# Swarming models with specular boundary condition and environmental noise

Razvan C. Fetecau  Department of Mathematics, Simon Fraser University, 8888
University Dr., Burnaby, BC V5A 1S6, Canada<EMAIL_ADDRESS>,  Hui Huang
Institute of Mathematics and Scientific Computing, Karl-Franzens-Universität
Graz , Universitätspl. 3, 8010 Graz, Austria<EMAIL_ADDRESS>and
Jinniao Qiu  Department of Mathematics and Statistics, University of Calgary,
2500 University Drive NW Calgary, AB, Canada, T2N 1N4<EMAIL_ADDRESS>
###### Abstract.

We investigate a general class of models for swarming/self-collective
behaviour in domains with boundaries. The model is expressed as a stochastic
system of interacting particles subject to both reflecting boundary condition
and common environmental noise. We rigorously derive its corresponding
macroscopic mean-field equation, which is a new type of stochastic partial
differential equation due to the presence of common noise. The approach relies
on a compactness argument, in which we first establish the tightness of the
empirical measures associated with the particle system and then demonstrate
that the time marginal of the limit measure is a solution to the mean-field
equation.

Keywords: Mean-field limit, swarming models, tightness, weak convergence,
stochastic partial differential equations

##  1\. Introduction

Large systems of interacting particles are widely used in mathematical models
of collective behaviour (or swarming), an area that has attracted a great deal
of attention in recent years. In such models, the particles (also referred to
as individuals or agents) can represent a wide variety of entities such as
biological organisms (flocks of birds, schools of fish, aggregation of
bacteria) [Couzin_etal, carrillo2010particle], nano-particles [HoPu2005],
swarming robots [JiEgerstedt2007] or a group of opinions [MotschTadmor2014].
Understanding the emergence of self-collective/coordinated behaviour from
basic interaction principles has been a key focus of research in disciplines
such as biology, engineering, physics or social sciences
[Theraulaz:Deneubourg, Cucker:Smale, PhysD_pref2013].

Aggregation models typically consider one or several of the three basic
interaction mechanisms: repulsion, attraction, and orientation (or alignment),
where these interactions are assumed to act in different zones with respect to
a reference individual; e.g., short-range repulsion, long-range attraction and
intermediate range alignment [Couzin_etal, carrillo2010particle]. The
qualitative behaviour of the solutions and its equilibria depend entirely on
the specific interaction forces (and the specific parameters) that are being
considered. The various models that have been proposed and studied in recent
literature have demonstrated emergence of self-organized behaviours such as
flocking, milling or double-milling [Cucker:Smale, Ha:Tadmor, DOrsogna_etal,
CarrilloDOrsognaPanferov].

Particularly relevant to our paper is the interplay between particle-based
(Lagrangian) models and mean-field models with (stochastic) partial
differential equation descriptions. While for the first class of models the
trajectories of all individuals are being tracked, mean-field models simply
govern the evolution of a population density field. There exists a vast amount
of theoretical and applied research on both classes of models, as well as on
the connection between the two via the mean-field limit. We refer to the
review articles [carrillo2014derivation, carrillo2017review] and the
references therein for expanded discussions on these aspects.

In this paper we consider both descriptions of aggregation behaviour: a system
of stochastic differential equations (SDE) for the particle evolution, as well
as its corresponding mean-field model, representing a statistical description
of the interacting system as the number of particles approaches infinity.
Specifically, we consider a class of the so called second-order aggregation
models (derived from Newton’s second law) in domains with boundaries. We note
that in spite of the extensive literature on aggregation models in free space,
the role of boundaries in collective behaviour has remained largely
unaddressed [BeTo2011, wu2015nonlocal, CarrilloSlepcevWu2016,
fetecau2017swarm, fetecau2017swarming, fetecau2022zero,
fetecau2019propagation, MeFe2020]. The presence of boundaries is important in
many realistic physical/biological applications of swarming models, where the
environment involves an obstacle or an impenetrable wall that limits the
movement of individuals (e.g., tank walls in experiments on collective
behaviour of fish [Couzin-fish] or the ground in models for locust swarming
[ToDoKeBe2012]).

For the classical particle system of aggregation models, which is only subject
to idiosyncratic noises that are independent from one particle to another, the
effect of the idiosyncratic noises averages out, leading to the deterministic
nature of the mean-field equation. In addition to such idiosyncratic noises
and the specular boundary condition, this paper studies particle systems that
also allow for common/environmental noises. In such case, the limiting density
function satisfies a stochastic partial differential equation (SPDE) over a
bounded domain, which is new to the best of our knowledge. Common
environmental noises (such as temperature, light and sound) together with
specular boundary conditions are intrinsic to realistic settings such as
culturing bacteria. We refer to [bensoussan2013mean, carmona2016mean,
coghi2016propagation, choi2019cucker, huang2021microscopic] for different
models with common noise in the whole space.

We introduce now the stochastic aggregation model investigated in this paper.
Throughout this paper, unless stated otherwise, let
$(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\geq 0},\mathbb{P})$ be a filtered
probability space satisfying usual conditions on which are defined
$d$-dimensional Wiener processes $\\{(B_{t}^{i})_{t\geq 0}\\}_{i=1}^{\infty}$
that are independent of each other, as well as the Wiener process
$(\overline{W}_{t})_{t\geq 0}$. For each integer $N>0$, denote by

$(\mathcal{X}_{t}^{N},\mathcal{V}_{t}^{N}):=((X_{t}^{1,N},V_{t}^{1,N})\cdots,(X_{t}^{N,N},V_{t}^{N,N}))\in\mathbb{R}^{2dN}$
(1.1)

the positions and velocities of $N$ particles in $\mathbb{R}^{d}$. Consider a
domain $D\subset\mathbb{R}^{d}$ and the evolution of the particle system
governed by the following stochastic differential equations with reflecting
boundary conditions and common noise:

$\displaystyle\begin{cases}X_{t}^{i,N}=X_{0}^{i,N}+\int_{0}^{t}V_{s}^{i,N}\,\mathrm{d}s,\quad
i=1,\cdots,N\,,\\\\[2.0pt]
V_{t}^{i,N}=V_{0}^{i,N}+\int_{0}^{t}\frac{1}{N}\sum\limits_{j=1}^{N}H(X_{s}^{i,N}-X_{s}^{j,N},V_{s}^{i,N}-V_{s}^{j,N})\,\mathrm{d}s+\sqrt{2\sigma}B_{t}^{i}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+K_{t}^{i,N}\,,\\\
K_{t}^{i,N}=-\sum\limits_{0<s\leq t}2(V_{s^{-}}^{i,N}\cdot
n(X_{s}^{i,N}))n(X_{s}^{i,N})\textbf{1}_{\\{X_{s}^{i,N}\in\partial
D\\}}.\end{cases}$ (1.2)

Here, $H:\mathbb{R}^{2d}\rightarrow\mathbb{R}$ is a suitable function which
models the pairwise interactions between particles,
$\sigma,\overline{\sigma}>0$ are the noise strengths, $\textbf{1}_{A}$ denotes
the characteristic function of a set $A$, and $n$ is the outward normal unit
vector of $\partial D$.

The boundary we consider in this paper can be represented as the 0-level set
of a signed distance function $\ell$ with $\ell(x)=\mbox{dist}(x,\partial D)$.
This means that

$\partial D=\left\\{x\in{\mathbb{R}}^{d}:~{}\ell(x)=0\right\\}\,.$ (1.3)

We also assume for simplicity that $\ell>0$ in the interior of $D$ and
$\ell<0$ in the exterior. Moreover the gradient $\nabla\ell$ is then the
inward unit normal on $\partial D$ wherever $\ell$ is defined, namely
$\nabla\ell(x)=-n(x)$ when $x\in\partial D$. For example, when $D$ is a disk
with radius $1$ and center $0$, it can be represented by using the distance
function $\ell(x)=1-|x|$. The càdlàg (right-continuous with left limit)
process $(K_{t}^{i})_{t\geq 0}$ confines the component $(X_{t}^{i})_{t\geq 0}$
in $\overline{D}$ by reflecting the velocity of the outgoing particles. The
initial data $\\{(X_{0}^{i},V_{0}^{i})\\}_{i=1}^{N}$ are independently,
identically, distributed (i.i.d.) with a common given probability density
$f_{0}(x,v)$. Throughout this study we will be assuming that $H$ is uniformly
bounded and continuous, which guarantees the existence of a weak solution to
the particle system (1.2) (see Theorem 2.4).

Set up in free space ($D=\mathbb{R}^{d}$) and without common noise, system
(1.2) includes several classes of aggregation models extensively studied in
the literature. Specifically, for $H(x)=\nabla U(x)$ (so $H$ is only spatially
dependent), with $U$ being an interaction potential modelling attraction and
repulsion, system(1.2) is a variant of the model from [DOrsogna_etal,
Chuang_etal], with additional noise modelled by the Wiener process. Models in
this class capture a variety of swarming behaviours. Also, for $H(x,v)=a(x)v$,
with $a(x)$ playing the role of a communication rate, system (1.2) represents
the popular flocking model first introduced by Cucker and Smale [Cucker:Smale]
(for the model with noise see [Cucker:Mordecki, HaLeeLevy2009]). Here $H$
models alignment of a reference individual’s velocity with the velocities of
the other agents. The key interest in such models is the emergence of flocking
behaviour through this velocity averaging process [Ha:Tadmor,
CarrilloFornasierRosadoToscani].

The main objective of this paper is to investigate the mean-field limit for
system (1.2). To the best of our knowledge this is the first rigorous
investigation of swarming models of type (1.2) in domains with boundaries. We
also note that various numerical studies in such domains, but without common
noise, are available, see for instance [Armbruster2017a, Armbruster2017b].
Specifically, we consider the following stochastic kinetic equation that
corresponds to model (1.2) in the limit $N\to\infty$:

$\displaystyle
df=\left[(\sigma+\overline{\sigma})\triangle_{v}f-v\cdot\nabla_{x}f-\nabla_{v}\cdot((H\ast
f)f)\right]\,dt-\sqrt{2\overline{\sigma}}\nabla_{v}f\,d\overline{W}_{t},\,(t,x,v)\in(0,T]\times
D\times\mathbb{R}^{d},$ (1.4a) $\displaystyle f(0,x,v)=f_{0}(x,v),$ (1.4b)
$\displaystyle\gamma(f)(t,x,v)=\gamma(f)\big{(}t,x,v-2(v\cdot
n(x))n(x)\big{)},\quad(x,v)\in\partial D\times\mathbb{R}^{d},\quad t>0,$
(1.4c)

where $\gamma(f)$ stands for the trace of $f$ (see Definition 2.1) and $\ast$
stands for the convolution with respect to $(x,v)\in D\times\mathbb{R}^{d}$,.
i.e.,

$H\ast f(t,x,v)=\iint_{D\times{\mathbb{R}}^{d}}H(x-y,v-w)f(t,y,w)\,{\rm
d}y\,\mathrm{d}w\,.$ (1.5)

The approach in our work is based on a tightness argument which is technically
different from [huang2022mean, li2019mean, liu2016propagation] due to the
presence of the specular boundary conditions and common noises; in particular,
the specular boundary conditions make it hard to analyze the stability of the
solution to system (1.2). Fix the time horizon $[0,T]$. We first prove that
the sequence of empirical measures $\\{f^{N}\\}_{N\geq 2}$, where
$f^{N}:=\frac{1}{N}\sum\limits_{i=1}^{N}\delta_{(X^{i,N},V^{i,N})}$ are
$\mathcal{P}(\mathcal{C}([0,T];\overline{D})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d}))$-valued
variables, is tight. Here we denote by ${\mathbb{D}}([0,T];{\mathbb{R}}^{d})$
the space of all ${\mathbb{R}}^{d}$-valued càdlàg functions equipped with the
Skorokhod topology. For $z\in{\mathbb{D}}([0,T];{\mathbb{R}}^{d})$, $\Delta
z_{t}$ and $z_{t^{-}}$ denote the jump and the left-hand limit of $z$ at time
$t$. Then Prokhorov’s theorem indicates that there exists a subsequence of
$\\{f^{N}\\}_{N\geq 2}$ converging in law to a random measure $f$. The main
difficulty lies in the identification of the time marginal of the limit
measure $f$ as a weak solution to the mean-field SPDE (1.4). Our approach
combines certain probabilistic and stochastic arguments with some analysis on
(S)PDEs. The main result, which combines Theorems 4.2 and 4.4, can be
summarized as follows.

######  Theorem 1.1.

Under certain assumptions, for each $N\geq 2$, assume that
$\\{(X_{t}^{i,N},V_{t}^{i,N})_{t\in[0,T]}\\}_{i=1}^{N}$ is a solution to the
particle system (1.2) with $f_{0}^{\otimes N}$-distributed initial data
$\\{(X_{0}^{i},V_{0}^{i})\\}_{i=1}^{N}$. Then the sequence of empirical
measures
$\left\\{f^{N}=\frac{1}{N}\sum\limits_{i=1}^{N}\delta_{(X^{i,N},V^{i,N})}\right\\}_{N\geq
2}$ is tight, and any limit (denoted by $f$) of $\\{f^{N}\\}_{N\geq 2}$ is a
stochastic probability measure on the path space
$\mathcal{C}([0,T];\overline{D})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})$.
Moreover the time marginal of $f$ is a weak solution to the mean-field SPDE
(1.4).

