# Mean reflected Mckean-Vlasov stochastic differential equation

Liu Xiangdong∗<EMAIL_ADDRESS>Hong Shaopeng<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate a class of mean reflected McKean-Vlasov
stochastic differential equation, which extends the equation proposed by [1]
by allowing the solution’s distribution to not only constrain its behavior,
but also affect the diffusion and drift coefficients. We establish the
existence and uniqueness results of this class, investigate the propagation of
chaos, and examine the stability properties with respect to the initial
condition, coefficients, and driving process. Moreover, we provide a rigorous
proof of a Fredlin-Wentzell type large deviations principle using the weak
convergence method. We also demonstrate the existence of an invariant measure
and employ the coupling by change of measure method to prove log-Harnack
inequality and shift Harnack inequality. Our study sheds light on the
properties and behaviors of these reflected Mckean-Vlasov stochastic
differential equation and contributes to the ongoing research in this field.

###### keywords:

Mckean-Vlasov stochastic differential equation , mean reflected , particle
system , stability analysis , large derivation principle , invariant measure ,
Harnack inequality

###### MSC:

[2020] 60H10

organization=Department of Statistic and Data Science, School of
Economics,addressline=Jinan University, city=Guangzhou, postcode=510632,
state=Guangdong, country=PR China

##  1 Introduction

In this paper, we consider the following _mean reflected Mckean-Vlasov
stochastic differential equation_ (MR-MVSDE for short).

$\left\\{\begin{array}[]{l}X_{t}=\xi+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}+K_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}=0,\quad
t\geq 0.\end{array}\right.$ (1.1)

Where $\mathcal{L}_{X_{s}}$ is the law of $X_{s}$, $(B_{t})_{t\geq 0}$is a
standard Brownian motion defined on some complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$. We will always assume that h is
nondecreasing and that the law of $\xi$ is such that
$\mathbb{E}\left[h(\xi)\right]\geq 0$. For the precise conditions on
$b,\sigma,h$, we refer the reader to Section 2. The solution to (1.1) is the
couple of continuous processes $(X,K)$, $K$ being needed to ensure that the
constraint is satisfied, in a minimal way according to the last condition
namely the Skorokhod condition.

The Mckean-Vlasov stochastic differential equation (Mckean-Vlasov SDE for
short), also referred to as the mean-field stochastic differential equation or
distribution-dependent stochastic differential equation, was first introduced
in the literature by [2] and [3]. In recent years, it has garnered significant
attention from scholars. The Mckean-Vlasov SDE is used to describe the
limiting behavior of mean-field interactive particle systems as $n$ goes to
infinity. We refer to [4] for a detailed exposition and references. Currently,
the Mckean-Vlasov SDE plays a crucial role in various fields such as mean
field games, financial derivatives pricing, measure value processes, see e.g.
[5, 6, 7, 8] and references therein.

The reflected SDE was introduced by [9] and [10]. Since then, it has been
widely researched, see e.g. [11, 12, 13, 14]. This type of equation restricts
the path of the solution to a given domain. The reflected Mckean-Vlasov SDE,
first proposed by [15], is well-posed, and its solution satisfies the
propagation of chaos. Recently, [16] studied the Mckean-Vlasov SDE with self-
stabilizing coefficients on unbounded convex regions, replacing the classical
Wasserstein Lipschitz condition. Their work proved the existence and
uniqueness of the solution, demonstrated chaos propagation, and examined the
Freidlin-Wentzell type Large Deviations Principle, as well as an Eyring-
Kramer’s law for the exit-time from subdomains that lie within the interior of
the reflecting domain. In a similar vein, [17] investigated the Mckean-Vlasov
SDE with oblique reflection on non-smooth time-dependent domains. Their
research proved the existence and uniqueness of the solution, demonstrated
chaos propagation, and examined the Freidlin-Wentzell type Large Deviations
Principle.

In their work, [18] introduced the concept of mean reflective backward
stochastic differential equation (MR-BSDE), which places restrictions not on
the path of the solution, but on its distribution. They proved that, under
certain conditions, MR-BSDE has a unique solution. Further research on MR-BSDE
can be found in [19, 20, 21], and their respective references. [1] studied the
mean reflected stochastic differential equation (MR-SDE for short),
established the well-posedness of the solution, and analyzed the propagation
of chaos. They also proposed a numerical simulation method based on particle
approximation techniques. [22] extended this work to Lévy driven MR-SDE. [23]
examined the MR-SDE with two reflective walls and solved the Skorokhod problem
with the mean minimum condition. They also studied the stability of the
solution. In a related study, [24] investigated SDE with normal reflection in
conditional law and BSDE with normal constraint in law. Using the penalty
method, they established the well-posedness of these two methods and proved
the forward chaos propagation and backward propagation of chaos [25].
Additionally, they studied a related obstacle problem on the Wasserstein
space. Overall, these studies contribute to a better understanding of MR-BSDE
and MR-SDE and their applications in various fields.

The large deviation principle (LDP) is primarily concerned with asymptotic
estimates of the probability of tail events $A$ , especially whether
$\mathbb{P}\left(X_{\varepsilon}\in A\right)$ converges to zero at an
exponential rate as $\varepsilon\rightarrow 0$. In this paper, we focus on the
Freidlin-Wentzell type of LDP. The study of large deviations for stochastic
differential equations originates from [26], who first studied the large
deviation principle for diffusion processes. Recently, the weak convergence
method [27] based on the variational representation of Brownian motion [28]
has become an important and effective tool for proving LDP. Using this method,
many LDP for stochastic differential equations and stochastic partial
differential equations have been established, see e.g. [29, 30, 31, 32, 33,
34]. For more on the weak convergence method and LDP, see monograph [35]. [36]
proposed sufficient conditions for verifying LDP that are easier to verify and
used these conditions to prove LDP for quasilinear stochastic partial
differential equations with reflection. Currently, there are relatively few
studies on McKean-Vlasov SDE. Using the classical exponential equivalence
method,[37] and [38] proved LDP for McKean-Vlasov SDE. For reflected McKean-
Vlasov SDE, [16] used the exponential equivalence method to prove the large
deviation principle. Notably, [39] proposed a new sufficient condition to
verify the criteria of the weak convergence method, and show correct
controlled equation for the McKean-Vlasov SDE with jumps, which was used to
prove the large deviation principle and moderate deviation principle for the
McKean-Vlasov SDE with jumps. [40] using weak convergence method shows the LDP
of MR-SDE with jump.

The invariant probability measure is an important tool for studying the long-
term behavior of SDE. The principle of invariant probability measure for
classical SDE or distribution independent SDE has been extensively studied, as
seen in [41, 42, 43, 44, 45]. However, due to the fact that the McKean-Vlasov
SDE is not a Markov process, the classical proof method cannot be used. [46]
first established the $\mathcal{W}_{2}$-exponential convergence of the
solution of the McKean-Vlasov SDE, and subsequently proved the uniqueness and
existence of the invariant probability measure. [47] studied the exponential
ergodicity of McKean-Vlasov SDE with additive pure jump noise in the relative
entropy and (weighted) Wasserstein distances. [48] studied the exponential
ergodicity of singular McKean-Vlasov SDE with or without reflection in the
relative entropy and (weighted) Wasserstein distances. [49] studied the
Donsker-Varadhan large deviation of the invariant measure for path-dependent
McKean-Vlasov SPDE.

