ according to the Arzéla-Ascoli theorem.
On the other hand, by Chebyshev’s inequality, we have

$\displaystyle\sup_{\mu\in H_{M}}\mu(B^{c}_{\tilde{M}})$
$\displaystyle=\sup_{\mu\in
H_{M}}\mu\big{(}\\{f\in\mathcal{X},G_{\alpha}(f)>\tilde{M}\\}\big{)}$
$\displaystyle\leq\sup_{\mu\in H_{M}}\frac{\mu(G_{\alpha}(f))}{\tilde{M}}$
$\displaystyle\leq\frac{M}{\tilde{M}}.$

Then for any $\varepsilon>0$, there exists a constant $\tilde{M}$ depending on
$\varepsilon$ such that

$\sup_{\mu\in H_{M}}\mu(B^{c}_{\tilde{M}})\leq\varepsilon.$

This shows that $H_{M}$ is relatively compact in
$\mathcal{P}\big{(}\mathcal{X}\big{)}$. On the other hand, we have

(4.2) $\displaystyle\sup_{N\in\mathbb{N}}\mathbb{P}(\bar{\mu}^{N}\in
H^{c}_{M})$
$\displaystyle=\sup_{N\in\mathbb{N}}\mathbb{P}(\bar{\mu}^{N}(G_{\alpha}(f))>M)\leq\frac{\sup_{N\in\mathbb{N}}\mathbb{E}[\bar{\mu}^{N}(G_{\alpha}(f))]}{M}.$

Next, we show that $\mathbb{E}[\bar{\mu}^{N}(G_{\alpha}(f))]$ is uniformly
bounded with respect to $N$. From the controlled RSDE (LABEL:II-eq2.3), for
any $0\leq s\leq t\leq T$, we have

$\displaystyle\bar{X}^{i,N}(t)-\bar{X}^{i,N}(s)$ $\displaystyle=$
$\displaystyle\int^{t}_{s}b(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}r$
$\displaystyle+\int^{t}_{s}\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))h^{N}_{i}(r)\mathrm{d}r$
$\displaystyle+\int^{t}_{s}\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}W^{i}(r)$
$\displaystyle-\int_{s}^{t}\mathbf{n}(\bar{X}^{i,N}(r))\mathrm{d}|\bar{K}^{i,N}|(r).$

Applying the Itô’s formula to (4), we have

$\displaystyle|\bar{X}^{i,N}(t)-\bar{X}^{i,N}(s)|^{2}$ $\displaystyle=$
$\displaystyle
2\int_{s}^{t}\langle\bar{X}^{i,N}(r)-\bar{X}^{i,N}(s),b(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\rangle\mathrm{d}r$
$\displaystyle+2\int_{s}^{t}\langle\bar{X}^{i,N}(r)-\bar{X}^{i,N}(s),\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))h^{N}_{i}(r)\rangle\mathrm{d}r$
$\displaystyle+2\int_{s}^{t}\langle\bar{X}^{i,N}(r)-\bar{X}^{i,N}(s),\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}W^{i}(r)\rangle$
$\displaystyle+\int_{s}^{t}\|\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\|^{2}\mathrm{d}r$
$\displaystyle+2\int_{s}^{t}\langle\bar{X}^{i,N}(s)-\bar{X}^{i,N}(r),\mathbf{n}(\bar{X}^{i,N}(r))\rangle\mathrm{d}|\bar{K}^{i,N}|(r)$
$\displaystyle=:$ $\displaystyle
I^{s,t}_{1}+I^{s,t}_{2}+I^{s,t}_{3}+I^{s,t}_{4}+I^{s,t}_{5}.$

Now, it follows from the definition of $G_{\alpha}$ that

$\displaystyle\mathbb{E}[G^{2}_{\alpha}(\bar{X}^{i,N}(\cdot))]$
$\displaystyle=\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{|\bar{X}^{i,N}(t)-\bar{X}^{i,N}(s)|^{2}}{|t-s|^{2\alpha}}\Big{]}$
$\displaystyle\leq\sum_{i=1}^{5}\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{i}}{|t-s|^{2\alpha}}\Big{]}.$

Note that there exists a positive constant $\tilde{L}$ such that
$\mathcal{D}\subset B(0,\tilde{L})$.