Very pertinent to our paper are the works of Bossy and Jabir
[bossy2011confined, bossy2015lagrangian, bossy2018particle] on Lagrangian
stochastic equations in domains with boundaries. Of particular interest is the
comprehensive analysis of a related Lagrangian stochastic model presented in
[bossy2015lagrangian]. The specular boundary conditions utilized in our paper
are similar to those employed in [bossy2015lagrangian]. However, our approach
is distinct in that the martingale method employed by Bossy and Jabir is
limited to a mean-field kinetic equation that is deterministic, and hence, it
is ineffective for our purpose, due to the presence of the additional common
noise $\overline{W}$, which results in mean-field limits characterized by the
stochastic kinetic equation (1.4).

The summary of the paper is as follows. In Section 2 we provide preliminaries
and background. In particular, we provide background on the existence of
solutions to the particle system (1.2); see Theorem 2.4. Then in Section 3 we
prove the tightness of the empirical measures $\\{f^{N}\\}_{N\geq 2}$
associated to the particle system (1.2) through the Aldous criteria; see
Theorem 3.1. Lastly in Section 4, we verify that the limit measure $f$ of a
subsequence of $\\{f^{N}\\}_{N\geq 2}$ is a solution to the mean-field SPDE
(1.4); see Theorems 4.2-4.4.

.

##  2\. Preliminaries

In this section we present some notations and definitions, along with some
background needed for subsequent sections.

### Notations

Throughout this paper, we will denote any generic constants as $C$, which may
change from line to line. For all $t\in(0,T]$ we define the phase space
$Q_{t}:=(0,t)\times D\times\mathbb{R}^{d}$ and the sets:

$\displaystyle\Sigma^{+}:=\\{(x,v)\in\partial D\times\mathbb{R}^{d}:~{}v\cdot
n(x)>0\\},$ $\displaystyle\Sigma_{t}^{+}:=(0,t)\times\Sigma^{+},$
$\displaystyle\Sigma^{-}:=\\{(x,v)\in\partial D\times\mathbb{R}^{d}:~{}v\cdot
n(x)<0\\},$ $\displaystyle\Sigma_{t}^{-}:=(0,t)\times\Sigma^{-},$ (2.1)
$\displaystyle\Sigma^{0}:=\\{(x,v)\in\partial D\times\mathbb{R}^{d}:~{}v\cdot
n(x)=0\\},$ $\displaystyle\Sigma_{t}^{0}:=(0,t)\times\Sigma^{0}.$

Furthermore, we denote

$\Sigma:=\Sigma^{+}\cup\Sigma^{0}\cup\Sigma^{-}=\partial
D\times\mathbb{R}^{d},$

and $\Sigma_{t}:=(0,t)\times\partial D\times\mathbb{R}^{d}\,.$ Denoting by
$\,\mathrm{d}\sigma_{\partial D}$ the surface measure on $\partial D$, we set
the product measure on $\Sigma_{t}$:

$\displaystyle\,\mathrm{d}\lambda_{\Sigma_{t}}:=\,{\rm
d}s\times\,\mathrm{d}\sigma_{\partial D}(x)\times\,{\rm d}v\,.$

Let $\mathbb{B}$ be a Banach space equipped with norm
$\|\cdot\|_{\mathbb{B}}$. For $p\in[1,\infty]$, denote by
$\mathcal{L}_{\mathscr{F}}^{p}({\mathbb{B}})$ the collection of all the
${\mathbb{B}}$-valued, $(\mathscr{F}_{t})_{t\in[0,T]}$-adapted processes
$\\{\mathcal{X}_{t}\\}_{t\in[0,T]}$ such that

$\|\mathcal{X}\|_{\mathcal{L}_{\mathscr{F}}^{p}({\mathbb{B}})}:=\|\mathcal{X}\|_{L^{p}(\Omega\times[0,T];\mathbb{B})}<\infty.$

Obviously,
$(\mathcal{L}_{\mathscr{F}}^{p}({\mathbb{B}}),\|\cdot\|_{\mathcal{L}_{\mathscr{F}}^{p}({\mathbb{B}})})$
is a Banach space.

Next we define the trace function $\gamma(f)$ at the boundary of the solution
process $(f_{t})_{t\in[0,T]}$ to SPDE (1.4).

######  Definition 2.1.

(Trace) Given any function $f\in L^{1}([0,T]\times
D\times\mathbb{R}^{d};\,\mathbb{R})$, its trace $\gamma(f)$ is a function
defined on $[0,T]\times\partial D\times\mathbb{R}^{d}$ that satisfies

$\displaystyle\int_{(0,t)\times\partial
D\times\mathbb{R}^{d}}\gamma(f)(s,x,v)\varphi(s,x,v)\,\mathrm{d}\lambda_{\Sigma_{t}}=\lim_{\delta\rightarrow
0^{+}}\frac{1}{\delta}\int_{0<\ell(x)\leq\delta}\int_{(0,t)\times\mathbb{R}^{d}}f(s,x,v)\varphi(s,x,v)\mathrm{d}x\mathrm{d}s\mathrm{d}v\,,$
(2.2)

for a.e. $t\in(0,T)$ and all
$\varphi\in\mathcal{C}_{c}^{2}\left((0,t)\times\overline{D}\times{\mathbb{R}}^{d}\right)$.

######  Remark 2.1.

Notice that if
$f\in\mathcal{C}([0,T]\times\overline{D}\times\mathbb{R}^{d};\,\mathbb{R})$,
then $\gamma(f)(t,\partial D,{\mathbb{R}}^{d})=f(t,\partial
D,{\mathbb{R}}^{d})$, which is the classical notation of the restriction of a
function to the boundary of its domain.

Then we introduce the definition of solutions to SPDE (1.4):

######  Definition 2.2.

(Weak solution) For any $T>0$, a family of random functions
$\\{f_{t}(\omega)\,:\,t\in[0,T],\omega\in\Omega\\}$ lying in
$\mathcal{L}^{\infty}_{\mathscr{F}}(L^{1}(D\times\mathbb{R}^{d}))$ is a weak
solution to (1.4) if for all $t\in(0,T]$ it holds that

$\displaystyle\iint_{D\times\mathbb{R}^{d}}\varphi(x,v)f(t,x,v)\,{\rm
d}x\,{\rm d}v-\iint_{D\times\mathbb{R}^{d}}\varphi(x,v)f_{0}(x,v)\,{\rm
d}x\,{\rm d}v$
$\displaystyle\qquad=\int_{Q_{t}}\left(v\cdot\nabla_{x}\varphi+\nabla_{v}\varphi\cdot
H\ast f+(\sigma+\overline{\sigma})\triangle_{v}\varphi\right)f(s,x,v)\,{\rm
d}x\,{\rm d}v\,{\rm d}s$
$\displaystyle\qquad\quad+\int_{Q_{t}}\sqrt{2\overline{\sigma}}\nabla_{x}\varphi
f(s,x,v)\,\,{\rm d}x\,{\rm d}v\cdot\mathrm{d}\overline{W}_{s}\quad\mbox{ a.s.
}$ (2.3)

for all $\varphi\in C_{c}^{\infty}(D\times{\mathbb{R}}^{d})$, and it satisfies
the boundary condition

$\gamma(f)(s,x,v)=\gamma(f)(s,x,v-2(v\cdot
n(x))n(x)),\qquad\lambda_{\Sigma_{T}}-a.e.\mbox{ on }\Sigma_{T}\quad\mbox{
a.s.}$ (2.4)

The set of hypotheses for the main theorem below is summarized in the
following:

######  Assumption 1.

We make the following assumptions on the initial density $f_{0}$, the
interaction kernel $H$ and the domain $D$:

  1. (1)

The initial data $\\{(X_{0}^{i},V_{0}^{i})\\}_{i=1}^{N}$ is assumed to be
i.i.d. and distributed according to a given initial density
$f_{0}\in\mathcal{P}_{2}(D\times{\mathbb{R}}^{d})\cap
L^{1}(D\times{\mathbb{R}}^{d})$, which has its support in
$D\times\mathbb{R}^{d}$.

  2. (2)

$H:~{}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ is a bounded
continuous function.

  3. (3)

$D$ is an open bounded subset of ${\mathbb{R}}^{d}$ with $\mathcal{C}^{3}$
boundary $\partial D$. Moreover, there exists an open neighborhood $\Gamma$ of
$\partial D$ such that $\ell\in\mathcal{C}^{3}(\Gamma)$.

###  2.1. Well-posedness for the microscopic particle system (1.2)

Generally, there are two different notions of solutions for SDEs: weak
solution and strong solution. The main difference between them is that a
strong solution satisfies the SDE with a given Wiener process on some given
filtered probability space, whereas for weak solutions we have to choose the
Wiener processes and the filtered probability space as part of the solution
satisfying the SDE. We refer to [karatzas2012brownian, Chapter 5] for detailed
discussions on weak and strong solutions.

For the existence and uniqueness of a strong solution to SDE (1.2), we need
the well-posedness of a sequence of hitting times for the linear confined
Langevin process.

######  Theorem 2.3.

([bossy2015lagrangian, Theorem 2.1]) Under Assumption 1, for each $T>0$ and
$B=B^{i}$, $i=1,2,\dots$, there exists a unique strong solution to the
following linear confined Langevin equation

$\displaystyle\begin{cases}Y_{t}=y_{0}+\int_{0}^{t}U_{s}\,\mathrm{d}s,\quad
U_{t}=u_{0}+\sqrt{2\sigma}B_{t}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+\widetilde{K}_{t},\\\\[2.0pt]
\widetilde{K}_{t}=-\sum\limits_{0<s\leq t}2(U_{s^{-}}\cdot
n(Y_{s}))n(Y_{s})\textbf{1}_{\\{Y_{s}\in\partial D\\}},\quad\forall
t\in[0,T],\end{cases}$ (2.5)

with any
$(y_{0},u_{0})\in(D\times\mathbb{R}^{d})\cup(\Sigma\setminus\Sigma^{0})$.
Furthermore the sequence of hitting times

$\tau_{n}=\inf\\{\tau_{n-1}<t\leq T;\,Y_{t}\in\partial D\\},\mbox{ for }n\geq
1,\tau_{0}=0,$ (2.6)

is well-defined and grows to infinity with probability 1.

Here, by saying the sequence of hitting times $\tau_{n}$ is well-defined, we
mean that the time-valued random variables $\tau_{n}$ are stopping times (i.e,
for each $n\geq 0$, the event $\\{\tau_{n}\leq t\\}$ belongs to
$\mathscr{F}_{t}$), which guarantees that for each $(\mathscr{F}_{t})_{t\geq
0}$-adapted process $X$, the confined process
$X_{t}\textbf{1}_{(0,\tau_{n}]}(t)$ is also $(\mathscr{F}_{t})_{t\geq
0}$-adapted. With the convention that $\inf~{}\emptyset=\infty$, the (well-
defined) stopping times $\tau_{n}$ growing to $\infty$ with probability 1
means that for all $t_{0}>0$, it holds

$\mathbb{P}(\lim\limits_{n\to\infty}\tau_{n}\leq t_{0})\rightarrow 0\,.$ (2.7)

In particular, it is proved in [bossy2018particle, Appendix A.2] that

$\sum_{n\in\mathbb{N}}\mathbb{P}(\tau_{n}\leq T)<+\infty\,.$ (2.8)

We also note that the jumps
$\Delta\widetilde{K}_{t}:=\widetilde{K}_{t}-\widetilde{K}_{t^{-}}$ and $\Delta
U_{t}:=U_{t}-U_{t^{-}}$ at the time $t$ satisfy

$\Delta U_{t}=-2(U_{t^{-}}\cdot
n(Y_{t}))n(Y_{t})\textbf{1}_{\\{Y_{t}\in\partial
D\\}}=\Delta\widetilde{K}_{t}\,.$ (2.9)

Furthermore, it holds that

$|U_{t}|=|U_{t^{-}}|.$ (2.10)

######  Remark 2.2.

The original work [bossy2015lagrangian, Theorem 2.1] only gives the existence
of a weak solution and the pathwise uniqueness for SDE (2.5). One may expect
to obtain the existence and uniqueness of a strong solution through some
generalized Yamada-Watanabe-Engelbert theorem as addressed in
[barczy2015yamada], but in fact, the unique strong solution has been already
constructed in [bossy2015lagrangian, Proof of Theorem 2.1]. Alternatively, in
view of the trivial linear form of SDE (2.5), we can construct the strong
solution step by step as follows. Starting from $n=0$, $\tau_{0}=0$ and
$(Y_{0},U_{0})\in(D\times\mathbb{R}^{d})\cup\Sigma^{-}$ (w.l.o.g. as in
[bossy2015lagrangian, Section 2.1]), solve the following SDE:

$\displaystyle Y^{n}_{t}=Y_{\tau_{n}\wedge T}+\int_{\tau_{n}\wedge
T}^{t}U^{n}_{s}\,\mathrm{d}s,\quad U^{n}_{t}=U_{\tau_{n}\wedge
T}+\sqrt{2\sigma}(B_{t}-B_{\tau_{n}\wedge
T})+\sqrt{2\overline{\sigma}}(\overline{W}_{t}-\overline{W}_{\tau_{n}\wedge
T}),\quad\tau_{n}<t\leq T,$

and set

$\tau_{n+1}=\inf\\{\tau_{n}<t\leq T;\,Y^{n}_{t}\in\partial
D\\},\quad(Y_{t},\,U_{t})=(Y_{t}^{n},\,U^{n}_{t}),\quad\text{for
}\tau_{n}<t<\tau_{n+1}\wedge T\,,$

and

$Y_{\tau_{n+1}\wedge T}=Y_{\tau_{n+1}\wedge T}^{n},\quad\quad
U_{\tau_{n+1}\wedge T}=U^{n}_{\tau_{n+1}\wedge T}-2(U^{n}_{\tau_{n+1}\wedge
T}\cdot n(Y^{n}_{\tau_{n+1}\wedge T}))\,n(Y^{n}_{\tau_{n+1}\wedge
T})\textbf{1}_{\\{Y^{n}_{\tau_{n+1}\wedge T}\in\partial D\\}}\,,$

where we denoted by $\tau_{n}\wedge T:=\min\\{\tau_{n},T\\}$. We observe that
the stopping times $\tau_{n}$ are greater than or equal to the stopping times
$\theta_{n}$ constructed in [bossy2015lagrangian, Lemma 2.3] and coincide with
the ones defined in Theorem 2.3. Consequently, this shows their well-
posedness, as proved in [bossy2015lagrangian, Theorem 2.1]. As $\tau_{n}$
grows to infinity with probability 1, the above construction does not only
indicate the existence of a strong solution, but also confirms the uniqueness.

######  Remark 2.3.

We make a comment here about the assumption that $D$ has $C^{3}$ boundary (see
Assumption 1). This assumption comes from the technique used in the proof of
[bossy2015lagrangian, Theorem 2.1]. Specifically, the authors there use a
boundary straightening technique to reduce (locally) the problem to a half-
space setting ${\mathbb{R}}^{d-1}\times(0,\infty)$, so they can use results
from [bossy2011confined]. It is the necessary smoothness of the “straightening
map” which requires that $\partial D$ is $C^{3}$.

The existence in law for the solution to the particle system (1.2) simply
follows from Girsanov’s transformation [pardoux2014stochastic, Theorem 2.51]
and Theorem 2.3.

######  Theorem 2.4.

Let $N\geq 2$ and $T>0$. Under Assumption 1, there exists at least one weak
solution to the particle system 1.2 up to time $T$.

###### Proof.

We concatenate all the particles’ positions and velocities
$\\{(X^{i,N},V^{i,N})\\}_{i=1}^{N}$ in vectors
$(\mathcal{X}^{N},\mathcal{V}^{N})$ (see equation (1.1)) and consider

$\displaystyle\begin{cases}\mathcal{X}_{t}^{N}&=\mathcal{X}_{0}+\int_{0}^{t}\mathcal{V}_{s}^{N}ds\\\\[2.0pt]
\mathcal{V}_{t}^{N}&=\mathcal{V}_{0}+\sqrt{2\sigma}\mathcal{B}_{t}^{N}+\sqrt{2\overline{\sigma}}\overline{\mathcal{W}}_{t}^{N}+\mathcal{K}_{t}^{N},~{}\mbox{
with }\mathcal{K}_{t}^{N}:=(K_{t}^{1,N},\dots,K_{t}^{N,N})\,,\end{cases}$

where $\mathcal{B}_{t}^{N}:=\\{B^{1}_{t},\dots,B^{N}_{t}\\}$ and
$\overline{\mathcal{W}}_{t}^{N}:=\\{\overline{W}_{t},\dots,\overline{W}_{t}\\}$
are of size $Nd\times 1$, and for which the well-posedness is ensured by
Theorem 2.3. Next we define a ${\mathbb{R}}^{dN}$-valued process
$\mathcal{H}^{N}:=(\mathcal{H}^{N}_{1},\dots,\mathcal{H}^{N}_{N})$ with

$\mathcal{H}^{N}_{i}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N})=\frac{1}{N}\sum\limits_{j=1}^{N}H(X_{s}^{i,N}-X_{s}^{j,N},V_{s}^{i,N}-V_{s}^{j,N}),$
(2.11)

and set

$\mathcal{Y}_{t}^{N}:=\mathcal{B}_{t}^{N}-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}\mathcal{H}^{N}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N})\,\mathrm{d}s.$
(2.12)

This implies that
$(\mathcal{X}_{t}^{N},\mathcal{V}_{t}^{N},\mathcal{Y}_{t}^{N})$ satisfies

$\displaystyle\begin{cases}\mathcal{X}_{t}^{N}&=\mathcal{X}_{0}+\int_{0}^{t}\mathcal{V}_{s}^{N}ds\\\\[2.0pt]
\mathcal{V}_{t}^{N}&=\mathcal{V}_{0}+\int_{0}^{t}\mathcal{H}^{N}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N})ds+\sqrt{2\sigma}\mathcal{Y}_{t}^{N}+\sqrt{2\overline{\sigma}}\overline{\mathcal{W}}_{t}^{N}+\mathcal{K}_{t}^{N},\end{cases}$

which will have the same form as (1.2) if one can justify that
$\mathcal{Y}_{t}^{N}$ is a Wiener process. Indeed, define $(Z_{t}^{N})_{t\geq
0}$ by

$Z_{t}^{N}:=\exp\left(\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}\langle\mathcal{H}^{N}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N}),d\mathcal{B}_{s}^{N}\rangle-\frac{1}{4\sigma}\int_{0}^{t}|\mathcal{H}^{N}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N})|^{2}ds\right).$
(2.13)

Clearly,

$\mathbb{E}\left[\exp\left(\frac{1}{4\sigma}\int_{0}^{T}|\mathcal{H}^{N}(\mathcal{X}_{s}^{N},\mathcal{V}_{s}^{N})|^{2}ds\right)\right]\leq\exp(\|H\|_{\infty}^{2}TN/(4\sigma)),$
(2.14)

which together with the classical exponential martingale theorem concludes
that the process $(Z_{t}^{N})_{t\in[0,T]}$ is a positive martingale with
respect to $(\mathscr{F}_{t})_{t\in[0,T]}$ and $\mathbb{E}[Z_{t}]=1$ for all
$t\in[0,T]$. Then by Girsanov’s Theorem [oksendal2013stochastic, Theorem
8.6.4], the processes $\mathcal{Y}_{t}^{N}$ defined in (2.12) is a
$dN$-dimensional Wiener process under the new probability measure $\mathbb{Q}$
defined by

$\mathbb{Q}(A)=\int_{A}Z_{T}^{N}\mathrm{d}\mathbb{P},~{}\mbox{ for
}A\in\mathscr{F}_{T}.$ (2.15)

This concludes the proof. ∎

######  Remark 2.4.

(Uniqueness) Let Assumption 1 hold. Assume further that the interaction force
$H$ is uniformly Lipschitz or satisfies a certain kind of so-called weak
monotonicity condition as in [ren2007stochastic]. Then the particle system
(1.2) has pathwise uniqueness and thus admits a unique strong solution; the
proof is analogous to that of Remark 2.2.

##  3\. Tightness of the empirical measures associated to the particle system
(1.2)

Let
$\mathcal{E}:=\mathcal{C}([0,T];\overline{D})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})$
be the sample space and we equip $\mathcal{E}$ with the Skorokhod topology, so
that $\mathcal{E}$ is a closed subset of
${\mathbb{D}}([0,T];{\mathbb{R}}^{2d})$. Now we treat
$(X_{t}^{i,N},V_{t}^{i,N})_{t\in[0,T]}$ as canonical process on $\mathcal{E}$.
Let
$f^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(X^{i,N},V^{i,N})}\in\mathcal{P}(\mathcal{E})$
be the empirical measure associated to (1.2), which is a random measure. In
the following we shall prove that $\\{\mathcal{L}(f^{N})\\}_{N\geq 2}$ is
tight by using Aldous’s Tightness criterion [huang2022mean, Lemma 2.2]:

######  Lemma 3.1.

Let $\\{X^{n}\\}_{n\in\mathbb{N}}$ be a sequence of random variables defined
on a probability space $(\Omega,\mathscr{F},\mathbb{P})$ and valued in
${\mathbb{D}}([0,T];{\mathbb{R}}^{d})$. The sequence
$\\{\mathcal{L}(X^{n})\\}_{n\in\mathbb{N}}$ is tight in
${\mathbb{D}}([0,T];{\mathbb{R}}^{d})$ if the following hold:

$(Con1)$ For all $t\in[0,T]$, $\\{\mathcal{L}(X^{n}_{t})\\}_{n\in\mathbb{N}}$
is tight in ${\mathbb{R}}^{d}$.

$(Con2)$ For all $\varepsilon>0$, $\eta>0$, there exists $\delta_{0}>0$ and
$n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ and for all discrete-
valued $\sigma(X^{n}_{s};s\in[0,T])$-stopping times $\beta$ with
$0\leq\beta+\delta_{0}\leq T$, it holds that

$\sup_{\delta\in[0,\delta_{0}]}\mathbb{P}\left(|X^{n}_{\beta+\delta}-X^{n}_{\beta}|\geq\eta\right)\leq\varepsilon\,.$
(3.1)

Now the tightness comes from justifying $(Con1)$ and $(Con2)$.

######  Theorem 3.1.

Under Assumption 1, the sequences $\\{\mathcal{L}(f^{N})\\}_{N\geq 2}$ and
$\\{\mathcal{L}(X^{1,N},V^{1,N})\\}_{N\geq 2}$ are tight.

###### Proof.

As shown in Sznitman [sznitman1991topics], the exchangeability of particles in
the system (1.2) induces the equivalence between the tightness of
$\\{\mathcal{L}(f^{N})\\}_{N\geq 2}$ and
$\\{\mathcal{L}((X^{1,N},V^{1,N}))\\}_{N\geq 2}$. Next we apply Aldous’s
Tightness criterion to $\\{(X^{1,N},V^{1,N})\\}_{N\geq 2}$, a sequence of
random variables valued in $\mathcal{E}$.

We first check $(Con1)$. Applying Itô’s formula for semimartingales
[tankov2003financial, Proposition 8.19] gives us

$|X_{t}^{1,N}|^{2}=|X_{0}^{1}|^{2}+\int_{0}^{t}2X_{s}^{1,N}\cdot
V_{s}^{1,N}\,{\rm
d}s\leq|X_{0}^{1}|^{2}+\int_{0}^{t}\left(|X_{s}^{1,N}|^{2}+|V_{s}^{1,N}|^{2}\right)\,{\rm
d}s,$ (3.2)

for $0\leq t\leq T$, and

$\displaystyle|V_{t}^{1,N}|^{2}=|V_{0}^{1}|^{2}+\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot
H\ast f_{s}^{N}(X_{s}^{1,N},V_{s}^{1,N})\,{\rm
d}s+\sqrt{2\sigma}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}B_{s}^{1}+\sqrt{2\overline{\sigma}}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}\overline{W}_{s}$
$\displaystyle+2d(\sigma+\overline{\sigma})t+\sum_{0<s\leq{t}}2V_{s^{-}}^{1,N}\cdot\Delta
K_{s}^{1,N}\textbf{1}_{\\{X_{s}^{1,N}\in\partial D\\}}+\sum_{0<s\leq
t}\left(|V_{s}^{1,N}|^{2}-|V_{s^{-}}^{1,N}|^{2}-2V_{s^{-}}^{1,N}\Delta
V_{s}^{1,N}\right)\textbf{1}_{\\{X_{s}^{1,N}\in\partial D\\}}$
$\displaystyle\leq$
$\displaystyle|V_{0}^{1}|^{2}+\int_{0}^{t}|V_{s}^{1,N}|^{2}\,{\rm
d}s+\sqrt{2\sigma}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}B_{s}^{1}++\sqrt{2\overline{\sigma}}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}\overline{W}_{s}+\left(\left\lVert
H\,\right\rVert_{\infty}^{2}+2d(\sigma+\overline{\sigma})\right)t,$ (3.3)

where according to (2.9) we have used the fact that

$\sum_{0<s\leq{t}}2V_{s^{-}}^{1,N}\cdot\Delta
K_{s}^{1,N}\textbf{1}_{\\{X_{s}^{1,N}\in\partial
D\\}}=\sum_{0<s\leq{t}}2V_{s^{-}}^{1,N}\cdot\Delta
V_{s}^{1,N}\textbf{1}_{\\{X_{s}^{1,N}\in\partial D\\}}$

and

$|V_{s}^{1,N}|^{2}=|V_{s^{-}}^{1,N}|^{2}\,.$

Notice that the Burkholder-Davis-Gundy inequality gives that

$\mathbb{E}\left[\sup_{t\in[0,r]}\left|\sqrt{2\sigma}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}B_{s}^{1}\right|^{2}\right]\leq
16\sigma\mathbb{E}\left[\int_{0}^{r}|V_{s^{-}}^{1,N}|^{2}\,\mathrm{d}s\right],\quad\text{for
}r\in[0,T],$

and

$\mathbb{E}\left[\sup_{t\in[0,r]}\left|\sqrt{2\overline{\sigma}}\int_{0}^{t}2V_{s^{-}}^{1,N}\cdot\,\mathrm{d}\overline{W}_{s}\right|^{2}\right]\leq
16\overline{\sigma}\mathbb{E}\left[\int_{0}^{r}|V_{s^{-}}^{1,N}|^{2}\,\mathrm{d}s\right],\quad\text{for
}r\in[0,T],$

which together with estimates (3.2) and (3.3) yield that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,r]}\big{(}|X_{t}^{1,N}|^{2}+|V_{t}^{1,N}|^{2}\big{)}\right]$
$\displaystyle\leq\mathbb{E}\left[|X_{0}^{1}|^{2}+|V_{0}^{1}|^{2}\right]+\mathbb{E}\left[\int_{0}^{t}\big{(}|X_{s}^{1,N}|^{2}+(2+16\sigma+16\overline{\sigma})|V_{s}^{1,N}|^{2}\big{)}\,{\rm
d}s\right]+\left(\left\lVert
H\,\right\rVert_{\infty}^{2}+2d(\sigma+\overline{\sigma})\right)r.$