The Harnack inequality is an important inequality for studying the properties
of solutions. [50] first proposed a dimension-free Harnack inequality. If we
denote $\left\\{P_{t}\right\\}_{t\geq 0}$ is the transition semigroup of a
Markov process , the dimension-free Harnack inequality states that

$\left(P_{t}f\right)^{\alpha}(x)\leq
P_{t}f^{\alpha}(x+y)\exp\left\\{C_{\alpha}(y,t)\right\\},x,y\in\mathbb{R}^{d},$

where $\alpha>1$,
$C_{\alpha}:\mathbb{R}^{d}\times(0,\infty)\rightarrow\mathbb{R}_{+}$ satisfies
$C_{\alpha}(0,t)=0$, and $x,y\in\mathbb{R}^{d}$. [51] proved the dimension-
free Harnack inequality for a large class of stochastic differential equations
with multiplicative noise by constructing a coupling with unbounded time-
dependent drift. This coupling method, also known as the variable measure
coupling method, has been used to prove the Harnack inequality for many SDE
and SPDE, see e.g., [52, 53, 54, 55, 56] and monograph [57]. For the McKean-
Vlasov SDE, due to technical difficulties, we can only prove the Harnack
inequality when the noise coefficient is independent of the distribution. [46]
used the variable measure coupling method to prove the Harnack inequality for
the McKean-Vlasov SDE when the noise coefficient is independent of the
distribution. [58] studied the Harnack inequality when the drift contains a
locally standard integrable term and a Lipschitz continuous term in the
spatial variable and is Lipschitz continuous in the distribution variable with
respect to a weighted variation distance. [59] studied the Harnack inequality
for McKean-Vlasov SDE, where the drift satisfies a standard integrability
condition in time-space and may be discontinuous in the distance induced by
any Dini function. [60] developed a novel coupling method to effectively prove
the shift Harnack inequality. Utilizing this new coupling method, we prove the
shift Harnack inequality for the McKean-Vlasov SDE when the noise coefficient
is independent of the spatial variable. [61] proved the shift Harnack
inequality for the McKean-Vlasov SPDE in a separable Hilbert space with a Dini
continuous drift. [62] used Zvonkin’s transform to prove the Harnack
inequality and shift Harnack inequality for the McKean-Vlasov SDE when the
coefficients are Dini continuous in the spatial variable.

_Our contributions._

To the best of our knowledge, there are few results on the MR-MVSDE. The aim
of this paper is to investigate this type of MR-MVSDE. In Section 3, we first
establish the existence and uniqueness of a strong solution $(X,K)$ for (1.1),
followed by a study of its properties, including: (i) the relationship between
(1.1) and a PDE with Neumann boundary conditions in the Wasserstein space,
i.e., the Feynman-Kac formula (Proposition 3.3); (ii) $L^{p}$ estimates for
the solution (Proposition 3.4); (iii) the absolute continuity of the Stieljes
measure $dK$ with respect to the Lebesgue measure, under certain regularity
assumptions on $h$, and an explicit expression for the density (Proposition
3.6). In Section 4, we study propagation of chaos (Theorem 4.1), i.e., we
prove the convergence of an interacting particle system (see (4.2)) to (1.1).

In Section 5, we investigate the impact of small perturbations on the system.
Specifically, we study the effects of perturbations in the initial condition
(Theorem 5.1), the coefficients (Theorem 5.2), and the driving process
(Theorem 5.3). We then use Proposition 5.7 and Proposition 5.9 to prove a
large deviation principle for (1.1) (Theorem 5.6). In Section 6, we focus on
the long-time behavior of solutions and the Harnack inequality. We prove the
existence and uniqueness of an invariant measure for (1.1) (Theorem 6.2).
Using coupling method, we prove log-Harnack inequality (Theorem 6.4) and shift
Harnack inequality (Theorem 6.5) for (1.1).

The paper is organised in the following way: Section 2 introduces the
Wasserstein metric and presents some Assumptions and preliminary results.
Section 3 shows that the MR-MVSDE admits a unique strong solution $(X,K)$.
Then we study the density of $K$. Section 4 shows that system (1.1) can be
seen as the limit of an interacting particles system with oblique reflection
of mean field type. Section 5 studies the effect of small noise on the system.
Section 6 proves the existence of invariant probability measure and the
Harnack inequality.

Throughout the paper, we adopt the following convention:
$\mathcal{B}^{+}_{b}(\mathbb{R})$ refers to the space of nonnegative and
bounded measurable functions on $\mathbb{R}$. The symbol $C$ denotes various
positive constants whose specific values may vary in different contexts.

##  2 Preliminary

In this section, we primarily focus on the Wasserstein space and some
preliminary results.

###  2.1 The Wasserstein space

Let $\mathcal{P}\left(\mathbb{R}\right)$ be the space of probability measures
on $\mathbb{R}$ and for any $p>1$ denote by
$\mathcal{P}_{p}\left(\mathbb{R}\right)$ the subspace of
$\mathcal{P}\left(\mathbb{R}\right)$ of probability measures with finite
moment of order $p$. Now, we can defined $p$-Wasserstein distance For
$\mu,\nu\in\mathcal{P}_{p}\left(\mathbb{R}\right)$, define the $p$-Wasserstein
distance $\mathcal{W}_{p}(\mu,\nu)$ by

$\mathcal{W}_{p}(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\left[\int_{\mathbb{R}\times\mathbb{R}}|x-y|^{p}d\pi(x,y)\right]^{1/p}$

where $\Pi(\mu,\nu)$ denotes the set of probability measures on
$\mathbb{R}\times\mathbb{R}$ whose first and second marginals are,
respectively, $\mu$ and $\nu$. An equivalent and more probabilistic definition
is

$\mathcal{W}_{p}(\mu,\nu)=\left(\inf_{X\sim\mu,Y\sim\nu}\mathbb{E}\left[\left|X-Y\right|^{p}\right]\right)^{\frac{1}{p}}\leq\mathbb{E}\left[\left|X-Y\right|^{p}\right]^{\frac{1}{p}}$

In following result, we mainly using $1$-Wasserstein distance and
$2$-Wasserstein distance. More detail about Wasserstein metric can see [63]
and references therein.

Then, we introduce the L-derivative (Lions’ derivative). A function
$f:\mathcal{P}_{2}\left(\mathbb{R}\right)\mapsto\mathbb{R}$ is differential at
$\mu\in\mathcal{P}_{2}\left(\mathbb{R}\right)$, if for lift map
$\tilde{f}(\gamma):=f\left(\mathcal{L}_{\gamma}\right),\gamma\in$
$L^{2}\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}\right)$, there exists
some $\zeta\in L^{2}\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}\right)$
with $\mathcal{L}_{\zeta}=\mu$ such that $\tilde{f}$ is Fréchet differentiable
at $\zeta$, that is, there exists a linear continuous mapping
$D_{\mu}\tilde{f}(\zeta)$ :
$L^{2}\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}\right)\mapsto\mathbb{R}$
such that for any $\eta\in
L^{2}\left(\Omega,\mathcal{F},\mathbb{P};\mathbb{R}\right)$

$\tilde{f}(\zeta+\eta)-\tilde{f}(\zeta)=D_{\mu}\tilde{f}(\zeta)(\eta)+o\left(|\eta|_{L^{2}}\right),\quad|\eta|_{L^{2}}\rightarrow
0$

In order to obtain the Feynman-Kac formula for (1.1), it is necessary to make
certain regularity assumptions on function $f$ and apply a special version of
Itô’s lemma, as demonstrated in the work of [24].