Therefore, combining Assumption 2.1(A3), we obtain that

(4.4) $\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{1}}{|t-s|^{2\alpha}}\Big{]}\leq
4L\tilde{L}\mathbb{E}[\sup_{0\leq s<t\leq T}|t-s|^{1-2\alpha}],$

and

(4.5) $\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{4}}{|t-s|^{2\alpha}}\Big{]}\leq L^{2}\mathbb{E}[\sup_{0\leq
s<t\leq T}|t-s|^{1-2\alpha}].$

By Hölder inequality, we have

$\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{2}}{|t-s|^{2\alpha}}\Big{]}$ $\displaystyle\leq$
$\displaystyle 2\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{\int_{s}^{t}|\bar{X}^{i,N}(r)-\bar{X}^{i,N}(s)|\cdot|\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))h^{N}_{i}(r)|\mathrm{d}r}{|t-s|^{2\alpha}}\Big{]}$
$\displaystyle\leq$ $\displaystyle 4\tilde{L}\mathbb{E}\Big{[}\sup_{0\leq
s<t\leq
T}\frac{(\int_{s}^{t}\|\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\|^{2}\mathrm{d}r)^{\frac{1}{2}}\cdot(\int_{s}^{t}|h^{N}_{i}(r)|^{2}\mathrm{d}r)^{\frac{1}{2}}}{|t-s|^{2\alpha}}\Big{]}$
$\displaystyle\leq$ $\displaystyle 4L\tilde{L}\mathbb{E}\Big{[}\sup_{0\leq
s<t\leq
T}|t-s|^{\frac{1}{2}-2\alpha}\cdot(\int_{s}^{t}|h^{N}_{i}(r)|^{2}\mathrm{d}r)^{\frac{1}{2}}\Big{]}.$

Since the domain $\mathcal{D}$ is convex, we have $I^{s,t}_{5}\leq 0$ and
hence

(4.7) $\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{5}}{|t-s|^{2\alpha}}\Big{]}\leq 0.$

Set

$Z^{i}(t):=\int_{0}^{t}\langle\bar{X}^{i,N}(r),\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}W^{i}(r)\rangle,$

and

$Y^{i}(t):=\int_{0}^{t}\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}W^{i}(r)$

Then, we have

(4.8)
$I^{s,t}_{3}=2(Z^{i}(t)-Z^{i}(s))-2\langle\bar{X}^{i,N}(s),Y^{i}(t)-Y^{i}(s)\rangle.$

We now proceed to give an estimate for $|Z^{i}(t)-Z^{i}(s)|$. By the
Burkholder–Davis–Gundy inequality, for each $p>2$ and $0\leq s\leq t\leq T$,

$\displaystyle\mathbb{E}[|Z^{i}(t)-Z^{i}(s)|^{2p}]$ $\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{s}^{t}\langle\bar{X}^{i,N}(r),\sigma(r,\bar{X}^{i,N}(r),\bar{\mu}^{N}(r))\mathrm{d}W^{i}(r)\rangle\Big{|}^{2p}\Big{]}$
$\displaystyle\leq$ $\displaystyle C\tilde{L}^{2p}L^{2p}|t-s|^{p},$

where $C$ is a positive constant depending on $d_{1}$ and $p$. Applying the
Garsia-Rodemich-Rumsey lemma (see Corollary 1.2 in [20]), there exists a
random variable $A_{i}$ such that with probability one, for all $0\leq s\leq
t\leq T$,

$|Z^{i}(t,\omega)-Z^{i}(s,\omega)|\leq A_{i}(\omega)|t-s|^{\frac{p-2}{2p}},$

where $\mathbb{E}[A_{i}^{2p}]\leq C\tilde{L}^{2p}L^{2p}.$ By the similar
arguments, there exists also a random variable $B_{i}$ such that with
probability one, for all $0\leq s\leq t\leq T$,

$|Y^{i}(t,\omega)-Y^{i}(s,\omega)|\leq B_{i}(\omega)|t-s|^{\frac{p-2}{2p}},$

where $\mathbb{E}[B_{i}^{2p}]\leq CL^{2p}.$ It follows from (4.8) that

(4.9) $\displaystyle\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{I^{s,t}_{3}}{|t-s|^{2\alpha}}\Big{]}$ $\displaystyle\leq
C\mathbb{E}\Big{[}\sup_{0\leq s<t\leq
T}\frac{|Z^{i}(t)-Z^{i}(s)|+|Y^{i}(t)-Y^{i}(s)|}{|t-s|^{2\alpha}}\Big{]}$
$\displaystyle\leq C\mathbb{E}[(A_{i}+B_{i})\sup_{0\leq s<t\leq
T}|t-s|^{\frac{p-2}{2p}-2\alpha}].$

Let $p=4$ and $\alpha=\frac{1}{8}$. It follows from (4.4)–(4.9) that

$\displaystyle\mathbb{E}[G^{2}_{\frac{1}{8}}(\bar{X}^{i,N}(\cdot))]$
$\displaystyle\leq\Big{\\{}(4L\tilde{L}+L^{2})T^{\frac{3}{4}}+C\mathbb{E}[A_{i}+B_{i}]+4L\tilde{L}T^{\frac{1}{4}}\mathbb{E}[(\int_{0}^{T}|h^{N}_{i}(r)|^{2}\mathrm{d}r)^{\frac{1}{2}}]\Big{\\}}$
$\displaystyle\leq
C\big{(}1+\mathbb{E}[\int_{0}^{T}|h^{N}_{i}(r)|^{2}\mathrm{d}r]\big{)},$

for some positive constant $C$ independent of $N$.