Then, by Gronwall’s inequality, we find

$\displaystyle\sup_{N\geq
2}\mathbb{E}\left[\sup_{t\in[0,T]}\big{(}|X_{t}^{1,N}|^{2}+|V_{t}^{1,N}|^{2}\big{)}\right]$
$\displaystyle\leq
e^{(2+16\sigma+16\overline{\sigma})T}\left(\mathbb{E}\left[|X_{0}^{1}|^{2}+|V_{0}^{1}|^{2}\right]+\left(\left\lVert
H\,\right\rVert_{\infty}^{2}+2d(\sigma+\overline{\sigma})\right)T\,\right)$
$\displaystyle=:C_{1},$ (3.4)

and thus, $(Con1)$ is fulfilled.

Now we check $(Con2)$. For given $N\geq 2$, let $\beta$ be a
$\sigma(X^{1,N},V^{1,N})$-stopping time with discrete values such that
$\beta+\delta_{0}\leq T$. Then we observe that for all
$\delta\in[0,\delta_{0}]$, it holds that

$\displaystyle\mathbb{E}[|X_{\beta+\delta}^{1,N}-X_{\beta}^{1,N}|]=\mathbb{E}\left[\left|\int_{\beta}^{\beta+\delta}V_{s}^{1,N}\mathrm{d}s\right|\right]\leq\delta\mathbb{E}\left[\sup_{t\in[0,T]}|V_{t}^{1,N}|\right]\leq\delta\sqrt{C_{1}}\,.$
(3.5)

Moreover, since
$\mathbb{E}[|B_{\beta+\delta}^{1}-B_{\beta}^{1}|]=\mathbb{E}\left[\mathbb{E}[|B_{\beta+\delta}^{1}-B_{\beta}^{1}|\big{|}\mathscr{F}_{\beta}]\right]=\mathbb{E}\left[\mathbb{E}[|B^{1}_{\delta}|\big{|}\mathscr{F}_{0}]\right]=\mathbb{E}[|B^{1}_{\delta}|]\leq\sqrt{\delta}$
it is easy to see that

$\displaystyle\mathbb{E}[|V_{\beta+\delta}^{1,N}-V_{\beta}^{1,N}|]$
$\displaystyle\leq\mathbb{E}\left[\left|\int_{\beta}^{\beta+\delta}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})ds\right|\right]+\sqrt{2\sigma}\mathbb{E}[|B_{\beta+\delta}^{1}-B_{\beta}^{1}|]$
$\displaystyle\quad+\sqrt{2\overline{\sigma}}\mathbb{E}[|\overline{W}_{\beta+\delta}-\overline{W}_{\beta}|]+\mathbb{E}[|K_{\beta+\delta}^{1,N}-K_{\beta}^{1,N}|]$
$\displaystyle\leq\delta\|H\|_{\infty}+(\sqrt{2\sigma}+\sqrt{2\overline{\sigma}})\sqrt{\delta}+\mathbb{E}[|K_{\beta+\delta}^{1,N}-K_{\beta}^{1,N}|]\,.$
(3.6)

Consider the following probability transformation:

$\frac{\rm{d}\mathbb{Q}}{\rm{d}\mathbb{P}}=Z_{T}:=\exp\left(-\frac{1}{\sqrt{2\sigma}}\int_{0}^{T}H\ast
f_{s}^{N}(X_{s}^{1,N},V_{s}^{1,N})\rm{d}B^{1}_{s}-\frac{1}{4\sigma}\int_{0}^{T}\big{|}H\ast
f_{s}^{N}(X_{s}^{1,N},V_{s}^{1,N})\big{|}^{2}\rm{d}s\right).$ (3.7)

Applying Girsanov’s theorem analogously to the proof of Theorem 2.4, we have
the process $(X^{1,N},\,V^{1,N},\,K^{1,N})$ under $\mathbb{Q}$ and the process
$(Y,\,U,\,\widetilde{K})$ in (2.5) under $\mathbb{P}$ being identically
distributed. Thus, it follows that

$\displaystyle\mathbb{E}\left[\left|K_{\beta+\delta}^{1,N}-K_{\beta}^{1,N}\right|\right]\leq\sqrt{\mathbb{E}\left[\left|K_{\beta+\delta}^{1,N}-K_{\beta}^{1,N}\right|^{2}Z_{T}\right]\cdot\mathbb{E}\left[1/Z_{T}\right]}$
$\displaystyle\quad\leq
C(T,\sigma,\|H\|_{\infty})\sqrt{\mathbb{E}_{\mathbb{Q}}\left[\left|K_{\beta+\delta}^{1,N}-K_{\beta}^{1,N}\right|^{2}\right]}=C(T,\sigma,\|H\|_{\infty})\sqrt{\mathbb{E}\left[\left|\widetilde{K}_{\beta+\delta}-\widetilde{K}_{\beta}\right|^{2}\right]}\,.$
(3.8)

Analogous to (3.4), there exists $\widetilde{C}_{1}>0$ such that

$\displaystyle\sup_{N\geq
2}\mathbb{E}\left[\sup_{t\in[0,T]}\big{(}|Y_{t}|^{2}+|U_{t}|^{2}\big{)}\right]$
$\displaystyle\leq\widetilde{C}_{1},$ (3.9)

with $\widetilde{C}_{1}$ being independent of $N$. Notice that for
$\tau\in\\{\beta,\,\beta+\delta\\}$,

$\displaystyle\mathbb{E}\left[\left|\widetilde{K}_{\tau}\right|\right]=\mathbb{E}\left[\left|-\sum_{0<s\leq\tau}2(U_{s^{-}}\cdot
n(Y_{s}))n(Y_{s})\textbf{1}_{\\{Y_{s}\in\partial
D\\}}\right|\right]=\mathbb{E}\left[\left|U_{\tau}-U_{0}-\int_{0}^{\tau}\sqrt{2\sigma}B_{\tau}+\sqrt{2\overline{\sigma}}\,\overline{W}_{\tau}\right|\right]$
$\displaystyle\leq$
$\displaystyle(\sqrt{2\sigma}+\sqrt{2\overline{\sigma}})\sqrt{T}+2\mathbb{E}\left[\sup_{t\in[0,T]}|U_{t}|\right]<\infty\,.$
(3.10)

So by the Dominated Convergence Theorem and the fact that $t\to\sum_{0<s\leq
t}2(U_{s^{-}}\cdot n(Y_{s}))n(Y_{s})\textbf{1}_{\\{Y_{s}\in\partial D\\}}$ is
right-continuous, one immediately gets that

$\displaystyle\lim_{\delta\to
0}\mathbb{E}[|\widetilde{K}_{\beta+\delta}-\widetilde{K}_{\beta}|]=\lim_{\delta\to
0}\mathbb{E}\left[\left|-\sum_{\beta<s\leq\beta+\delta}2(U_{s^{-}}\cdot
n(Y_{s}))n(Y_{s})\textbf{1}_{\\{Y_{s}\in\partial D\\}}\right|\right]=0\,.$
(3.11)

This together with (3.8) and (3.6) implies that

$\lim_{\delta_{0}\rightarrow 0}\sup_{N\geq
2}\sup_{\delta\in[0,\delta_{0}]}\mathbb{E}\left[\left|V_{\beta+\delta}^{1,N}-V_{\beta}^{1,N}\right|\right]=0,$
(3.12)

which combined with (3.5) justifies $(Con2)$ by applying Chebyshev’s
inequality directly. ∎

As a consequence of Theorem 3.1, we obtain the joint tightness of
$\\{\mathcal{L}(f^{N},X^{N},V^{N})\\}_{N\geq 2}$, which implies the following
result by Prokhorov’s theorem:

######  Corollary 3.2.

For any fixed $i=1,\dots,N$, there exists a subsequence of
$\\{(f^{N},X^{i,N},V^{i,N})\\}_{N\geq 2}$ (denoted w.l.o.g. by itself) and a
random triplet
$(f,\overline{X}^{i},\overline{V}^{i})\in\mathcal{P}(\mathcal{E})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})$
such that

$(f^{N},X^{i,N},V^{i,N})\rightharpoonup(f,\overline{X}^{i},\overline{V}^{i})\mbox{
in law as }N\to\infty\,.$ (3.13)

##  4\. Identification of the limit measure

First, for each $\psi\in\mathcal{C}_{c}^{\infty}(D\times{\mathbb{R}}^{d})$, we
define the following functional on $\mathcal{P}(\mathcal{E})$:

$\displaystyle F_{\psi}(\nu)$
$\displaystyle:=\int_{\mathcal{E}}\psi({\mathbf{x}}_{t},{\mathbf{v}}_{t})\,\nu(\mathrm{d}{\mathbf{x}},\mathrm{d}{\mathbf{v}})-\int_{\mathcal{E}}\psi({\mathbf{x}}_{0},{\mathbf{v}}_{0})\,\nu(\mathrm{d}{\mathbf{x}},\mathrm{d}{\mathbf{v}})-\int_{0}^{t}\int_{\mathcal{E}}\nabla_{x}\psi({\mathbf{x}}_{s},{\mathbf{v}}_{s})\cdot{\mathbf{v}}_{s}\,\nu(\mathrm{d}{\mathbf{x}},\mathrm{d}{\mathbf{v}})\,{\rm
d}s$
$\displaystyle\quad-\int_{0}^{t}\int_{\mathcal{E}}\left(\nabla_{v}\psi\cdot
H\ast\nu({\mathbf{x}}_{s},{\mathbf{v}}_{s})+(\sigma+\overline{\sigma})\Delta_{v}\psi({\mathbf{x}}_{s},{\mathbf{v}}_{s})\right)\,\nu(\mathrm{d}{\mathbf{x}},\mathrm{d}{\mathbf{v}})\,{\rm
d}s$
$\displaystyle\quad-\sqrt{2\overline{\sigma}}\int_{0}^{t}\int_{\mathcal{E}}\nabla_{v}\psi({\mathbf{x}}_{s},{\mathbf{v}}_{s})\,\nu(\mathrm{d}{\mathbf{x}},\mathrm{d}{\mathbf{v}})\mathrm{d}\overline{W}_{s}$
$\displaystyle=\int_{D\times\mathbb{R}^{d}}\psi(x,v)\,\nu_{t}(\,{\rm
d}x,\,{\rm d}v)-\int_{D\times\mathbb{R}^{d}}\psi(x,v)\nu_{0}(\,{\rm d}x,\,{\rm
d}v)-\int_{0}^{t}\int_{D\times\mathbb{R}^{d}}\nabla_{x}\psi(x,v)\cdot
v\,\nu_{s}(\,{\rm d}x,\,{\rm d}v)\,{\rm d}s$
$\displaystyle\quad-\int_{0}^{t}\int_{D\times\mathbb{R}^{d}}\left(\nabla_{v}\psi\cdot
H\ast\nu(x,v)+(\sigma+\overline{\sigma})\Delta_{v}\psi(x,v)\right)\nu_{s}(\,{\rm
d}x,\,{\rm d}v)\,{\rm d}s$
$\displaystyle\quad-\sqrt{2\overline{\sigma}}\int_{0}^{t}\int_{D\times\mathbb{R}^{d}}\nabla_{v}\psi(x,v)\,\nu_{s}(\,{\rm
d}x,\,{\rm d}v)\cdot\mathrm{d}\overline{W}_{s},$ (4.1)

for all $\nu\in\mathcal{P}(\mathcal{E})$. Then we have the following estimate.