###### Defination 2.1

We say $f$ is partially $C^{2}$ if the map $y\rightarrow$
$\mathrm{D}_{\mu}f(\mu)(y)$ is continuously differentiable with a derivative
$(\mu,y)\rightarrow\partial_{y}\mathrm{D}_{\mu}f(\mu)(y)$ jointly continuous
on $\mathcal{P}_{2}\left(\mathbb{R}\right)\times\mathbb{R}$

######  Lemma 2.2 (Itô’s lemma)

Consider

$dX_{t}=b_{t}dt+\sigma_{t}dB_{t}+dK_{t}.$

where $K_{t}$ is a deterministic, continuous and nondecreasing process with
$K_{0}=0$, $\left(b_{t}\right)$ and $\left(\sigma_{t}\right)$ are
progressively measurable with values in $\mathbb{R}$ and satisfy

$\mathbb{E}\left[\int_{0}^{T}\left(\left|b_{s}\right|^{2}+\left|\sigma_{s}\right|^{4}\right)ds\right]<+\infty$

If $f$ is partially $C^{2}$, we have

$\begin{gathered}f\left(\mathcal{L}_{X_{t}}\right)=f\left(\mathcal{L}_{X_{t}}\right)+\int_{0}^{t}\mathbb{E}\left[\mathrm{D}_{\mu}f\left(\mathcal{L}_{X_{t}}\right)\left(X_{s}\right)b_{s}\right]ds+\int_{0}^{t}\mathbb{E}\left[\mathrm{D}_{\mu}f\left(\mathcal{L}_{X_{t}}\right)\left(X_{s}\right)\right]dK_{s}\\\
+\frac{1}{2}\int_{0}^{t}\mathbb{E}\left[\left(\sigma_{s}^{2}\partial_{y}\mathrm{D}_{\mu}f\left(\mathcal{L}_{X_{t}}\right)\left(X_{s}\right)\right)\right]ds,\end{gathered}$

###  2.2 Assumptions and preliminary results

We make following assumptions about the coefficients:

Assumption. (A.1)

  1. 1.

Lipschitz assumption: there exists a constant $K>0$ such that for all
$x,y\in\mathbb{R}$, $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R})$ we have

$|b(x,\mu)-b(y,\nu)|+|\sigma(x,\mu)-\sigma(y,\nu)|\leq
K\left(|x-y|+\mathcal{W}_{2}(\mu,\nu)\right)$

  2. 2.

The random variable $\xi$ is square integrable, i.e.
$\mathbb{E}\left[\xi^{2}\right]<\infty$.

###### Remark 2.1

The Assumption (A1) implies that for any $t\in[0,T]$, the coefficients $b$ and
$\sigma$ satisfy

$|b(x,\mu)|+|\sigma(x,\mu)|\leq C(1+K)$

Assumption. (A.2)

  1. 1.

The function $h:\mathbb{R}\rightarrow\mathbb{R}$ is increasing and bi-
Lipschitz. There exist $0<m\leq M$ such that

$\forall x\in\mathbb{R},\forall y\in\mathbb{R},\quad m|x-y|\leq|h(x)-h(y)|\leq
M|x-y|$

  2. 2.

The initial condition $\xi$ satisfies $\mathbb{E}\left[h(\xi)\right]\geq 0$

Assumption. (A.3) There exists $p>4$ such that $\xi\in L^{p}$, i.e.
$\mathbb{E}\left[|\xi|^{p}\right]<\infty$.

Assumption. (A.4) The function $h$ is twice continuously differentiable with
bounded derivatives.

Assumption. (A.5) For some constants $C_{2}>C_{1}\geq 0$,

$2(x-y)(b(x,\mu)-b(y,\nu))+|\sigma(x,\mu)-\sigma(y,\nu)|\leq
C_{1}\mathcal{W}_{2}(\mu,\nu)^{2}-C_{2}|x-y|^{2}$

We now introduce function $H$, $\bar{G}_{0}$ and $G$ and some properties from
[1].

$H:\mathbb{R}\times\mathcal{P}_{1}(\mathbb{R})\ni(x,v)\mapsto\int h(x+z)v(dz)$
(2.1)

Let $\bar{G}_{0}$ be the inverse function in space of $H$ evaluated at 0 ,

$\bar{G}_{0}:\mathcal{P}_{1}(\mathbb{R})\ni
v\mapsto\inf\\{x\in\mathbb{R}:H(x,v)\geq 0\\}$ (2.2)

and let $G_{0}$ denote the positive part of $\bar{G}_{0}$,

$\displaystyle G_{0}:\mathcal{P}_{1}(\mathbb{R})\ni\nu\mapsto\inf\\{x\geq
0:H(x,v)\geq 0\\}$ (2.3)

###### Lemma 2.3

Under Assumption (A.2) we have:

  1. 1.

For all $\nu$ in $\mathcal{P}_{1}(\mathbb{R})$, the mapping
$H(\cdot,\nu):\mathbb{R}\ni x\mapsto H(x,v)$ is a bi-Lipschitz function,
namely:

$\forall x,y\in\mathbb{R},\quad m|x-y|\leq|H(x,v)-H(y,v)|\leq M|x-y|.$

  2. 2.

For all $x$ in $\mathbb{R}$, the mapping
$H(x,\cdot):\mathcal{P}_{1}(\mathbb{R})\ni\nu\mapsto H(x,v)$ satisfies the
following Lipschitz estimate:

$\forall
v,v^{\prime}\in\mathcal{P}_{1}(\mathbb{R}),\quad\left|H(x,v)-H\left(x,v^{\prime}\right)\right|\leq\left|\int
h(x+\cdot)\left(dv-dv^{\prime}\right)\right|$

###### Lemma 2.4

Under Assumption (A.2), the mapping
$G_{0}:\mathcal{P}_{1}(\mathbb{R})\ni\nu\mapsto G_{0}(\nu)$ is Lipschitz
continuous in the following sense:

$\left|G_{0}(v)-G_{0}\left(v^{\prime}\right)\right|\leq\frac{1}{m}\left|\int
h\left(\bar{G}_{0}(v)+\cdot\right)\left(dv-dv^{\prime}\right)\right|$

where we recall that $\bar{G}_{0}(v)$ is the inverse of $H(\cdot,v)$ at point
0 . In particular,

$\left|G_{0}(v)-G_{0}\left(v^{\prime}\right)\right|\leq\frac{M}{m}\mathcal{W}_{1}\left(v,v^{\prime}\right).$

##  3 Existence, uniqueness and relative properties of the solution

###  3.1 Existence and Uniqueness of (1.1)

We now introduce the definition of the solution to equation.

###### Defination 3.1

A couple of processes $(X,K)$ is said to be a flat deterministic solution to
(1.1) if $(X,K)$ satisfy (1.1) with $K$ being a non-decreasing continuous
deterministic function with $K_{0}=0$.