Therefore, using the condition (4.1), for each $N\in\mathbb{N}$,

$\displaystyle\mathbb{E}[\bar{\mu}^{N}(G_{\frac{1}{8}}(\cdot))]$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}[G_{\frac{1}{8}}(\bar{X}^{i,N}(\cdot))]$
$\displaystyle\leq\frac{1}{N}\sum_{i=1}^{N}(1+\mathbb{E}[G^{2}_{\frac{1}{8}}(\bar{X}^{i.N}(\cdot))])$
$\displaystyle\leq\tilde{C},$

where $\tilde{C}$ is independent on $N$. In combination with (4.2), it follows
that for any $\varepsilon>0$, there exists a positive constant $M$ depending
on $\varepsilon$ such that

$\sup_{N\in\mathbb{N}}\mathbb{P}(\bar{\mu}^{N}\in H^{c}_{M})\leq\varepsilon.$

This implies that the family of the laws of
$\\{\bar{\mu}^{N},N\in\mathbb{N}\\}$ is tight in
$\mathcal{P}\big{(}\mathcal{P}\big{(}\mathcal{X}\big{)}\big{)}$. ∎

###  4.1. Laplace Upper Bound

In this subsection, we establish the Laplace upper bound (2.6) in Step 1,
namely, we will show that for any sequence $\\{h^{N},N\in\mathbb{N}\\}$ with
$h^{N}\in\mathcal{H}_{N}$,

(4.10)
$\displaystyle\begin{split}&\liminf_{N\mapsto\infty}\Big{\\{}\frac{1}{2}\mathbb{E}\Big{[}\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}|^{2}\mathrm{d}t\Big{]}+\mathbb{E}\big{[}F(\bar{\mu}^{N})\big{]}\Big{\\}}\\\
&\geq\inf_{\Theta\in\mathcal{P}_{\infty}}\Big{\\{}\frac{1}{2}\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}}(\mathrm{d}r)+F(\Theta_{\mathcal{X}})\Big{\\}}.\end{split}$

For a sequence $\\{h^{N}\in\mathcal{H}_{N},N\in\mathbb{N}\\}$, we may assume
that

$\mathbb{E}\left[\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}(t)|^{2}\mathrm{d}t\right]\leq
4\|F\|_{\infty},$

since otherwise the inequality (4.10) is automatic.

For each $N\in\mathbb{N}$, define a $\mathcal{P}(\mathcal{Z})$-valued random
variable by

(4.11) $Q^{N}_{\omega}(A\times R\times
B):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\bar{X}^{i,N}(\cdot,\omega)}(A)\cdot\delta_{\rho^{i,N}_{\omega}}(R)\cdot\delta_{W^{i}(\cdot,\omega)}(B)$

for any $A\times R\times B\in\mathcal{B}(\mathcal{Z})$ and $\omega\in\Omega$,
where $\bar{X}^{i,N}$ is the solution of equation (LABEL:II-eq2.3) with
$h^{N}=(h^{N}_{1},...,h^{N}_{N})$ and

$\rho^{i,N}_{\omega}(B\times
I):=\int_{I}\delta_{h^{N}_{i}(t,\omega)}(B)\mathrm{d}t,\quad
B\in\mathcal{B}(\mathbb{R}^{d_{1}}),\quad
I\in\mathcal{B}([0,T]),\quad\omega\in\Omega.$

######  Lemma 4.2.

The family $\\{Q^{N},N\in\mathbb{N}\\}$ of $\mathcal{P}(\mathcal{Z})$-valued
random variables is tight.

###### Proof.

By Lemma 4.1, we know that the family of the first marginals of
$\\{Q^{N},N\in\mathbb{N}\\}$ is tight. The rest of the proof is the same as
that of Lemma 5.1 in [5], we omit the details. ∎

The following result is crucial for the proof the Laplace upper bound.