######  Proposition 4.1.

Let Assumption 1 hold. For any $N\geq 2$ and $T>0$, assume that
$(\mathcal{X}_{t}^{N},\mathcal{V}_{t}^{N})_{t\in[0,T]}$ is a solution to the
particle system (1.2) with $f_{0}^{\otimes N}$-distributed initial data
$(\mathcal{X}_{0},\mathcal{V}_{0})$. There exists a constant $C>0$ depending
only on $\sigma,T$, and $\|\nabla_{v}\psi\|_{\infty}$ such that

$\mathbb{E}\left[|F_{\psi}(f^{N})|^{2}\right]\leq\frac{C}{N}\,,$

where $f^{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{(X^{i,N},V^{i,N})}$ is the
empirical measure associated with (1.2).

###### Proof.

Using the definition of $F_{\psi}$ one has

$\displaystyle F_{\psi}(f^{N})$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})-\frac{1}{N}\sum_{i=1}^{N}\psi(X_{0}^{i,N},V_{0}^{i,N})-\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\nabla_{x}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot
V_{s}^{i,N}\,{\rm d}s$
$\displaystyle\quad-\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot
H\ast f^{N}(X_{s}^{i,N},V_{s}^{i,N})\,{\rm
d}s-\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}(\sigma+\overline{\sigma})\Delta_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\,{\rm
d}s$
$\displaystyle\quad-\sqrt{2\overline{\sigma}}\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}\overline{W}_{s}\,.$

For each $i=1,\dots,N$ and $\psi\in\mathcal{C}_{c}(D\times{\mathbb{R}}^{d})$,
applying Itô’s formula, it gives

$\displaystyle\psi(X_{t}^{i,N},V_{t}^{i,N})$
$\displaystyle=\psi(X_{0}^{i,N},V_{0}^{i,N})+\int_{0}^{t}\nabla_{x}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot
V_{s}^{i,N}\,{\rm d}s$
$\displaystyle\quad+\int_{0}^{t}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot
H\ast f_{s}^{N}(X_{s}^{i,N},V_{s}^{i,N})\,{\rm d}s$
$\displaystyle\quad+\sqrt{2\sigma}\int_{0}^{t}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}B_{s}^{i}+\sqrt{2\overline{\sigma}}\int_{0}^{t}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}\overline{W}_{s}$
$\displaystyle\quad+\int_{0}^{t}(\sigma+\overline{\sigma})\Delta_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\,{\rm
d}s\,.$ (4.2)

This implies that

$F_{\psi}(f^{N})=\sqrt{2\sigma}\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}B_{s}^{i}\,.$
(4.3)

Then using the fact that $B^{i}$ is independent of $B^{j}$ when $i\neq j$ it
holds

$\displaystyle\mathbb{E}\left[|F_{\psi}(f^{N})|^{2}\right]=\frac{2\sigma}{N^{2}}\mathbb{E}\left[\left|\int_{0}^{t}\sum_{i=1}^{N}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}B_{s}^{i}\right|^{2}\right]$
$\displaystyle=$
$\displaystyle\frac{2\sigma}{N^{2}}\sum_{i=1}^{N}\mathbb{E}\left[\left|\int_{0}^{t}\nabla_{v}\psi(X_{s}^{i,N},V_{s}^{i,N})\cdot\mathrm{d}B_{s}^{i}\right|^{2}\right]\leq
C(\sigma,T,\|\nabla_{v}\psi\|_{\infty})\frac{1}{N}\,.$ (4.4)

This completes the proof. ∎

By Skorokhod’s lemma, using Corollary 3.2, we may find a common probability
space, w.l.o.g., still denoted by $(\Omega,\mathscr{F},\mathbb{P})$, on which
for any fixed $i$ the processes
$\\{(f^{N},X^{i,N},V^{i,N},B^{i},\overline{W})\\}_{N\geq 2}$ converge to some
process $(f,\overline{X}^{i},\overline{V}^{i},B^{i},\overline{W})$ as a random
variable valued in
$\mathcal{P}(\mathcal{E})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})$
almost surely. In particular, we have that for all $t\in[0,T]$, and
$\phi\in\mathcal{C}_{b}(D\times\mathbb{R}^{d})$,

$\displaystyle\lim_{N\rightarrow\infty}|X_{t}^{i,N}-\overline{X}_{t}^{i}|+|V_{t}^{i,N}-\overline{V}_{t}^{i}|=0\quad\mbox{
a.s. }$ (4.5)

and

$\lim_{N\rightarrow\infty}\iint_{D\times{\mathbb{R}}^{d}}\phi(x,v)(f^{N}_{t}-f_{t})(\,{\rm
d}x,\,{\rm d}v)=0\quad\mbox{ a.s.},$ (4.6)

which also implies

$\lim_{N\rightarrow\infty}\mathbb{E}\left[\left|\iint_{D\times{\mathbb{R}}^{d}}\phi(x,v)(f^{N}_{t}-f_{t})(\,{\rm
d}x,\,{\rm d}v)\right|^{2}\right]=0\,.$ (4.7)

Then our main result states that the limit process $f$ satisfies the mean-
field stochastic equation (1.4).

######  Theorem 4.2.

Assume the same assumptions as in Remark 2.4. The limit $f$ is
$(\mathscr{F}_{t}^{\overline{W}})_{t\geq 0}$-adapted, and it satisfies the
following stochastic integral equation

$\displaystyle\langle\psi(x,v),f_{t}(\,{\rm d}x,\,{\rm
d}v)\rangle-\langle\psi(x,v),f_{0}(\,{\rm d}x,\,{\rm
d}v)\rangle-\int_{0}^{t}\langle\nabla_{x}\psi(x,v)\cdot v,f_{s}(\,{\rm
d}x,\,{\rm d}v)\rangle\,{\rm d}s$
$\displaystyle\quad-\int_{0}^{t}\langle\nabla_{v}\psi\cdot H\ast
f(x,v)+(\sigma+\overline{\sigma})\Delta_{v}\psi(x,v),f_{s}(\,{\rm d}x,\,{\rm
d}v)\rangle\,{\rm d}s$
$\displaystyle\quad-\sqrt{2\overline{\sigma}}\int_{0}^{t}\langle\nabla_{v}\psi(x,v),f_{s}(\,{\rm
d}x,\,{\rm d}v)\rangle\cdot\mathrm{d}\overline{W}_{s}=0\quad\mbox{ a.s.},$
(4.8)

for all $\psi\in\mathcal{C}_{c}^{\infty}(D\times{\mathbb{R}}^{d})$. Here,
$(\mathscr{F}_{t}^{\overline{W}})_{t\geq 0}$ is the filtration generated by
the Wiener process $\overline{W}$.

###### Proof.

Step 1. We prove the equality (4.8). For each
$\psi\in\mathcal{C}_{c}^{\infty}(D\times{\mathbb{R}}^{d})$, using the
convergence result in (4.7) one has

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\left[|(\langle\psi,f_{t}^{N}\rangle-\langle\psi,f_{0}^{N}\rangle)-(\langle\psi,f_{t}\rangle-\langle\psi,f_{0}\rangle)|\right]=0.$
(4.9)

Further we notice that for all $s\in[0,t]$,

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}[|\langle\nabla_{x}\psi\cdot
v,f_{s}^{N}-f_{s}\rangle|]+\mathbb{E}[|\langle\Delta_{v}\psi,f_{s}^{N}-f_{s}\rangle|]=0.$

Thus by Dominated Convergence Theorem one obtains

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\left[\left|\int_{0}^{t}\langle\nabla_{x}\psi\cdot
v,f_{s}^{N}-f_{s}\rangle+(\sigma+\overline{\sigma})\langle\Delta_{v}\psi,f_{s}^{N}-f_{s}\rangle\,{\rm
d}s\right|\right]=0.$ (4.10)

Since $H\in\mathcal{C}_{b}$ it is easy to see that for all $s\in[0,t]$

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\left[|\langle\nabla_{v}\psi\cdot
H\ast f_{s}^{N},f_{s}^{N}\rangle-\langle\nabla_{v}\psi\cdot H\ast
f_{s},f_{s}\rangle|\right]$
$\displaystyle\leq\lim_{N\rightarrow\infty}\mathbb{E}\left[|\langle\nabla_{v}\psi\cdot
H\ast
f_{s}^{N},f_{s}^{N}-f_{s}\rangle|\right]+\lim_{N\rightarrow\infty}\mathbb{E}\left[|\langle\nabla_{v}\psi\cdot
H\ast(f_{s}^{N}-f_{s}),f_{s}\rangle|\right]=0\,.$

So using Dominated Convergence Theorem again it yields that

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\left[\left|\int_{0}^{t}\langle\nabla_{v}\psi\cdot
H\ast f_{s}^{N},f_{s}^{N}\rangle-\langle\nabla_{v}\psi\cdot H\ast
f_{s},f_{s}\rangle\,{\rm d}s\right|\right]=0\,.$ (4.11)

Similarly we check

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\left[\left|\sqrt{2\overline{\sigma}}\int_{0}^{t}\langle\nabla_{v}\psi,f_{s}^{N}-f_{s}\rangle\cdot\mathrm{d}\overline{W}_{s}\right|\right]$
$\displaystyle\leq\lim_{N\rightarrow\infty}\sqrt{2\overline{\sigma}}\left(\mathbb{E}\left[\int_{0}^{t}|\langle\nabla_{v}\psi,f_{s}^{N}-f_{s}\rangle|^{2}\,{\rm
d}s\right]\right)^{1/2}=0$ (4.12)

Collecting estimates (4.9)-(4) one concludes that

$\lim_{N\rightarrow\infty}\mathbb{E}\left[|F_{\psi}(f^{N})-F_{\psi}(f)|\right]=0\,.$
(4.13)

Then we have

$\displaystyle\mathbb{E}[|F_{\psi}(f)|]$
$\displaystyle\leq\mathbb{E}\left[|F_{\psi}(f^{N})-F_{\psi}(f)|\right]+\mathbb{E}[|F_{\psi}(f^{N})|]$
$\displaystyle\leq\mathbb{E}\left[|F_{\psi}(f^{N})-F_{\psi}(f)|\right]+\frac{C}{\sqrt{N}}\to
0,\quad\mbox{ as }N\to\infty.$

This implies $F_{\psi}(f)=0$ a.s., namely (4.8) holds and the limit $f$ is
$(\mathscr{F}_{t}^{\overline{W},\mathcal{B}})_{t\geq 0}$-adapted where
$(\mathscr{F}_{t}^{\overline{W},\mathcal{B}})_{t\geq 0}$ denotes the
filtration generated by $\overline{W}$ and $B^{i}$, $i=1,2,\dots$.

Step 2. To prove that $f$ is $(\mathscr{F}_{t}^{\overline{W}})_{t\geq
0}$-adapted, it sufficient to show that $f$ is independent of the Wiener
processes $B^{i}$ for any $i=1,2,\dots$.

Without any loss of generality, we prove $f$ is independent of $B^{1}$ first,
by introducing a $N-1$-particle system (see (4.15) below). For any
$t\in[0,T]$, $\xi\in L^{\infty}(\Omega,\mathscr{F}_{T}^{B^{1}};{\mathbb{R}})$
and $\Phi\in\mathcal{C}_{c}^{\infty}({\mathbb{R}})$, let us consider

$\displaystyle\mathbb{E}[\Phi(\langle\psi,f_{t}^{N}\rangle)\xi]=\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=1}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]+\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=1}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]-\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]+C(\|\xi\|_{\mathcal{L}^{\infty}},\|\nabla\Phi\|_{\infty},\|\psi\|_{\infty})\frac{1}{N}\,,$
(4.14)

where we have used the mean-value theorem in the last inequality.