Given this definition, we have the following result.

###### Theorem 3.2

Under Assumption (A.1) and (A.2), the MR-MVSDE (1.1) has a unique
deterministic flat solution $(X,K)$. Moreover,

$\forall t\geq 0,\quad K_{t}=\sup_{s\leq t}\inf\left\\{x\geq
0:\mathbb{E}\left[h\left(x+U_{s}\right)\right]\geq 0\right\\}=\sup_{s\leq
t}G_{0}\left(\mathcal{L}_{U_{s}}\right),$ (3.1)

where $\left(U_{t}\right)_{0\leq t\leq T}$ is the process defined by :

$\displaystyle
U_{t}=\xi+\int_{0}^{t}b(X_{s},\mathcal{L}_{X_{s}})ds+\int_{0}^{t}\sigma(X_{s},\mathcal{L}_{X_{s}})dB_{s}.$
(3.2)

With these notations, denoting by $\mathcal{L}_{U_{s}}$ the family of marginal
laws of $\left(U_{t}\right)_{0\leq t\leq T}$ we have

$K_{t}=\sup_{s\leq t}G_{0}(\mathcal{L}_{U_{s}}).$

###### Proof 1

The proof for the case of MR-SDE is given in [1]. We present here the proof of
the MR-MVSDE.

Let us consider the set $\mathcal{C}^{2}=\left\\{X\text{ is
}\mathcal{F}-\text{adapted continuous process},\mathbb{E}\left(\sup_{0\leq
t\leq T}|X_{t}|^{2}<\infty\right)\right\\}$. And let
$\widetilde{X}\in\mathcal{C}^{2}$ be a given process. We set

$\tilde{U}_{t}=\xi+\int_{0}^{t}b\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)ds+\int_{0}^{t}\sigma\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)dB_{s},\quad\operatorname{Law}\left(\tilde{U}_{s}\right)=:\mathcal{L}_{\tilde{U}_{s}}$
(3.3)

and define the function $K$ by setting

$K_{t}=\sup_{s\leq t}\inf\left\\{x\geq
0:\mathbb{E}\left[h\left(x+\tilde{U}_{s}\right)\right]\geq
0\right\\}=G_{0}\left(\tilde{\mu}_{s}\right).$

The function $K$ being given, let us define the process $X$ by the formula

$\displaystyle
X_{t}=\xi+\int_{0}^{t}b\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)ds+\int_{0}^{t}\sigma\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)dB_{s}+K_{t}.$
(3.4)

Let us check $(X,K)$ is the solution of (1.1) with $U_{t}$ replaced by
$\tilde{U}_{t}$. First, based on the definition of $K$, we have
$\mathbb{E}\left[h(X_{t})\right]\geq 0$,
$K_{t}=G_{0}\left(\mathcal{L}_{\tilde{U}_{t}}\right),dK_{t}-a.e.$ and
$G_{0}(\mathcal{L}_{\tilde{U}_{t}})>0,dK_{t}-a.e.$ Then we obtain

$\displaystyle\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}$
$\displaystyle=\int_{0}^{t}\mathbb{E}\left[h\left(\tilde{U}_{s}+K_{s}\right)\right]dK_{s}$
$\displaystyle=\int_{0}^{t}\mathbb{E}\left[h\left(\tilde{U}_{s}+G_{0}\left(\mathcal{L}_{\tilde{U}_{s}}\right)\right)\right]dK_{s}$
$\displaystyle=\int_{0}^{t}\mathbb{E}\left[h\left(\tilde{U}_{s}+G_{0}\left(\mathcal{L}_{\tilde{U}_{s}}\right)\right)\right]\mathbbm{1}_{\left\\{G_{0}\left(\mathcal{L}_{\tilde{U}_{s}}\right)>0\right\\}}dK_{s}.$

Moreover, since $h$ is continuous, we have
$\mathbb{E}\left[h\left(\tilde{U}_{s}+G_{0}\left(\mathcal{L}_{\tilde{U}_{s}}\right)\right)\right]=0$
as soon as $G_{0}\left(\mathcal{L}_{\tilde{U}_{s}}\right)>0$, so that

$\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}=0$

Second, consider the map $\Phi$ which associates to $\tilde{X}$ to the
solution $X$ of (1.1). We will show $\Phi$ is a contraction. Let
$\tilde{X},\tilde{X}^{\prime}\in\mathcal{C}^{2}$ and define $\tilde{U}$, $K$
and $\tilde{U}^{\prime}$, $K^{\prime}$ as above, using the some Brownian
motion. We have from Cauchy-Schwarz and Burkholder-Davis-Gundy (B-D-G)
inequality

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}|\tilde{X}_{t}-\tilde{X}_{t}^{\prime}|\right]$ (3.5) $\displaystyle\leq
3\mathbb{E}\left[\sup_{t\leq
T}\left\\{\left|\int_{0}^{t}\left(b\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)-b\left(\tilde{X}^{\prime}_{s},\mathcal{L}_{\tilde{X}^{\prime}_{s}}\right)\right)ds\right|^{2}+\left|\int_{0}^{t}\left(\sigma\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)-\sigma\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)\right)dB_{s}\right|^{2}\right.\right.$
$\displaystyle\left.\left.+\left|K_{t}-K^{\prime}_{t}\right|^{2}\right\\}\right]$
$\displaystyle\leq 3\mathbb{E}\left[\left.\sup_{t\leq
T}t\int_{0}^{t}\left(b\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)-b\left(\tilde{X}_{s}^{\prime},\mathcal{L}_{\tilde{X}_{s}^{\prime}}\right)\right)ds\right|^{2}\right]+3\mathbb{E}\left[\sup_{t\leq
T}\left|\int_{0}^{t}\left(\sigma\left(\tilde{X},\mathcal{L}_{\tilde{X}}\right)-\sigma\left(\tilde{X}^{\prime},\mathcal{L}_{\tilde{X}^{\prime}}\right)\right)dB_{s}\right|^{2}\right]$
$\displaystyle+3\sup_{t\leq T}\left|K_{t}-K^{\prime}_{t}\right|^{2}$
$\displaystyle:=3(I_{1}+I_{2}+I_{3})$