######  Theorem 4.3.

Let $\\{Q^{N_{j}},j\in\mathbb{N}\\}$ be a weakly convergent subsequence of
$\\{Q^{N},N\in\mathbb{N}\\}$, with $Q$ being a
$\mathcal{P}(\mathcal{Z})$-valued random variable defined on some probability
space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$ such that
$Q^{N_{j}}\stackrel{{\scriptstyle j\rightarrow\infty}}{{\longrightarrow}}Q$ in
distribution. Then, $Q_{\omega}\in\mathcal{P}_{\infty}$ for
$\tilde{\mathbb{P}}$-almost all $\omega\in\tilde{\Omega}$, where
$\mathcal{P}_{\infty}$ is the family (defined in Section 2) of weak solutions
of the controlled reflected McKean-Vlasov equations:

(4.12) $\displaystyle\left\\{\begin{aligned}
&\mathrm{d}\bar{X}(t)=b(t,\bar{X}(t),\nu_{\Theta}(t))\mathrm{d}t+\int_{\mathbb{R}^{d_{1}}}\sigma(t,\bar{X}(t),\nu_{\Theta}(t))y\rho_{t}(\mathrm{d}y))\mathrm{d}t\\\
&\quad\quad\quad\quad+\sigma(t,\bar{X}(t),\nu_{\Theta}(t)))\mathrm{d}W(t)-\mathrm{d}\bar{K}(t),\\\
&|\bar{K}|(t)=\int^{t}_{0}\mathbf{1}_{\partial\mathcal{D}}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\quad\bar{K}(t)=\int^{t}_{0}\mathbf{n}(\bar{X}(s))\mathrm{d}|\bar{K}|(s),\\\
&\nu_{\Theta}(t)=\mbox{Law}(\bar{X}(t)),\quad\bar{X}(t)\in\overline{\mathcal{D}},\quad
t\in[0,T].\\\ \end{aligned}\right.$

###### Proof.

Set $I:=\\{N_{j},j\in\mathbb{N}\\}$ and write $(Q^{n})_{n\in I}$ for
$(Q^{N_{j}})_{j\in\mathbb{N}}$, then $Q^{n}\rightarrow Q$ in distribution.
Note that by Fatou’s Lemma (see Theorem A.3.12 in [13]) and Assumption
2.1(A2), it is easy to see that for $\tilde{\mathbb{P}}$-a.s.
$\omega\in\tilde{\Omega}$, $Q_{\omega}$ satisfies conditions (i) and (iii) in
the definition of $\mathcal{P}_{\infty}$. To complete the proof, we need to
prove that for $\tilde{\mathbb{P}}$-a.s. $\omega\in\tilde{\Omega}$,
$Q_{\omega}$ is a weak solution to equation (4.12)

According to Theorem 3.3, a probability measure
$\Theta\in\mathcal{P}(\mathcal{Z})$ with
$\Theta(\\{(\phi,r,w)\in\mathcal{Z}:w(0)=0\\})=1$ is a weak solution of
equation (4.12) if for all $f\in
C^{1,2,2}_{0}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$ with
$\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq 0$ on
$[0,T]\times\partial{\mathcal{D}}\times\mathbb{R}^{d_{1}}$, $M^{\Theta}_{f}$
(defined in (3.3) and (3.4)) is a sub-martingale under $\Theta$ with respect
to the canonical filtration $(\mathcal{G}_{t})$, i.e., the following holds

(4.13)
$\mathbb{E}_{\Theta}[\Psi\cdot(M^{\Theta}_{f}(t_{1})-M^{\Theta}_{f}(t_{0}))]\geq
0$

for all $t_{0},t_{1}\in[0,T]$ with $t_{0}\leq t_{1}$, and
$\mathcal{G}_{t_{0}}$-measurable $\Psi\in C_{b}^{+}(\mathcal{Z})$. Here
$C_{b}^{+}(\mathcal{Z})$ denotes the space of nonnegative functions in
$C_{b}(\mathcal{Z})$.

Note that it is enough to show that (4.13) holds for any countable collection
of times $t_{0}$ and $t_{1}$ that is dense in $[0,T]$, any countable
collection of $\Psi\in C_{b}^{+}(\mathcal{Z})$ that generates the
$\sigma$-algebras $\mathcal{G}_{t_{0}}$, and the countable collection of test
function $f$ that is dense in
$C^{1,2,2}_{0}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}})$
with $\langle\nabla_{x}f(t,x,z),n(x)\rangle\leq 0$ on
$[0,T]\times\partial\mathcal{D}\times\mathbb{R}^{d_{1}}$. Thus, there is a
countable collection $\mathcal{I}\subset[0,T]^{2}\times
C_{b}^{+}(\mathcal{Z})\times\\{f\in
C_{0}^{1,2,2}([0,T]\times\overline{\mathcal{D}}\times\mathbb{R}^{d_{1}}):\langle\nabla_{x}f(t,x,z),\mathbf{n}(x)\rangle\leq
0\ \mbox{on}\ [0,T]\times\partial\mathcal{D}\times\mathbb{R}^{d_{1}}\\}$ of
test parameters such that if (4.13) holds for all
$(t_{0},t_{1},\Psi,f)\in\mathcal{I}$, then $\Theta$ is a weak solution of
equation (4.12).