Next we investigate the term
$\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]$.
One introduces the following dynamics with $N-1$ particles:

$\displaystyle\begin{cases}\widehat{X}_{t}^{i,N}=X_{0}^{i}+\int_{0}^{t}\widehat{V}_{s}^{i,N}\,\mathrm{d}s,\quad
i=2,\dots,N\\\\[2.0pt]
\widehat{V}_{t}^{i,N}=V_{0}^{i}+\int_{0}^{t}\frac{1}{N}\sum\limits_{j\geq
2}^{N}H(\widehat{X}_{s}^{i,N}-\widehat{X}_{s}^{j,N},\widehat{V}_{s}^{i,N}-\widehat{V}_{s}^{j,N})\,\mathrm{d}s+\sqrt{2\sigma}B_{t}^{i}+\sqrt{2\overline{\sigma}}\overline{W}_{t}+\widehat{K}_{t}^{i,N},\\\\[2.0pt]
\widehat{K}_{t}^{i,N}=-\sum\limits_{0<s\leq t}2(\widehat{V}_{s^{-}}^{i,N}\cdot
n(\widehat{X}_{s}^{i,N}))n(\widehat{X}_{s}^{i,N})\textbf{1}_{\\{\widehat{X}_{s}^{i,N}\in\partial
D\\}}.\end{cases}$ (4.15)

By the uniqueness of solution in Remark 2.4,
$\\{(\widehat{X}^{i,N},\widehat{V}^{i,N},{B}^{i},\overline{W})\\}_{i=2}^{N}$
has the same distribution as
$\\{(X^{i,N},V^{i,N},{B}^{i},\overline{W})\\}_{i=1}^{N-1}$. Then we use
Girsanov’s transformation to separate the interaction of particles
$i~{}(i=2,\dots,N)$ with particle 1 from the rest of the interactions in the
original $N$-particle system (1.2), and we include it in a new Wiener process.
Specifically, we consider

$\displaystyle\frac{d\mathbb{Q}}{d\mathbb{P}}=\exp\bigg{\\{}-\frac{1}{\sqrt{2\sigma}}\int_{0}^{T}\frac{1}{N}\sum\limits_{i=2}^{N}H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})\,\mathrm{d}B_{s}^{i}$
$\displaystyle-\frac{1}{4\sigma}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s\bigg{\\}}=:\frac{1}{M_{T}}.$ (4.16)

Then, $(X^{i,N},V^{i,N})$ for $i=2,\dots,N$ satisfies

$\displaystyle\begin{cases}X_{t}^{i,N}=X_{0}^{i}+\int_{0}^{t}V_{s}^{i,N}\,\mathrm{d}s,\\\\[2.0pt]
V_{t}^{i,N}=V_{0}^{i}+\displaystyle\int_{0}^{t}\frac{1}{N}\sum\limits_{j\geq
2}^{N}H(X_{s}^{i,N}-X_{s}^{j,N},V_{s}^{i,N}-V_{s}^{j,N})\,\mathrm{d}s+\sqrt{2\sigma}\mathring{B}_{t}^{i}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+K_{t}^{i,N},\end{cases}$
(4.17)

where

$\mathring{B}_{t}^{i}:=B_{t}^{i}+\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}\frac{1}{N}H(X^{i,N}_{s}-X^{1,N}_{s},V^{i,N}_{s}-V^{1,N}_{s})ds$
(4.18)

is a Wiener process under $\mathbb{Q}$.

Hence by the uniqueness of the solution in Remark 2.4, the joint process of
$(X^{1,N},V^{1,N},B^{1})$ and
$\\{(X^{i,N},V^{i,N},\mathring{B}^{i},\overline{W})\\}_{i=2}^{N}$ under
$\mathbb{Q}$ has the same distribution as the joint process of
$(X^{1,N},V^{1,N},B^{1})$ and
$\\{(\widehat{X}^{i,N},\widehat{V}^{i,N},{B}^{i},\overline{W})\\}_{i=2}^{N}$
under $\mathbb{P}$. This indicates that

$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]$
$\displaystyle=\mathbb{E}_{\mathbb{Q}}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi
M_{T}\right]$
$\displaystyle=\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\xi
M_{T}^{\prime}\right],$ (4.19)

where

$\displaystyle M_{T}^{\prime}$
$\displaystyle:=\exp\bigg{\\{}\frac{1}{\sqrt{2\sigma}}\int_{0}^{T}\frac{1}{N}\sum\limits_{i=2}^{N}H(\widehat{X}_{s}^{i,N}-X_{s}^{1,N},\widehat{V}_{s}^{i,N}-V_{s}^{1,N})\,\mathrm{d}{B}_{s}^{i}$
$\displaystyle-\frac{1}{4\sigma}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(\widehat{X}_{s}^{i,N}-X_{s}^{1,N},\widehat{V}_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s\bigg{\\}}\,,$ (4.20)

which under $\mathbb{P}$ has the same distribution as $M_{T}$ under
$\mathbb{Q}$. This is because

$\displaystyle M_{T}$
$\displaystyle=\exp\bigg{\\{}\frac{1}{\sqrt{2\sigma}}\int_{0}^{T}\frac{1}{N}\sum\limits_{i=2}^{N}H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})\,\mathrm{d}B_{s}^{i}$
$\displaystyle+\frac{1}{4\sigma}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s\bigg{\\}}$
$\displaystyle=\exp\bigg{\\{}\frac{1}{\sqrt{2\sigma}}\int_{0}^{T}\frac{1}{N}\sum\limits_{i=2}^{N}H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})\,\mathrm{d}\mathring{B}_{s}^{i}$
$\displaystyle-\frac{1}{4\sigma}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(X_{s}^{i,N}-X_{s}^{1,N},V_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s\bigg{\\}}\,.$ (4.21)

Now we observe that

$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(X_{t}^{i,N},V_{t}^{i,N})\right)\xi\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\xi\right]+\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\xi(M_{T}^{\prime}-1)\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\right]\mathbb{E}[\xi]+\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\xi(M_{T}^{\prime}-1)\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\right]\mathbb{E}[\xi]+\|\Phi\|_{\infty}\cdot\|\xi\|_{\mathcal{L}^{\infty}}\mathbb{E}\left[\left|M_{T}^{\prime}-1\right|\right],$
(4.22)

where we have used the fact that
$\\{(\widehat{X}^{i,N},\widehat{V}^{i,N})\\}_{i=2}^{N}$ is independent of
$B^{1}$ (so is $\xi$) and Hölder’s inequality. Due to the boundedness of $H$,
the standard theory of linear stochastic differential equations (see
[karatzas2012brownian]) indicates that

$\displaystyle\mathbb{E}\left[|M_{T}^{\prime}|^{2}\right]$ $\displaystyle\leq
C(T,\sigma,\|H\|_{\infty}),$
$\displaystyle\mathbb{E}\left[\left|\ln(M_{T}^{\prime})\right|^{2}\right]$
$\displaystyle\leq\mathbb{E}\bigg{[}\frac{1}{\sigma}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(\widehat{X}_{s}^{i,N}-X_{s}^{1,N},\widehat{V}_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s$
$\displaystyle\quad\quad+\frac{1}{\sigma^{2}}\bigg{|}\int_{0}^{T}\frac{1}{N^{2}}\sum\limits_{i=2}^{N}|H(\widehat{X}_{s}^{i,N}-X_{s}^{1,N},\widehat{V}_{s}^{i,N}-V_{s}^{1,N})|^{2}\,{\rm
d}s\bigg{|}^{2}\bigg{]}$
$\displaystyle\leq\frac{C(T,\sigma,\|H\|_{\infty})}{N}.$

Thus, recalling the fact that $|e^{x}-1|\leq|x|(e^{x}+1)$, we have further

$\displaystyle\mathbb{E}\left[\left|M_{T}^{\prime}-1\right|\right]\leq\sqrt{\mathbb{E}\left[|\ln(M_{T}^{\prime})|^{2}\right]\cdot\mathbb{E}\left[2\cdot|M_{T}^{\prime}|^{2}+2\right]}\leq\frac{C(T,\sigma,\|H\|_{\infty})}{\sqrt{N}}.$
(4.23)

Combining (4.23), (4) and (4) yields that

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}[\Phi(\langle\psi,f_{t}^{N}\rangle)\xi]$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbb{E}\left[\Phi\left(\frac{1}{N}\sum_{i=2}^{N}\psi(\widehat{X}_{t}^{i,N},\widehat{V}_{t}^{i,N})\right)\right]\mathbb{E}[\xi]$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbb{E}\left[\Phi\left(\frac{N-1}{N}\cdot\frac{1}{N-1}\sum_{i=1}^{N-1}\psi(X_{t}^{i,N-1},V_{t}^{i,N-1})\right)\right]\mathbb{E}[\xi]$
$\displaystyle=\mathbb{E}[\Phi(\langle\psi,f_{t}\rangle)]\mathbb{E}[\xi]\,.$
(4.24)

On the other hand it holds that

$\lim_{N\rightarrow\infty}\mathbb{E}[\Phi(\langle\psi,f_{t}^{N}\rangle)\xi]=\mathbb{E}[\Phi(\langle\psi,f_{t}\rangle)\xi]\,.$
(4.25)

This means that

$\mathbb{E}[\Phi(\langle\psi,f_{t}\rangle)\xi]=\mathbb{E}[\Phi(\langle\psi,f_{t}\rangle)]\mathbb{E}[\xi]$
(4.26)

holds for all $\xi\in
L^{\infty}(\Omega,\mathscr{F}_{T}^{B^{1}};{\mathbb{R}})$. Thus we have that
$f$ is independent of $B^{1}$, and similarly one can obtain that $f$ is
actually independent of the whole family of Wiener processes
$\\{B^{i}\\}_{i=1}^{\infty}$. Therefore one has proved that $f$ is
$(\mathscr{F}_{t}^{\overline{W}})_{t\geq 0}$-adapted. ∎

We are left to verify the boundary condition for $f$. To do this, let us
introduce the following auxiliary system

$\displaystyle\begin{cases}\widetilde{X}_{t}^{i,N}=\widetilde{X}_{0}^{i,N}+\int_{0}^{t}\widetilde{V}_{s}^{i,N}\,\mathrm{d}s,\quad
i=1,\dots,N,\\\\[3.0pt]
\widetilde{V}_{t}^{1,N}=\widetilde{V}_{0}^{1}+\sqrt{2\sigma}B_{t}^{1}+\sqrt{2\overline{\sigma}}\overline{W}_{t}+\widetilde{K}_{t}^{1,N},\\\\[3.0pt]
\widetilde{V}_{t}^{i,N}=\widetilde{V}_{0}^{i,N}+\int_{0}^{t}H\ast\widetilde{f}^{N}_{s}(\widetilde{X}_{s}^{i,N},\widetilde{V}_{s}^{i,N})\,\mathrm{d}s+\sqrt{2\sigma}B_{t}^{i}+\sqrt{2\overline{\sigma}}~{}\overline{W}_{t}+\widetilde{K}_{t}^{i,N},\quad
i=2,\dots,N,\\\\[3.0pt] \widetilde{K}_{t}^{i,N}=-\sum\limits_{0<s\leq
t}2(\widetilde{V}_{s^{-}}^{i,N}\cdot
n(\widetilde{X}_{s}^{i,N}))n(\widetilde{X}_{s}^{i,N})\textbf{1}_{\\{\widetilde{X}_{s}^{i,N}\in\partial
D\\}}\quad i=2,\dots,N,\end{cases}$ (4.27)

where
$H\ast\widetilde{f}^{N}_{s}(\widetilde{X}_{s}^{i,N},\widetilde{V}_{s}^{i,N}):=\frac{1}{N}\sum\limits_{j=1}^{N}H(\widetilde{X}_{s}^{i,N}-\widetilde{X}_{s}^{j,N},\widetilde{V}_{s}^{i,N}-\widetilde{V}_{s}^{j,N})$
with
$\widetilde{f}^{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{(\widetilde{X}_{t}^{i,N},\widetilde{V}_{t}^{i,N})}$
being the empirical measure. In the particle system (4.27), particle 1 does
not interact with particles $i=2,\dots,N$, but particles $i=2,\dots,N$ do have
a full interaction with all the remaining particles (including particle 1). In
a similar way to Corollary 3.2, we may obtain the tightness of
$\\{(\widetilde{f}^{N},\widetilde{X}_{s}^{i,N},\widetilde{V}_{s}^{i,N})\\}_{i=1}^{N}$
and thus Skorokhod’s lemma allows us to assume that on the common probability
space, w.l.o.g., still denoted by $(\Omega,\mathscr{F},\mathbb{P})$, for any
fixed $i$ the processes
$\\{((\widetilde{f}^{N},\widetilde{X}_{s}^{i,N},\widetilde{V}_{s}^{i,N},f^{N},X^{i,N},V^{i,N},B^{i},\overline{W})\\}_{N\geq
2}$ converge to some process
$(\widetilde{f},\overline{\widetilde{X}}^{i},\overline{\widetilde{V}}^{i},f,\overline{X}^{i},\overline{V}^{i},B^{i},\overline{W})$
as a random variable valued in
$\mathcal{P}(\mathcal{E})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})\times\mathcal{P}(\mathcal{E})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times{\mathbb{D}}([0,T];{\mathbb{R}}^{d})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})\times\mathcal{C}([0,T];{\mathbb{R}}^{d})$
almost surely. Note here that
$(\widetilde{X}_{t}^{1,N},\widetilde{V}_{t}^{1,N})$ is independent of $N$, so
we will just denote it by $(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$. We
shall use (4.27) to find a stochastic process such that the limit $f$ is its
conditional density given $\mathscr{F}^{\overline{W}}$.