From the Lipschitz assumption, we have

$\displaystyle I_{1}$ $\displaystyle\leq
T\mathbb{E}\left[\int_{0}^{T}\left|b\left(\tilde{X},\mathcal{L}_{\tilde{X}_{s}}\right)-b\left(\tilde{X}^{\prime},\mathcal{L}_{\tilde{X}^{\prime}}\right)ds\right|^{2}\right]$
(3.6) $\displaystyle\leq
CT\mathbb{E}\left[\int_{0}^{T}\left|\tilde{X}-\tilde{X}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{\tilde{X}_{s}},\mathcal{L}_{\tilde{X}^{\prime}}\right)^{2}ds\right]$
$\displaystyle\leq CT\left\\{T\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}^{\prime}_{t}\right|\right]+\mathbb{E}\left[\int_{0}^{T}\mathcal{W}_{2}\left(\mathcal{L}_{\tilde{X}_{s}},\mathcal{L}_{\tilde{X}^{\prime}}\right)^{2}ds\right]\right\\}$
$\displaystyle\leq 2CT^{2}\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}^{\prime}_{t}\right|\right]$ $\displaystyle
I_{2}$ $\displaystyle\leq
C\mathbb{E}\left[\int_{0}^{T}\left|\sigma\left(\tilde{X},\mathcal{L}_{\tilde{X}_{s}}\right)-\sigma\left(\tilde{X}^{\prime},\mathcal{L}_{\tilde{X}^{\prime}}\right)\right|^{2}ds\right]$
(3.7) $\displaystyle\leq CT\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}^{\prime}_{t}\right|\right]+CT\mathbb{E}\left[\int_{0}^{T}\mathcal{W}_{2}\left(\mathcal{L}_{\tilde{X}_{s}},\mathcal{L}_{\tilde{X}^{\prime}}\right)^{2}ds\right]$
$\displaystyle\leq 2CT\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}^{\prime}_{t}\right|\right]$ $\displaystyle
I_{3}$ $\displaystyle\leq\frac{M}{m}\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{U}_{t}-\tilde{U}^{\prime}\right|^{2}\right]\leq
C\left(T^{2}C_{1}+TC_{2}\right)\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}_{t}^{\prime}\right|^{2}\right].$ (3.8)

Combining (3.5)-(3.8), we have

$\mathbb{E}\left[\sup_{t\leq
T}\left|X_{t}-X^{\prime}_{t}\right|^{2}\right]\leq
C(1+T)T\mathbb{E}\left[\sup_{t\leq
T}\left|\tilde{X}_{t}-\tilde{X}_{t}^{\prime}\right|^{2}\right].$

Hence, there exists a positive $\tilde{T}$, depending on $b,\sigma$ and $h$
only, such that for all $T<\tilde{T}$, the map $\Phi$ is a contraction. We
first deduce the existence and uniqueness of the solution on $[0,\tilde{T}]$
and then on $\mathbb{R}^{+}$by iterating the construction.

###### Remark 3.1

From this construction, we deduce that for all positive $r$:

$K_{t+r}-K_{t}=\sup_{s\in[0,r]}\inf\left\\{x\geq
0:\mathbb{E}\left[h\left(x+X_{t}+\int_{t}^{t+s}b\left(X_{u},\mathcal{L}_{\mathcal{X}_{u}}\right)du+\int_{t}^{t+s}\sigma\left(X_{u},\mathcal{L}_{\mathcal{X}_{u}}\right)dB_{u}\right)\right]\geq
0\right\\}$

Then, we connect our MR-MVSDE with a PDE on the Wasserstein space.

Let $\mathcal{O}:=\left\\{\mu\in\mathcal{P}_{2}(\mathbb{R}),\int
h(x)\mu(dx)>0\right\\}$. Given a bounded, continuous map
$G:\bar{\mathcal{O}}\rightarrow\mathbb{R}$ we consider the map
$u:[0,T]\times\bar{\mathcal{O}}\rightarrow\mathbb{R}$ defined by

$\displaystyle u(t,\mu)=\mathbb{E}\left[G(\mathcal{L}_{X_{t}})\right]$ (3.9)

######  Proposition 3.3 (Feynman-Kac formula)

Assume that $u$ is a classical solution to Neumann problem on the Wasserstein
space (3.10). Then, $u$ is given by (3.9).

$\left\\{\begin{array}[]{l}\text{ (i)
}\left(\partial_{t}+\mathcal{T}\right)u(t,\mu)=0\quad\text{ in
}(0,T)\times\mathcal{O},\\\ \text{ (ii)
}\int_{\mathbb{R}}\mathrm{D}_{\mu}u(t,\mu)(y)\mu(dy)=0,\quad\text{ in
}(0,T)\times\partial\mathcal{O},\\\ \text{ (iii) }u(T,\mu)=G(\mu),\quad\text{
in }\mathcal{O},\end{array}\right.$ (3.10)

where the operator $\mathcal{T}$ is given, for any smooth function
$\phi:[0,T]\times\mathcal{P}_{2}(\mathcal{R})\rightarrow\mathbb{R}$, by

$\displaystyle\mathcal{T}\phi(t,\mu):=\int_{\mathbb{R}}b(z,\mu)D_{\mu}\phi(t,\mu)(z)d\mu(z)+\frac{1}{2}\int_{\mathbb{R}}\sigma(z,\mu)^{2}\partial_{z}D_{\mu}\phi(t,\mu)(z)d\mu(z)$
(3.11)

###### Proof 2

Let us set $\mu_{s}:=\mathcal{L}_{X_{s}^{t_{0},X_{0}}}$. Then, by Lemma 2.2:

$u(T,\mu_{T})=u(t_{0},\mu_{t_{0}})+\int_{t_{0}}^{T}\left(\partial_{t}+\mathcal{T}\right)u(s,\mu_{s})+\int_{t_{0}}^{T}\int_{\mathbb{R}}D_{\mu}u(s,\mu_{s})(y)\mu_{s}(dy)dK_{s}$

where, by (3.10)-(ii),

$\int_{t_{0}}^{T}\int_{\mathbb{R}}D_{\mu}u(s,\mu_{s})(y)\mu_{s}(dy)dK_{s}=\int_{t_{0}}^{T}\mathbbm{1}_{\left\\{\mu_{s}\in\partial\mathcal{O}\right\\}}\left(\int_{\mathbb{R}}D_{\mu}u(s,\mu_{s})(y)\right)dK_{s}=0$

From the equation satisfied by $u$, we obtain

$u(t_{0},\mu_{0})=u(t_{0},\mu_{t_{0}})=\mathbb{E}\left[u(T,\mu_{T})\right]=\mathbb{E}\left[G(\mathcal{L}_{X_{T}})\right]$

###### Proposition 3.4

Suppose that Assumption (A.1) and (A.2) hold.

  1. 1.

For all $p$, there exists a positive $C_{p}$, depending on $T,b,\sigma$ and
$h$ such that

$\mathbb{E}\left[\sup_{t\leq T}|X_{t}|^{p}\right]\leq
C_{p}(1+\mathbb{E}\left[|\xi|^{p}\right])$

  2. 2.

There exists a positive constant $C$, depending on $T,b,\sigma$ and $h$ such
that

$\forall 0\leq s\leq t\leq T,\quad|K_{t}-K_{s}|\leq C|t-s|^{\frac{1}{2}}$

  3. 3.