To verify the sub-martingale property of $M^{\Theta}_{f}$ with
$\Theta=Q_{\omega}$, $\omega\in\tilde{\Omega}$, we introduce a continuous
function with compact support. For each $B\in(0,\infty)$, let
$g_{B}:\mathbb{R}^{d_{1}}\rightarrow\mathbb{R}^{d_{1}}$ be a continuous
function with compact support satisfying $g_{B}(y)=y$ for $|y|\leq B$ and
$|g_{B}(y)|\leq|y|+1$ for every $y\in\mathbb{R}^{d_{1}}$. Also, define the
operator $\mathcal{A}^{\Theta,B}_{s}$ by replacing $y$ on the right side of
(3.4) with $g_{B}(y)$ and define $M^{\Theta,B}_{f}$ by replacing
$\mathcal{A}^{\Theta}_{s}$ in (3.3) with $\mathcal{A}^{\Theta,B}_{s}$.

For each $(t_{0},t_{1},\Psi,f)\in\mathcal{I}$, define
$\Phi=\Phi_{(t_{0},t_{1},\Psi,f)}$ by

(4.14)
$\mathcal{P}(\mathcal{Z})\ni\Theta\mapsto\Phi(\Theta):=\bigg{(}\mathbb{E}_{\Theta}[\Psi\cdot(M^{\Theta}_{f}(t_{1})-M^{\Theta}_{f}(t_{0}))]\bigg{)}^{-},$

where for each $a\in\mathbb{R}$, $a^{-}:=\max\\{-a,0\\}$. Similarly, for each
$B\in(0,\infty)$, define $\Phi_{B}$ by replacing $M^{\Theta}_{f}$ in (4.14)
with $M^{\Theta,B}_{f}$.

Similarly to the proof of Lemma 3.2 in [3], we obtain that (i) $\forall
B\in(0,\infty)$, $\Phi_{B}$ is continuous on $\mathcal{P}(\mathcal{Z})$, (ii)
$\sup_{n}{\mathbb{E}}|\Phi(Q^{n})-\Phi_{B}(Q^{n})|\rightarrow 0$, and (iii)
$\tilde{\mathbb{E}}|\Phi(Q)-\Phi_{B}(Q)|\rightarrow 0$ as
$B\rightarrow\infty$. For fixed $B>0$, $\Phi_{B}(Q^{n})\rightarrow\Phi_{B}(Q)$
in distribution. The above properties yield that
$\Phi(Q^{n})\rightarrow\Phi(Q)$ in distribution as well. By the definition of
$Q^{n}$, for $\omega\in\Omega$, we obtain

$\displaystyle\Phi(Q^{n}_{\omega})$ $\displaystyle=$
$\displaystyle\Big{(}\mathbb{E}_{Q^{n}_{\omega}}\left[\Psi\cdot\big{(}M^{Q^{n}_{\omega}}_{f}(t_{1})-M^{Q^{n}_{\omega}}_{f}(t_{0})\big{)}\right]\Big{)}^{-}$
$\displaystyle=$
$\displaystyle\Big{(}\frac{1}{n}\sum_{i=1}^{n}\Psi\big{(}(\bar{X}^{i,n}(.,\omega),\rho^{i,n}_{\omega},W^{i}(.,\omega))\big{)}$
$\displaystyle\cdot\big{(}f(t_{1},\bar{X}^{i,n}(t_{1},\omega),W^{i}(t_{1},\omega))-f(t_{0},\bar{X}^{i,n}(t_{0},\omega),W^{i}(t_{0},\omega))$
$\displaystyle-\int_{t_{0}}^{t_{1}}f_{s}(s,\bar{X}^{i,n}(s,\omega),W^{i}(s,\omega))\mathrm{d}s$
$\displaystyle-\int_{t_{0}}^{t_{1}}\mathcal{A}^{Q^{n}_{\omega}}_{s}(f)(s,\bar{X}^{i,n}(s,\omega),h_{i}^{n}(s,\omega),W^{i}(s,\omega))\mathrm{d}s\big{)}\Big{)}^{-},$

where $\mathcal{A}^{Q^{n}_{\omega}}$ is defined in (3.4) with
$\bar{\mu}^{n}_{\omega}$ in place of $\nu_{\Theta}$.