######  Proposition 4.3.

Assume the same assumptions as in Remark 2.4. Let $f$ be the limit obtained in
Theorem 4.2. Then $f$ is the conditional density of a process
$(\widetilde{Y}_{t},\widetilde{U}_{t})$ under
$\mathscr{F}_{t}^{\overline{W}}$, where
$(\widetilde{Y}_{t},\widetilde{U}_{t})$ is the weak (also strong) solution to
the following SDE:

$\displaystyle\begin{cases}\widetilde{Y}=\widetilde{Y}_{0}+\int_{0}^{t}\widetilde{U}_{s}\,\mathrm{d}s\,,\\\\[2.0pt]
\widetilde{U}_{t}=\widetilde{U}_{0}+\int_{0}^{T}H\ast\widetilde{f}_{s}(\widetilde{Y}_{s},\widetilde{U}_{s})\,\mathrm{d}s+\sqrt{2\sigma}B_{t}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+\widetilde{K}_{t}\,,\\\\[2.0pt]
\widetilde{K}_{t}=-\sum\limits_{0<s\leq t}2(\widetilde{U}_{s^{-}}\cdot
n(\widetilde{Y}_{s}))n(\widetilde{Y}_{s})\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}\,,\end{cases}$ (4.28)

where $\widetilde{f}$ is the limit of $\\{\widetilde{f}^{N}\\}_{N\geq 2}$.

###### Proof.

The proof idea is to use Girsanov’s transformation to find a Brownian motion
$\widehat{B}^{1}$ under some probability measure $\widehat{\mathbb{P}}$ such
that
$(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1},\widehat{B}_{t}^{1},\widehat{\mathbb{P}})$
is a weak solution to (4.28), and it holds that

$\displaystyle\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})]\to\mathbb{E}_{\widehat{\mathbb{P}}}[\phi(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})]\quad\mbox{
as }N\to\infty$

for any $\phi\in\mathcal{C}_{b}(D\times\mathbb{R}^{d})$. Then by Corollary 3.2
one also has

$\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})]\to\mathbb{E}[\phi(\overline{X}_{t}^{1},\overline{V}_{t}^{1})]\quad\mbox{
as }N\to\infty\,,$

which implies that $(\overline{X}_{t}^{1},\overline{V}_{t}^{1})$ under
$\mathbb{P}$ has the same distribution as
$(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$ under $\widehat{\mathbb{P}}$.
Finally, it is claimed that $f$ is the conditional density of
$(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$ given
$\mathscr{F}_{t}^{\overline{W}}$. The proof is split into three steps:

Step 1. Fix $i=1$ in (1.2) and we have

$\displaystyle\begin{cases}X_{t}^{1,N}=X_{0}^{1}+\int_{0}^{t}V_{s}^{1,N}\,\mathrm{d}s,\\\\[2.0pt]
V_{t}^{1,N}=V_{0}^{1}+\int_{0}^{t}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})\,\mathrm{d}s+\sqrt{2\sigma}B_{t}^{1}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+K_{t}^{1,N}\,,\\\
\end{cases}$ (4.29)

where $H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N}):=\frac{1}{N}\sum\limits_{j=1}^{N}H(X_{s}^{1,N}-X_{s}^{j,N},V_{s}^{1,N}-V_{s}^{j,N})$.
We set

$\widetilde{B}_{t}^{1,N}:=B_{t}^{1}+\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})\,{\rm d}s,$ (4.30)

and define

$M_{t}^{1,N}:=\exp\left\\{-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})\,\mathrm{d}{B}_{s}^{1}-\frac{1}{4\sigma}\int_{0}^{t}|H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})|^{2}\,{\rm d}s\right\\}\,.$ (4.31)

Since

$|H\ast f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})|\leq\|H\|_{\infty}\quad\mbox{
a.s.},$

we obtain

$\mathbb{E}\left[\exp\left\\{\frac{1}{4\sigma}\int_{0}^{T}|H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})|^{2}\,{\rm
d}s\right\\}\right]\leq\exp\left\\{\frac{T}{4\sigma}\|H\|^{2}_{\infty}\right\\}\,.$
(4.32)

This yields that $(M_{t}^{1,N})_{t\in[0,T]}$ is a positive martingale.

Then by Girsanov’s Theorem, the stochastic process $\widetilde{B}_{t}^{1,N}$
defined in (4.30) is a $d$-dimensional Wiener process under the probability
measure $\mathbb{P}_{N}^{1}$ defined by

$\mathbb{P}_{N}^{1}(A):=\int_{A}M_{T}^{1,N}\,\mathrm{d}\mathbb{P},\quad\mbox{
for }A\in\mathscr{F}_{T}.$ (4.33)

This means that $(X^{1,N},V^{1,N})$ is the unique solution to

$\displaystyle\begin{cases}X_{t}^{1}=X_{0}^{1}+\int_{0}^{t}V_{s}^{1}\,{\rm
d}s,\\\\[3.0pt]
V_{t}^{1}=V_{0}^{1}+\sqrt{2\sigma}\widetilde{B}_{t}^{1,N}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+K_{t}^{1},\\\\[3.0pt]
K_{t}^{1}=-\sum\limits_{0<s\leq t}2(V_{s^{-}}^{1}\cdot
n(X_{s}^{i}))n(X_{s}^{1})\textbf{1}_{\\{X_{s}^{1}\in\partial
D\\}},\end{cases}$ (4.34)

under the new probability measure $\mathbb{P}_{N}^{1}$. Now the interactions
of particle 1 are included in the new Wiener process $\widetilde{B}^{1,N}$.

Step 2. Recall the fact that $(\widetilde{X}^{1},\widetilde{V}^{1})$ satisfies

$\displaystyle\begin{cases}\widetilde{X}_{t}^{1}=\widetilde{X}_{0}^{1}+\int_{0}^{t}\widetilde{V}_{s}^{1}\,\mathrm{d}s,\\\\[3.0pt]
\widetilde{V}_{t}^{1}=\widetilde{V}_{0}^{1}+\sqrt{2\sigma}B_{t}^{1}+\sqrt{2\overline{\sigma}}\overline{W}_{t}+\widetilde{K}_{t}^{1},\\\\[3.0pt]
\widetilde{K}_{t}^{1}=-\sum\limits_{0<s\leq t}2(\widetilde{V}_{s^{-}}^{1}\cdot
n(\widetilde{X}_{s}^{i}))n(\widetilde{X}_{s}^{1})\textbf{1}_{\\{\widetilde{X}_{s}^{1}\in\partial
D\\}},\end{cases}$ (4.35)

under the original probability measure $\mathbb{P}$. Thus for any
$\phi\in\mathcal{C}_{b}(D\times\mathbb{R}^{d})$, it holds that

$\displaystyle\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})]$
$\displaystyle=\int_{\Omega}\phi(X_{t}^{1,N}(\omega),V_{t}^{1,N}(\omega))\,\mathrm{d}\mathbb{P}(\omega)=\int_{\Omega}\phi(X_{t}^{1,N}(\omega),V_{t}^{1,N}(\omega))\frac{1}{M_{T}^{1,N}}\,\mathrm{d}\mathbb{P}_{N}^{1}(\omega)$
$\displaystyle=\int_{\Omega}\phi(\widetilde{X}_{t}^{1}(\omega),\widetilde{V}_{t}^{1}(\omega))\frac{1}{\widetilde{M}_{T}^{1,N}}\,\mathrm{d}\mathbb{P}(\omega)\,,$
(4.36)

where

$\widetilde{M}_{t}^{1,N}:=\exp\left\\{-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast\widetilde{f}^{N}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})\,\mathrm{d}{B}_{s}^{1}+\frac{1}{4\sigma}\int_{0}^{t}|H\ast\widetilde{f}^{N}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})|^{2}\,{\rm
d}s\right\\}\,.$ (4.37)

Here we have used the fact that

$\displaystyle M_{t}^{1,N}$
$\displaystyle=\exp\left\\{-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})\,\mathrm{d}{B}_{s}^{1}-\frac{1}{4\sigma}\int_{0}^{t}|H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})|^{2}\,{\rm d}s\right\\}$
$\displaystyle=\exp\left\\{-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})\,\mathrm{d}\widetilde{B}_{s}^{1,N}+\frac{1}{4\sigma}\int_{0}^{t}|H\ast
f^{N}_{s}(X_{s}^{1,N},V_{s}^{1,N})|^{2}\,{\rm d}s\right\\}\,,$ (4.38)

which under $\mathbb{P}_{N}^{1}$ has the same distribution as
$\widetilde{M}_{t}^{1,N}$ under $\mathbb{P}$.

Next we notice that

$\displaystyle|H\ast\widetilde{f}^{N}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})-H\ast\widetilde{f}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})|$
$\displaystyle=$ $\displaystyle\left|\iint
H(\widetilde{X}_{s}^{1}-x,\widetilde{V}_{s}^{1}-v)(\widetilde{f}^{N}_{s}-\widetilde{f}_{s})(\,\mathrm{d}x,\,\mathrm{d}v)\right|\to
0,\quad a.s.\mbox{ as }N\to\infty\,.$ (4.39)

This implies that $\widetilde{M}_{T}^{1,N}\to\widetilde{M}_{T}^{1}$,
$\mathbb{P}$ a.s., where

$\widetilde{M}_{t}^{1}:=\exp\left\\{-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast\widetilde{f}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})\,\mathrm{d}{B}_{s}^{1}+\frac{1}{4\sigma}\int_{0}^{t}|H\ast\widetilde{f}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})|^{2}\,{\rm
d}s\right\\}\,,$ (4.40)

which is obviously a martingale under $\mathbb{P}$. Indeed it is easy to see
that

$\displaystyle\,\mathrm{d}(\widetilde{M}_{t}^{1,N})^{-1}=(\widetilde{M}_{t}^{1,N})^{-1}(\frac{1}{\sqrt{2\sigma}}H\ast\widetilde{f}^{N}_{t}(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})\,\mathrm{d}{B}_{t}^{1})$
(4.41)

and

$\,\mathrm{d}(\widetilde{M}_{t}^{1})^{-1}=(\widetilde{M}_{t}^{1})^{-1}(\frac{1}{\sqrt{2\sigma}}H\ast\widetilde{f}_{t}(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})\,\mathrm{d}{B}_{t}^{1})\,.$
(4.42)

This yields that

$\displaystyle\,\mathrm{d}((\widetilde{M}_{t}^{1,N})^{-1}-(\widetilde{M}_{t}^{1})^{-1})$
$\displaystyle=((\widetilde{M}_{t}^{1,N})^{-1}-(\widetilde{M}_{t}^{1})^{-1})\left(\frac{1}{\sqrt{2\sigma}}H\ast\widetilde{f}^{N}_{t}\,\mathrm{d}{B}_{t}^{1}\right)$
$\displaystyle\quad+(\widetilde{M}_{t}^{1})^{-1}\left(\frac{1}{\sqrt{2\sigma}}H\ast(\widetilde{f}^{N}_{t}-\widetilde{f}_{t})\,\mathrm{d}{B}_{t}^{1}\right)\,.$
(4.43)

Hence it holds that

$\displaystyle\mathbb{E}[|(\widetilde{M}_{t}^{1,N})^{-1}-(\widetilde{M}_{t}^{1})^{-1}|^{2}]\leq
2\mathbb{E}[\int_{0}^{t}\frac{1}{2\sigma}|(\widetilde{M}_{s}^{1,N})^{-1}-(\widetilde{M}_{s}^{1})^{-1}|^{2}|H\ast\widetilde{f}^{N}_{s}|^{2}\,{\rm
d}s]$
$\displaystyle\quad+2\mathbb{E}[\int_{0}^{t}\frac{1}{2\sigma}|(\widetilde{M}_{s}^{1})^{-1}|^{2}|H\ast(\widetilde{f}^{N}_{s}-\widetilde{f}_{s}^{N})|^{2}\,{\rm
d}s]\,,$ (4.44)

which combining with (4) and using Gronwall’s inequality leads to

$\mathbb{E}[|(\widetilde{M}_{T}^{1,N})^{-1}-(\widetilde{M}_{T}^{1})^{-1}|^{2}]\to
0\quad\mbox{ as }N\to\infty\,.$ (4.45)

Letting $N\to\infty$ in (4) yields that

$\displaystyle\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})]\to\int_{\Omega}\phi(\widetilde{X}_{t}^{1}(\omega),\widetilde{V}_{t}^{1}(\omega))\frac{1}{\widetilde{M}_{T}^{1}}\,\mathrm{d}\mathbb{P}(\omega)=\mathbb{E}\left[\phi(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})\frac{1}{\widetilde{M}_{T}^{1}}\right]=\mathbb{E}_{\widehat{\mathbb{P}}}[\phi(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})]\,,$
(4.46)

where $\widehat{\mathbb{P}}$ satisfies

$\widehat{\mathbb{P}}(A)=\int_{A}\frac{1}{\widetilde{M}_{T}^{1}}\,\mathrm{d}\mathbb{P}\quad\mbox{
for }A\in\mathscr{F}_{T},$ (4.47)

and $(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$ under
$\widehat{\mathbb{P}}$ satisfies

$\displaystyle\begin{cases}\widetilde{X}_{t}^{1}=\widetilde{X}_{0}^{1}+\int_{0}^{t}\widetilde{V}_{s}^{1}\,\mathrm{d}s,\\\\[3.0pt]
\widetilde{V}_{t}^{1}=\widetilde{V}_{0}^{1}+\int_{0}^{T}H\ast\widetilde{f}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})ds+\sqrt{2\sigma}\widehat{B}_{t}^{1}+\sqrt{2\overline{\sigma}}\,\overline{W}_{t}+\widetilde{K}_{t}^{1},\\\\[3.0pt]
\widetilde{K}_{t}^{1}=-\sum\limits_{0<s\leq t}2(\widetilde{V}_{s^{-}}^{1}\cdot
n(\widetilde{X}_{s}^{1}))n(\widetilde{X}_{s}^{1})\textbf{1}_{\\{\widetilde{X}_{s}^{1}\in\partial
D\\}}.\end{cases}$ (4.48)

Here,

$\widehat{B}_{t}^{1}:=B_{t}^{1}-\frac{1}{\sqrt{2\sigma}}\int_{0}^{t}H\ast\widetilde{f}_{s}(\widetilde{X}_{s}^{1},\widetilde{V}_{s}^{1})\,{\rm
d}s$ (4.49)

is a Wiener process under the new probability measure $\widehat{\mathbb{P}}$.
Namely
$(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1},\widehat{B}_{t}^{1},\widehat{\mathbb{P}})$
is a weak solution to (4.28).