For all $p\geq 1$ such that $\xi\in L^{p}$ we have

$\forall 0\leq s\leq t\leq
T,\quad\mathbb{E}\left[\left|X_{t}-X_{s}\right|^{p}\right]\leq
C|t-s|^{\frac{p}{2}}$

###### Proof 3

(i) We have

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}\left|X_{t}\right|^{p}\right]$
$\displaystyle\leq
4^{p-1}\left\\{\mathbb{E}\left[|\xi|^{p}\right]+\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}\left|b\left(X_{s},\mathcal{L}_{X_{s}}\right)\right|ds\right)^{p}\right]\right.$
(3.12) $\displaystyle\left.+\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}\left|\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)\right|dB_{s}\right)^{p}\right]+K_{T}^{p}\right\\}$

We first deal with $K^{p}_{T}=\sup_{t\leq T}G_{0}(\mathcal{L}_{U_{t}})$. Apply
Lipschitz property of $G_{0}$, Lemma 2.4 and the definition of the Wasserstein
metric $\mathcal{W}_{1}$, we have

$\forall t\geq
0,\quad\left|G_{0}(\mathcal{L}_{U_{t}})\right|\leq\frac{M}{m}\mathbb{E}\left[|U_{t}-U_{0}|\right],$

since $G_{0}(\mathcal{L}_{U_{0}})=0$ as $\mathbb{E}\left[h(X_{0})\right]\geq
0$, and where $U$ is defined by (3.2). Therefore

$\displaystyle|K_{T}|^{p}=\left|\sup_{t\leq
T}G_{0}(\mathcal{L}_{U_{t}})\right|^{p}$ $\displaystyle\leq
2^{p-1}\left\\{\left(\frac{M}{m}\right)^{p}\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}\left|b(X_{s},\mathcal{L}_{X_{s}})\right|ds\right)^{p}\right]\right.$
(3.13) $\displaystyle\left.+\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}\left|\sigma(X_{s},\mathcal{L}_{X_{s}})\right|dB_{s}\right)^{p}\right]\right\\}$

Plugging (3.13) into (3.12), we deduce that there exists a constant $C_{p}\geq
0$ such that

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}|X_{t}|^{p}\right]$
$\displaystyle\leq C_{p}\mathbb{E}\left[|\xi|^{p}+\sup_{t\leq
T}\left(\int_{0}^{t}\left|b(X_{s},\mathcal{L}_{X_{s}})\right|ds\right)^{p}+\sup_{t\leq
T}\left|\int_{0}^{t}\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}\right|^{p}\right]$
(3.14) $\displaystyle\leq
C_{p}\left\\{\mathbb{E}\left[|\xi|^{p}\right]+T^{p-1}\mathbb{E}\left[\int_{0}^{T}(1+|X_{s}|)^{p}ds\right]+C\mathbb{E}\left[\int_{0}^{T}(1+|X_{s}|)^{2}ds\right]^{\frac{p}{2}}\right\\}$
$\displaystyle\leq
C_{p}(1+\mathbb{E}\left[|\xi|^{p}\right])+C\int_{0}^{T}\mathbb{E}\left[\sup_{t\leq
r}|X_{t}|^{p}\right]dr$

Apply Gronwall’s lemma, we deduce for all $p\geq 2$, there exists a constant
$C>0$ such that

$\mathbb{E}\left[\sup_{t\leq T}|X_{t}|^{p}\right]\leq
C(1+\mathbb{E}\left[|\xi|^{p}\right]).$

(ii) we observe that $\nu\longmapsto G_{0}(\nu)$ is Lipschitz continuous, we
have

$K_{t}-K_{s}=\sup_{r\leq t}G_{0}(\mathcal{L}_{U_{r}})-\sup_{r\leq
s}G_{0}(\mathcal{L}_{U_{r}})\leq\frac{M}{m}\mathcal{W}_{1}(\mathcal{L}_{U_{r}},\mathcal{L}_{U_{s}})\leq\frac{M}{m}\mathbb{E}\left[\left|U_{r}-U_{s}\right|\right],$

so the result follows from standard computations.

(iii) we use the decomposition

$X_{t}-X_{s}=U_{t}-U_{s}+K_{t}-K_{s}$

and the result of (ii), we complete this proof.

###  3.2 Density of $K_{t}$

Let $\mathcal{L}_{\mu}$ be the partial operator of second order defined by

$\displaystyle\mathcal{A}_{\mu}f(x)=b(x,\mu)\frac{\partial}{\partial
x}f(x)+\frac{1}{2}\sigma\left(x,\mu\right)^{2}\frac{\partial^{2}}{\partial
x^{2}}f(x)$ (3.15)

In order to prove Proposition 3.6, we need following lemma:

###### Lemma 3.5

If Assumption (A.1), (A.2) and (A.4) hold

  1. 1.

If $\varphi$ is a continuous function such that, for some $C\geq 0$ and $p\geq
1$,

$\forall x\in\mathbb{R},\quad|\varphi(x)|\leq C(1+|x|^{p})$

then the function $t\longmapsto\mathbb{E}\left[\varphi(X_{t})\right]$ is
continuous.

  2. 2.

The functions $t\longmapsto\mathbb{E}\left[h(X_{t})\right]$ and
$t\longmapsto\mathbb{E}\left[\mathcal{A}_{\mathcal{L}_{X_{r}}}h(X_{t})\right]$
are continuous.

###### Proof 4

(i) is the [22][Lemma 3].

Now, we prove (ii). Since Assumption (A.1), (A.2) and (A.4) hold, the mapping
$x\longmapsto\mathcal{A}_{\mathcal{L}_{X_{t}}}h(X_{t})$ is continuous and for
all $x\in\mathbb{R}$, there exists constabts $C_{1}$ and $C_{2}$ such that

$\displaystyle\left|b(x,\mu)h^{\prime}(x)\right|\leq C_{1}(1+|x|).$
$\displaystyle\left|\sigma(x,\mu)^{2}h^{\prime}(x)\right|\leq C_{2}(1+|x|).$

Then, using (i), we deduce that
$t\longmapsto\mathbb{E}\mathcal{A}_{\mathcal{L}_{X_{t}}}h(X_{t})$ is
continuous.

###### Proposition 3.6

Suppose Assumptions (A.1), (A.2) and (A.4) hold. Let $(X,K)$ be the unique
deterministic flat solution to (1.1). Then the process $K$ is Lipschitz
continuous and the Stielije measure $dK$ has the following density w.r.t the
Lebesgue measure:

$k:\mathbb{R}^{+}\ni
t\longmapsto\frac{\left(\mathbb{E}\left[\mathcal{A}_{\mathcal{L}_{X_{t}}}h\left(X_{t}\right)\right]\right)^{-}}{\mathbb{E}\left[h^{\prime}\left(X_{t}\right)\right]}\mathbbm{1}_{\mathbb{E}\left[h\left(X_{t}\right)\right]=0}$
(3.16)

###### Proof 5

Firstly, we show $K$ is Lipschitz continuous. To do so, we prove that
$s\mapsto\bar{G}_{0}(\mu_{s})$ is Lipschitz continuous on $[0,T]$. If $s\leq
t$, we have

$\displaystyle\left|\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)-\bar{G}_{0}\left(\mathcal{L}_{U_{t}}\right)\right|$
$\displaystyle\leq\frac{1}{m}\left|H\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right),\mathcal{L}_{U_{t}}\right)\right|$
$\displaystyle=\frac{1}{m}\left|\mathbb{E}\left[h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+U_{t}\right)\right]\right|$
$\displaystyle=\frac{1}{m}\left|\mathbb{E}\left[h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+U_{s}+\int_{s}^{t}b\left(X_{r},\mathcal{L}_{X_{r}}\right)dr+\int_{s}^{t}\sigma\left(X_{r},\mathcal{L}_{X_{r}}\right)dB_{r}\right)\right]\right|,$

the first equality since $H(\bar{G}_{0}(\mu_{t}),\mu_{t})=0$.