Applying the Itô’s formula, for each $i$, we have

$\displaystyle
f\big{(}t_{1},\bar{X}^{i,n}(t_{1}),W^{i}(t_{1})\big{)}-f\big{(}t_{0},\bar{X}^{i,n}(t_{0}),W^{i}(t_{0})\big{)}$
$\displaystyle\ \ \ \
-\int_{t_{0}}^{t_{1}}f_{s}(s,\bar{X}^{i,n}(s),W^{i}(s))\mathrm{d}s$
$\displaystyle\ \ \ \
-\int_{t_{0}}^{t_{1}}\mathcal{A}^{Q^{n}}_{s}(f)\big{(}s,\bar{X}^{i,n}(s),h_{i}^{n}(s),W^{i}(s)\big{)}\mathrm{d}s$
$\displaystyle=\int_{t_{0}}^{t_{1}}\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\sigma(s,\bar{X}^{i,n}(s),\bar{\mu}^{n}(s))\mathrm{d}W^{i}(s)\rangle$
$\displaystyle\ \ \ \
+\int_{t_{0}}^{t_{1}}\langle\nabla_{z}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}W^{i}(s)\rangle$
$\displaystyle\ \ \ \
-\int_{t_{0}}^{t_{1}}\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}\bar{K}^{i,n}_{s}\rangle.$

Keeping in mind that $\Psi$ is $\mathcal{G}_{t_{0}}$-measurable and non-
negative, we have

$\displaystyle\mathbb{E}[{\Phi^{2}(Q^{n})}]$ $\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\bigg{(}\Big{(}\mathbb{E}_{Q^{n}_{\omega}}\big{[}\Psi\cdot\big{(}M^{Q^{n}_{\omega}}_{f}(t_{1})-M^{Q^{n}_{\omega}}_{f}(t_{0})\big{)}\big{]}\Big{)}^{-}\bigg{)}^{2}\bigg{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\bigg{[}\bigg{(}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\int_{t_{0}}^{t_{1}}\Psi\cdot\big{(}\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\sigma(s,\bar{X}^{i,n}(s),\bar{\mu}^{n}(s))\mathrm{d}W^{i}(s)\rangle$
$\displaystyle+\langle\nabla_{z}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}W^{i}(s)\rangle-\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}\bar{K}^{i,n}_{s}\rangle\big{)}\Big{)}^{-}\bigg{)}^{2}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\bigg{[}\bigg{(}\Big{(}\frac{1}{n}\sum_{i=1}^{n}\int_{t_{0}}^{t_{1}}\Psi\cdot\big{(}\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\sigma(s,\bar{X}^{i,n}(s),\bar{\mu}^{n}(s))\mathrm{d}W^{i}(s)\rangle$
$\displaystyle+\langle\nabla_{z}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}W^{i}(s)\rangle\big{)}\Big{)}^{-}$
$\displaystyle+\Big{(}-\frac{1}{n}\sum_{i=1}^{n}\int_{t_{0}}^{t_{1}}\Psi\cdot\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}\bar{K}^{i,n}_{s}\rangle\Big{)}^{-}\bigg{)}^{2}\bigg{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n^{2}}\sum_{i=1}^{n}\mathbb{E}\bigg{[}\int_{t_{0}}^{t_{1}}\Big{|}\Psi\cdot\Big{(}\nabla_{z}f(s,\bar{X}^{i,n}(s),W^{i}(s))$
$\displaystyle+\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s))\sigma(s,\bar{X}^{i,n}(s),\bar{\mu}^{n}(s))\Big{)}\Big{|}^{2}\mathrm{d}s\bigg{]}$
$\displaystyle\stackrel{{\scriptstyle n\rightarrow\infty}}{{\longrightarrow}}$
$\displaystyle 0,$

where we have used the fact that

$-\frac{1}{n}\sum_{i=1}^{n}\int_{t_{0}}^{t_{1}}\Psi\cdot\langle\nabla_{x}f(s,\bar{X}^{i,n}(s),W^{i}(s)),\mathrm{d}\bar{K}^{i,n}_{s}\rangle\geq
0.$

Since $\mathcal{I}$ is countable, it follows that for
$\tilde{\mathbb{P}}$-almost all $\omega\in\tilde{\Omega}$, all
$(t_{0},t_{1},\Psi,f)\in\mathcal{T}$,

$\mathbb{E}_{Q_{\omega}}[\Psi\cdot(M^{Q_{\omega}}_{f}(t_{1})-M^{Q_{\omega}}_{f}(t_{0})]\geq
0,$

which implies that for $\tilde{\mathbb{P}}$-almost all
$\omega\in\tilde{\Omega}$, $Q_{\omega}$ is a weak solution of equation (4.12)
and we complete the proof of the theorem. ∎