Step 3. On the other hand, by Corollary 3.2 we also have

$\displaystyle\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})]\to\mathbb{E}[\phi(\overline{X}_{t}^{1},\overline{V}_{t}^{1})],\quad\mbox{
as }N\to\infty.$ (4.50)

This means that

$\mathbb{E}[\phi(\overline{X}_{t}^{1},\overline{V}_{t}^{1})]=\mathbb{E}_{\widehat{\mathbb{P}}}[\phi(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})].$
(4.51)

Namely, $(\overline{X}_{t}^{1},\overline{V}_{t}^{1})$ under $\mathbb{P}$ has
the same distribution as $(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$ under
$\widehat{\mathbb{P}}$.

In addition, one has for all $\xi\in
L^{\infty}(\Omega,\mathscr{F}_{T}^{\overline{W}};{\mathbb{R}})$,

$\displaystyle\mathbb{E}[\phi(\overline{X}_{t}^{1},\overline{V}_{t}^{1})\xi]$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbb{E}[\phi(X_{t}^{1,N},V_{t}^{1,N})\xi]=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}[\phi(X_{t}^{i,N},V_{t}^{i,N})\xi]$
$\displaystyle=\lim_{N\rightarrow\infty}\mathbb{E}[\langle\phi,f^{N}_{t}\rangle\xi]=\mathbb{E}[\langle\phi,f_{t}\rangle\xi]\,,$

where we have used (4.5)-(4.6) and the exchangeability of particles.
Furthermore, since $f$ is $(\mathscr{F}_{t}^{\overline{W}})_{t\geq
0}$-adapted, we have

$\mathbb{E}\left[\phi(\overline{X}_{t}^{1},\overline{V}_{t}^{1})\big{|}\mathscr{F}_{t}^{\overline{W}}\right]=\mathbb{E}[\langle\phi,f_{t}\rangle|\mathscr{F}_{t}^{\overline{W}}]=\langle\phi,f_{t}\rangle\,,$
(4.52)

which implies

$f_{t}(x,v)\,{\rm d}x\,{\rm
d}v=\mathbb{E}[(\overline{X}_{t}^{1},\overline{V}_{t}^{1})\in(\,{\rm
d}x,\,{\rm
d}v)|\mathscr{F}_{t}^{\overline{W}}]=\mathbb{E}_{\widehat{\mathbb{P}}}[(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})\in(\,{\rm
d}x,\,{\rm d}v)|\mathscr{F}_{t}^{\overline{W}}]\,.$ (4.53)

Namely $f$ is the conditional density of
$(\widetilde{X}_{t}^{1},\widetilde{V}_{t}^{1})$ given
$\mathscr{F}_{t}^{\overline{W}}$. ∎

The following result is an analogue to [bossy2018particle, Corollary 2.3]:

######  Lemma 4.1.

Let $f\in\mathcal{L}^{\infty}_{\mathscr{F}}(L^{1}(D\times\mathbb{R}^{d}))$ be
the limit process obtained in Theorem 4.2. Then its trace function $\gamma(f)$
satisfies that for all $0<t\leq T$,

$\displaystyle\mathbb{E}\left[\sum_{0<s\leq
t}\left(\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s})-\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\right)\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}\big{|}\mathscr{F}_{t}^{\overline{W}}\right]$ (4.54)
$\displaystyle\qquad=-\int_{\Sigma_{t}}v\cdot
n(x)\gamma(f)(s,x,v)\varphi(s,x,v)\,\mathrm{d}\lambda_{\Sigma_{t}}(s,x,v)\quad\mbox{a.s.}$

holds for all bounded measurable functions $\varphi$ on $\Sigma_{t}$, where
the processes $\widetilde{Y}$ and $\widetilde{U}$ are from Proposition 4.3.

###### Proof.

For any
$\varphi\in\mathcal{C}_{c}^{2}\left((0,t)\times\overline{D}\times{\mathbb{R}}^{d}\right)$,
according to Definition 2.1 we have

$\displaystyle-\int_{(0,t)\times\partial D\times\mathbb{R}^{d}}n(x)\cdot
v\gamma(f)(s,x,v)\varphi(s,x,v)\,\mathrm{d}\lambda_{\Sigma_{t}}(s,x,v)$
$\displaystyle=$ $\displaystyle\lim_{\delta\rightarrow
0^{+}}\frac{1}{\delta}\int_{0<\ell(x)\leq\delta}\int_{(0,t)\times\mathbb{R}^{d}}\nabla\ell(x)\cdot
vf_{s}(x,v)\varphi(s,x,v)\mathrm{d}x\mathrm{d}sdv\,,$ (4.55)

where we have also used the fact that $\nabla\ell(x)=-n(x)$ when $x\in\partial
D$. For a fixed $\delta>0$, we set
$\beta_{\delta}(y):=1-(\frac{\ell(y)}{\delta}\wedge 1)$ a.e. differentiable on
$\mathring{D}=\\{y\in{\mathbb{R}}^{d}:~{}\ell(y)>0\\}$. Applying Itô’s
formula, it gives that

$\displaystyle\beta_{\delta}(\widetilde{Y}_{t})\varphi(t,\widetilde{Y}_{t},\widetilde{U}_{t})=\beta_{\delta}(\widetilde{Y}_{0})\varphi(0,Y_{0},U_{0})+\int_{0}^{t}\beta_{\delta}(\widetilde{Y}_{s})\frac{\partial}{\partial
s}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s})\,{\rm
d}s+\int_{0}^{t}\beta_{\delta}(\widetilde{Y}_{s})\nabla_{x}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\widetilde{U}_{s}\,{\rm
d}s$
$\displaystyle+\int_{0}^{t}\beta_{\delta}(\widetilde{Y}_{s})\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot
H\ast\widetilde{f}_{s}(\widetilde{Y}_{s},\widetilde{U}_{s})\,{\rm
d}s++\sqrt{2\sigma}\int_{0}^{t}\beta_{\delta}(\widetilde{Y}_{s})\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\,\mathrm{d}B_{s}$
$\displaystyle+\sqrt{2\overline{\sigma}}\int_{0}^{t}\beta_{\delta}(\widetilde{Y}_{s})\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\,\mathrm{d}\overline{W}_{s}+\sum_{0<s\leq
t}\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\Delta\widetilde{K}_{s}\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}+\int_{0}^{t}(\sigma+\overline{\sigma})\beta_{\delta}(\widetilde{Y}_{s})\triangle_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\,{\rm
d}s$ $\displaystyle+\sum_{0<s\leq
t}\left(\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s})-\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})-\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\Delta\widetilde{U}_{s}\right)\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}$
$\displaystyle-\int_{0}^{t}\frac{1}{\delta}\nabla\ell(\widetilde{Y}_{s})\cdot\widetilde{U}_{s^{-}}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\textbf{1}_{\\{\ell(\widetilde{Y}_{s})\leq\delta\\}}\,{\rm
d}s\,.$ (4.56)

Moreover, according to (2.9), one has

$\sum_{0<s\leq
t}\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\Delta\widetilde{K}_{s}\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}=\sum_{0<s\leq
t}\nabla_{v}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\cdot\Delta\widetilde{U}_{s}\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}\,.$

Thus, taking expectation on both side of (4), letting $\delta\to 0^{+}$ (so
$\beta_{\delta}\to 0$) and using (4), we conclude:

$\displaystyle\mathbb{E}\left[\sum_{0<s\leq
t}\left(\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s})-\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\right)\textbf{1}_{\\{\widetilde{Y}_{s}\in\partial
D\\}}\big{|}\mathscr{F}_{t}^{\overline{W}}\right]$ $\displaystyle=$
$\displaystyle-\lim_{\delta\to
0^{+}}\frac{1}{\delta}\int_{0}^{t}\mathbb{E}[\nabla\ell(\widetilde{Y}_{s})\cdot\widetilde{U}_{s^{-}}\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})\textbf{1}_{\\{\ell(\widetilde{Y}_{s})\leq\delta\\}}\big{|}\mathscr{F}_{t}^{\overline{W}}]\,{\rm
d}s$ $\displaystyle=$ $\displaystyle-\lim_{\delta\rightarrow
0^{+}}\frac{1}{\delta}\int_{0<\ell(x)\leq\delta}\int_{(0,t)\times\mathbb{R}^{d}}\nabla\ell(x)\cdot
vf_{s}(x,v)\varphi(s,x,v)\mathrm{d}s\mathrm{d}xdv$ $\displaystyle=$
$\displaystyle-\int_{(0,t)\times\partial D\times\mathbb{R}^{d}}n(x)\cdot
v\gamma(f)(s,x,v)\varphi(s,x,v)\,\mathrm{d}\lambda_{\Sigma_{t}}(s,x,v)\,,$
(4.57)

for any
$\varphi\in\mathcal{C}_{c}^{2}\left((0,t)\times\overline{D}\times{\mathbb{R}}^{d}\right)$.
Thanks to (2.8), the property in (4) can be extended to any bounded measurable
function $\varphi$ on $\Sigma_{t}$. This completes the proof. ∎

Finally we can show that the limit process $f$ obtained in Theorem 4.2
satisfies the reflecting boundary condition as follows.

######  Theorem 4.4.

Let $f$ be the limit process obtained in Theorem 4.2. Then its trace function
$\gamma(f)$ satisfies

$\gamma(f)(s,x,v)=\gamma(f)(s,x,v-2(v\cdot
n(x))n(x)),\quad\,\mathrm{d}\lambda_{\Sigma_{t}}-a.e.\mbox{ on
}\Sigma_{t}\quad\mbox{a.s.},$ (4.58)

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

###### Proof.

In (4.54) choose $\varphi$ of the form

$\varphi(s,x,v)=\phi(s,x,v)+\phi(s,x,v-2(v\cdot n(x))n(x))\,.$

with $\phi$ being any bounded measurable function on $\Sigma_{t}$. Since

$\widetilde{U}_{s}=\widetilde{U}_{s^{-}}-2(\widetilde{U}_{s^{-}}\cdot
n(\widetilde{Y}_{s}))n(\widetilde{Y}_{s}),\quad\widetilde{U}_{s^{-}}=\widetilde{U}_{s}-\widetilde{2}(U_{s}\cdot
n(\widetilde{Y}_{s}))n(\widetilde{Y}_{s})\,,$

one has

$\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s})-\varphi(s,\widetilde{Y}_{s},\widetilde{U}_{s^{-}})=0\,.$

Then (4.54) implies that

$\int_{\Sigma_{t}}v\cdot
n(x)\gamma(f)(s,x,v)\left(\phi(s,x,v)+\phi(s,x,v-2(v\cdot
n(x))n(x))\right)\,{\rm d}s\,\mathrm{d}\sigma_{\partial
D}(x)\,\mathrm{d}v=0\,.$

Let us define

$\overline{v}:=v-2(v\cdot n(x))n(x)\,,$

then it holds that

$\overline{v}\cdot n(x)=-v\cdot n(x)\quad\mbox{ and
}\quad\det(D\overline{v}(v))=-1.$

It follows by a change of variable that

$\int_{\Sigma_{t}}\overline{v}\cdot
n(x)\left(\gamma(f)(s,x,\overline{v})-\gamma(f)(s,x,\overline{v}-2(\overline{v}\cdot
n(x))n(x))\right)\phi(s,x,\overline{v})\,{\rm d}s\,\mathrm{d}\sigma_{\partial
D}(x)\,\mathrm{d}\overline{v}=0\quad\mbox{a.s.}\,,$

which by vanishing lemma leads to (4.58). ∎

## References