Let $\bar{A}_{x,\mu}$ be the partial operator of second order defined by

$\bar{\mathcal{A}}_{y,\mu}f(x):=b(y,\mu)\frac{\partial}{\partial
x}f(x)+\frac{1}{2}\sigma\left(y,\mu\right)^{2}\frac{\partial^{2}}{\partial
x^{2}}f(x)$

Apply Itô’s lemma, we have

$\displaystyle\mathbb{E}\left[h\left(\bar{G}_{0}(\mathcal{L}_{U_{s}})+U_{t}\right)\right]$
$\displaystyle=\mathbb{E}\left[h\left(\bar{G}_{0}(\mathcal{L}_{U_{s}})+U_{s}\right)\right]+\int_{s}^{t}\mathbb{E}\left[\bar{\mathcal{A}}_{X_{r},\mathcal{L}_{X_{r}}}h\left(\bar{G}_{0}(\mathcal{L}_{U_{s}})+U_{r}\right)\right]dr$
(3.17)
$\displaystyle=H\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right),\mathcal{L}_{U_{s}}\right)+\int_{s}^{t}\mathbb{E}\left[\bar{\mathcal{A}}_{X_{r},\mathcal{L}_{X_{r}}}h\left(\bar{G}_{0}(\mathcal{L}_{U_{s}})+U_{r}\right)\right]dr$
$\displaystyle=\int_{s}^{t}\mathbb{E}\left[\bar{\mathcal{A}}_{X_{r},\mathcal{L}_{X_{r}}}h\left(\bar{G}_{0}(\mathcal{L}_{U_{s}})+U_{r}\right)\right]dr.$

We get the result follows from that $h$ has bounded derivatives and
$\sup_{t\leq T}|X_{t}|\in L^{2}$ for each $T>0$. So from [1], we deduce that
$K$ is Lipschitz continuous and has a bounded density on $[0,T]$ for each
$T>0$.

Then we show the density of $dK$ is (3.16). From Assumption (A.4) and apply
Itô’s lemma, for any $s\leq t$ we can get

$\displaystyle h(X_{t})-h(X_{s})$
$\displaystyle=\int_{s}^{t}b(X_{r},\mathcal{L}_{X_{r}})h^{\prime}(X_{r})dr+\int_{s}^{t}\sigma(X_{r},\mathcal{L}_{X_{r}})h^{\prime}(X_{r})dB_{r}+\frac{1}{2}\int_{s}^{t}\sigma(X_{r},\mathcal{L}_{X_{r}})^{2}h^{\prime}(X_{r})dr$
(3.18) $\displaystyle+\int_{s}^{t}h^{\prime}(X_{r})dK_{r}$
$\displaystyle=\int_{s}^{t}\mathcal{A}_{\mathcal{L}_{X_{r}}}h(X_{r})dr+\int_{s}^{t}h^{\prime}(X_{r})dK_{r}+\int_{s}^{t}\sigma\left(X_{r},\mathcal{L}_{X_{r}}\right)dB_{r}$

Then, we obtain

$\displaystyle\mathbb{E}\left[\int_{s}^{t}h^{\prime}(X_{r})dK_{r}\right]=\mathbb{E}h(X_{t})-\mathbb{E}h(X_{s})-\int_{s}^{t}\mathbb{E}\mathcal{A}_{\mathcal{L}_{X_{r}}}h(X_{r})dr$
(3.19)

Using (3.19), Lemma 3.5 , and the proof of [1][Proposition 2.7], we can get
the density of $dK$ w.r.t the Lebesgue measure is

$k_{t}=\frac{\left(\mathbb{E}\left[\mathcal{A}_{\mathcal{L}_{X_{t}}}h\left(X_{t}\right)\right]\right)^{-}}{\mathbb{E}\left[h^{\prime}\left(X_{t}\right)\right]}\mathbbm{1}_{\mathbb{E}\left[h\left(X_{t}\right)\right]=0}$

##  4 Propagation of Chaos

We can write the unique solution of the MR-MVSDE (1.1) as :

$\displaystyle
X_{t}=\xi+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}+\sup_{s\leq
t}G_{0}(\mathcal{L}_{U_{s}})$ (4.1)

where $\mu_{t}$ stands for the law of

$U_{t}=\xi+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}$

We are here intersted in the particle approximation of such a system. Our
candidates are the particles

$\displaystyle
X^{i}_{t}=\hat{\xi}+\int_{0}^{t}b\left(X_{s},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)dB^{i}_{s}+\sup_{s\leq
t}G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)$ (4.2)

where $B^{i}$ are independent Brownian motions,
$\hat{\xi}^{i}=\xi^{i}+G_{0}(\mathcal{L}_{U^{\xi,N}})$, $(\xi^{i})_{i}$ are
independent copies of $\xi$, $\mathcal{L}_{U^{N}_{s}}$ denotes the empirical
distribution at time $s$ of the particles

$U_{s}^{i}=\hat{\xi}^{i}+\int_{0}^{s}b\left(X_{r}^{i},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)dr+\int_{0}^{s}\sigma\left(X_{r}^{i},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)dB_{r}^{i},\quad
1\leq i\leq
N,\quad\mathcal{L}_{U^{N}_{s}}=\frac{1}{N}\sum_{i=1}^{N}\delta_{U_{s}^{i}}$

###### Remark 4.1

Similar [1] and [22], the previous system of interacting particles can be seen
as a multidimensional reflected SDE with oblique reflection. If $h$ is
concave, the set

$\mathcal{S}:=\left\\{\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{N}:h(x_{1})+\cdots+h(x_{n})\geq
0\right\\}$

is convex and the system

$\left\\{\begin{array}[]{l}X^{i}_{t}=\hat{\xi}+\int_{0}^{t}b\left(X_{s},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)dB^{i}_{s}+K^{N}_{t},\quad
1\leq i\leq N\\\ \frac{1}{N}\sum_{i=1}^{N}h(X^{i}_{t})\geq
0,\quad\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{t}h(X^{i}_{t})dK^{N}_{s}=0\end{array}\right.$
(4.3)

is the SDE with oblique reflection in convex $\mathcal{S}$.

######  Theorem 4.1 (Propagation of Chaos)

Let $T>0$, and suppose that Assumptions (A.1) and (A.2) hold.

  1. 1.

Under Assumption (A.3) hold, there exists a constant $C$ depending on $b$,
$\sigma$, $M$, $m$ and $T$ such that, for each
$j\in\left\\{1,\cdots,N\right\\}$,

$\mathbb{E}\left[\sup_{t\leq
T}\left|X^{j}_{t}-\tilde{X}^{j}_{t}\right|^{2}\right]\leq CN^{-\frac{1}{2}}$

  2. 2.