Now we are ready to complete the proof of the upper bound (2.6). By the
definition of $Q^{N}$, the left-hand side of (2.6) can be rewritten as

$\displaystyle\frac{1}{2}\mathbb{E}\bigg{[}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{T}|h_{i}^{N}(t)|^{2}\mathrm{d}t\bigg{]}+\mathbb{E}[F(\bar{\mu}^{N})]$
$\displaystyle=$
$\displaystyle\int_{\Omega}\bigg{[}\int_{\mathcal{R}_{1}}\bigg{(}\frac{1}{2}\int_{\mathcal{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\bigg{)}Q^{N}_{\omega,\mathcal{R}}(\mathrm{d}r)+F(Q^{N}_{\omega,\mathcal{X}})\bigg{]}\mathbb{P}(\mathrm{d}\omega).$

Noticing that the function $F$ is bounded and continuous, by Lemma 4.2 and
Theorem 4.3, Fatou’s Lemma, the Laplace upper bound is immediate.

###  4.2. Laplace lower Bound

In this subsection, we proves the Laplace lower bound (2.7) in Step 2. The
proof is very similar to the ones in [3] and [5]. For completeness, we give a
sketch.

Let $\Theta\in\mathcal{P}_{\infty}$. We will construct a sequence
$\\{h^{N},N\in\mathbb{N}\\}$ with $h^{N}\in\mathcal{H}_{N}$ on a common
stochastic basis satisfying (2.7):

$\displaystyle\limsup_{N\to\infty}\Big{\\{}\frac{1}{2}\mathbb{E}\big{[}\frac{1}{N}\sum^{N}_{i=1}\int^{T}_{0}|h^{N}_{i}|^{2}\mathrm{d}t\big{]}+\mathbb{E}[F(\bar{\mu}^{N})]\Big{\\}}$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\int_{\mathcal{R}_{1}}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}r(\mathrm{d}y\times\mathrm{d}t)\Theta_{\mathcal{R}}(\mathrm{d}r)+F(\Theta_{\mathcal{X}}).$

Recall $(\bar{X},\rho,W)$ from Section 3, which is the canonical process on
$(\mathcal{Z},\mathcal{B}(\mathcal{Z}),(\mathcal{G}_{t}),\Theta)$ and
$\bar{v}$ from Section 2, which is the map from $\mathcal{Z}$ to
$\mathcal{Z}^{0}$. Then we can disintegrate $\Theta\circ\bar{v}^{-1}$ as

$\Theta\circ\bar{v}^{-1}:=\nu_{0}(\mathrm{d}\phi_{0})\Theta_{\mathcal{W}}(\mathrm{d}w)\bar{\lambda}(\mathrm{d}r|\phi_{0},w).$

Define a product space $(\Omega_{\infty},\mathcal{F}^{\infty})$ as the
countably infinite product of
$(\mathcal{R}_{1}\times\mathcal{W},\mathcal{B}(\mathcal{R}_{1}\times\mathcal{W}))$
and write $(r^{\prime},w^{\prime})$ as the element of $\Omega_{\infty}$, where
$r^{\prime}=(r_{1},r_{2},\ldots)$ and $w^{\prime}=(w_{1},w_{2},\ldots)$. For
each $i\in\mathbb{N}$, define

$W^{i,\infty}(t,(r^{\prime},w^{\prime})):=w_{i}(t),\ \
\rho^{i,\infty}(r^{\prime},w^{\prime}):=r_{i},$

and

$h^{\infty}_{i}(t,(r^{\prime},w^{\prime})):=\int_{\mathbb{R}^{d_{1}}}y\rho^{i,\infty}_{(r^{\prime},w^{\prime}),t}(\mathrm{d}y),\
(r^{\prime},w^{\prime})\in\Omega_{\infty},\ t\in[0,T],$

where $\rho^{i,\infty}_{(r^{\prime},w^{\prime}),t}$ is the derivative measure
of $\rho^{i,\infty}(r^{\prime},w^{\prime})$ at time $t$. Furthermore, for each
$N\in\mathbb{N}$, define $\mathbb{P}_{N}\in\mathcal{P}(\Omega_{\infty})$ as

$\mathbb{P}_{N}(\mathrm{d}r,\mathrm{d}w):=\bigotimes_{i=1}^{N}\Theta_{\mathcal{W}}(\mathrm{d}w_{i})\bar{\lambda}(\mathrm{d}r_{i}|x^{i,N},w_{i})\bigotimes_{i=N+1}^{\infty}\Theta\circ(\rho,W)^{-1}(\mathrm{d}r_{i},\mathrm{d}w_{i}),$

and define a
$\mathcal{P}(\overline{\mathcal{D}}\times\mathcal{R}_{1}\times\mathcal{W})$-valued
random variable by

$\lambda^{N}(A\times R\times
B):=\frac{1}{N}\sum_{i=1}^{N}\delta_{x^{i,N}}(A)\delta_{\rho^{i,\infty}}(R)\delta_{W^{i,\infty}}(B),$

for any $A\times R\times
B\in\mathcal{B}(\overline{\mathcal{D}}\times\mathcal{R}_{1}\times\mathcal{W})$,
where $\\{x^{i,N}\\}$ are the initial values.

By construction, under $\mathbb{P}_{N}$,
$\\{W^{i,\infty},i\in\\{1,\ldots,N\\}\\}$ are independent Wiener processes. By
Assumption 2.1(A2), we obtain that

(4.15)
$\mathbb{P}_{N}\circ(\lambda^{N})^{-1}\rightarrow\delta_{\Theta\circ\bar{v}^{-1}}$

and

$\displaystyle\limsup_{N\rightarrow\infty}\mathbb{E}_{\mathbb{P}_{N}}\bigg{[}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{T}|h^{\infty}_{i}(t)|^{2}\mathrm{d}t\bigg{]}$
$\displaystyle=$
$\displaystyle\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}\int_{\mathcal{R}_{1}\times\mathcal{W}}\int_{0}^{T}|\int_{\mathbb{R}^{d_{1}}}yr_{t}(\mathrm{d}y)|^{2}\mathrm{d}t\bar{\lambda}(\mathrm{d}r|x^{i,N},w)\Theta_{\mathcal{W}}(\mathrm{d}w)$
$\displaystyle=$
$\displaystyle\mathbb{E}_{\Theta}\bigg{[}\int_{0}^{T}|\int_{\mathbb{R}^{d_{1}}}y\rho_{t}(\mathrm{d}y)|^{2}\mathrm{d}t\bigg{]}\leq\mathbb{E}_{\Theta}\bigg{[}\int_{\mathbb{R}^{d_{1}}\times[0,T]}|y|^{2}\rho(\mathrm{d}y\times\mathrm{d}t)\bigg{]}<\infty.$

In analogy with (4.11), for each $N\in\mathbb{N}$, define a
$\mathcal{P}(\mathcal{Z})$-valued random variable by

$\tilde{Q}^{N}(A\times R\times
B):=\frac{1}{N}\sum_{i=1}^{N}\delta_{\tilde{X}^{i,N}}(A)\cdot\delta_{\rho^{i,\infty}}(R)\cdot\delta_{W^{i,\infty}}(B),$

for any $A\times R\times B\in\mathcal{B}(\mathcal{Z})$, where
$(\tilde{X}^{1,N},\ldots,\tilde{X}^{N,N})$ is the solution of the system
(LABEL:II-eq2.3) with $h^{N}=(h^{\infty}_{1},\ldots,h^{\infty}_{N})$ and
$\tilde{\mu}^{N}(t)=\frac{1}{N}\sum_{i=1}^{N}\delta_{\tilde{X}^{i,N}(t)}$ for
each $t\in[0,T]$.

By (4.2) and Lemma 4.1, we know that $\\{\tilde{Q}^{N},N\in\mathbb{N}\\}$ is
tight. Let $\tilde{Q}$ be a limit point of
$\\{\tilde{Q}^{N},N\in\mathbb{N}\\}$ defined on some probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$. According to
Theorem 4.3 and its proof, $\tilde{Q}_{\omega}\in\mathcal{P}_{\infty}$ for
$\tilde{\mathbb{P}}$-almost all $\omega\in\tilde{\Omega}$. Moreover, by
(4.15), we derive that for $\tilde{\mathbb{P}}$-almost all
$\omega\in\tilde{\Omega}$,
$\tilde{Q}_{\omega}\circ\bar{v}^{-1}=\Theta\circ\bar{v}^{-1}$. Therefore, by
Assumption 2.1(A4), for $\tilde{\mathbb{P}}$-almost all
$\omega\in\tilde{\Omega}$, $\tilde{Q}_{\omega}=\Theta$. In combination with
(4.2) and $F$ in (2.7) is bounded and continuous, the Laplace lower bound is
immediate.

Acknowledgement. This work is partially supported by National Key R&D Program
of China (No.2022YFA1006001), National Natural Science Foundation of China
(Nos. 12131019, 11971456, 11721101)

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