Under Assumption (A.4) hold, there exists a constant $C$ depending on $b$,
$\sigma$, $M$, $m$ and $T$ such that, for each
$j\in\left\\{1,\cdots,N\right\\}$,

$\mathbb{E}\left[\sup_{t\leq
T}\left|X^{j}_{t}-\tilde{X}^{j}_{t}\right|^{2}\right]\leq CN^{-1}$

###### Proof 6

Let $t>0$, for $r\leq t$, we have

$\displaystyle\left|X_{r}^{j}-\bar{X}_{r}^{j}\right|\leq$
$\displaystyle\left|\hat{\xi}^{j}-\xi^{j}\right|$ (4.4)
$\displaystyle+\int_{0}^{r}\left|b\left(X_{s}^{j},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)-b\left(\bar{X}_{s}^{j},\mathcal{L}_{X_{s}}\right)\right|ds$
$\displaystyle+\left|\int_{0}^{r}\left(\sigma\left(X_{s}^{j},\frac{1}{n}\sum_{j=1}^{N}\delta_{x^{j}_{s}}\right)-\sigma\left(\bar{X}_{s}^{j},\mathcal{L}_{X_{s}}\right)\right)dB_{s}^{j}\right|$
$\displaystyle+\left|\sup_{s\leq
r}G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-\sup_{s\leq
r}G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|.$

From the inequality

$\displaystyle\left|\sup_{s\leq
r}G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-\sup_{s\leq
r}G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|$ $\displaystyle\leq\sup_{s\leq
r}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|\leq\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|$
$\displaystyle\leq\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)\right|+\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|,$

We have

$\displaystyle\sup_{r\leq t}\left|X^{j}_{r}-\bar{X}^{j}_{r}\right|\leq
I^{j}_{1}(r)+\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)\right|+\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)\right|,$
(4.5)

where, $I^{j}_{1}$ is defined by

$\displaystyle
I^{j}_{1}=\int_{0}^{t}\left|b\left(X^{j}_{s},\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}}\right)-b\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)\right|ds+\left|\int_{0}^{t}\sigma\left(X^{j}_{s},\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}}\right)-\sigma\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)\right|dB^{j}_{s}$
(4.6)

We have

$\displaystyle\mathbb{E}\left[|I^{j}_{1}(t)|^{2}\right]$ $\displaystyle\leq
C\mathbb{E}\left[t\int_{0}^{t}\left|b\left(X^{j}_{s},\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}}\right)-b\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)\right|^{2}ds\right]$
(4.7)
$\displaystyle+C\mathbb{E}\left[\int_{0}^{t}\left|\sigma\left(X^{j}_{s},\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}}\right)-\sigma\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)\right|^{2}ds\right]$
$\displaystyle\leq
Ct\int_{0}^{t}\mathbb{E}\left[\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|^{2}+\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]ds$
$\displaystyle+C\int_{0}^{t}\mathbb{E}\left[\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|^{2}+\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]ds$
$\displaystyle=C(1+t)\int_{0}^{t}\mathbb{E}\left[\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|^{2}\right]ds+C(1+t)\int_{0}^{t}\mathbb{E}\left[\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]ds$

We have

$\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)\right|\leq\frac{M}{m}\sup_{s\leq
t}\frac{1}{N}\sum_{i=1}^{N}\left|U_{s}^{i}-\bar{U}_{s}^{i}\right|\leq\frac{M}{m}\frac{1}{N}\sum_{i=1}^{N}\sup_{s\leq
t}\left|U_{s}^{i}-\bar{U}_{s}^{i}\right|$

Moreover, taking into account that the variables are exchangeable, the Cauchy-
Schwarz inequality implies

$\mathbb{E}\left[\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{U^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)\right|^{2}\right]\leq\frac{M^{2}}{m^{2}}\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\left[\sup_{s\leq
t}\left|U_{s}^{i}-\bar{U}_{s}^{i}\right|^{2}\right]=\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{s\leq
t}\left|U_{s}^{j}-\bar{U}_{s}^{j}\right|^{2}\right]$

Since

$\displaystyle
U^{j}_{s}-\bar{U}^{j}_{s}=\int_{0}^{s}b\left(X^{j}_{r},\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{r}}\right)-b\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)ds+\int_{0}^{s}\sigma\left(X^{j}_{r},\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i}_{r}}\right)-\sigma\left(\bar{X}^{j}_{s},\mathcal{L}_{\bar{X}^{j}_{s}}\right)dB^{j}_{s}$
(4.8)

We have

$\mathbb{E}\left[\sup_{s\leq
t}\left|G_{0}(\mathcal{L}_{U^{N}_{s}})-G_{0}(\mathcal{L}_{\bar{U}^{N}_{s}})\right|\right]\leq
C\frac{M^{2}}{m^{2}}(1+t)\left\\{\int_{0}^{t}\mathbb{E}\left[\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|^{2}\right]+\mathbb{E}\left[\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]ds\right\\}$

Next, dealing with the
$\mathbb{E}\left[\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]$terms.
By the triangle inequality, we get

$\displaystyle\mathbb{E}\left[\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)\right]\leq\mathbb{E}\left[\left(\frac{1}{N}\sum_{i=1}^{N}\left|X^{i}_{s}-\bar{X}^{i}_{s}\right|^{2}\right)^{\frac{1}{2}}+\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{\bar{X}^{i}_{s}},\mathcal{L}_{\bar{X}^{i}_{s}}\right)\right]$
(4.9)

Noting that the particles are exchangeable and from [5], we have

$\mathbb{E}\left[\mathcal{W}_{2}\left(\frac{1}{n}\sum_{i=1}^{N}\delta_{X^{i}_{s}},\mathcal{L}_{\bar{X}^{j}_{s}}\right)^{2}\right]\leq
C\mathbb{E}\left[\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|\right]+CN^{-\frac{1}{2}}$

Consequently, we have

$\displaystyle\mathbb{E}\left[\sup_{r\leq
t}\left|X^{j}_{r}-\bar{X}^{j}_{r}\right|^{2}\right]\leq
2A\int_{0}^{t}\mathbb{E}\left[\sup_{r\leq
s}\left|X^{j}_{s}-\bar{X}^{j}_{s}\right|^{2}\right]ds+4\mathbb{E}\left[\sup_{s\leq
t}\left|G_{0}(\mathcal{L}_{\bar{U}^{N}_{s}})-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|^{2}\right]+ATN^{-\frac{1}{2}},$
(4.10)

where $A=C(1+t)(1+M^{2}/m^{2})$. Apply Gronwall’s lemma, we deduce that

$\displaystyle\mathbb{E}\left[\sup_{r\leq
t}\left|X^{j}_{r}-\bar{X}^{j}_{r}\right|^{2}\right]\leq
2Ce^{A}\left\\{\mathbb{E}\left[\sup_{s\leq
t}\left|G_{0}(\mathcal{L}_{\bar{U}^{N}_{s}})-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|^{2}\right]+ATN^{-\frac{1}{2}}\right\\}$
(4.11)

In view of Lemma 2.4, we have

$\mathbb{E}\left[\sup_{s\leq
t}\left|G_{0}\left(\mathcal{L}_{\bar{U}^{N}_{s}}\right)-G_{0}\left(\mathcal{L}_{U_{s}}\right)\right|^{2}\right]\leq\frac{1}{m^{2}}\mathbb{E}\left[\sup_{s\leq
t}\left|\int
h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\cdot\right)\left(d\mathcal{L}_{\bar{U}^{N}_{s}}-d\mathcal{L}_{U_{s}}\right)\right|^{2}\right]$

which leads to

$\mathbb{E}\left[\sup_{r\leq
t}\left|X_{r}^{j}-\bar{X}_{r}^{j}\right|^{2}\right]\leq
Ce^{A}\left\\{\frac{1}{m^{2}}\mathbb{E}\left[\sup_{s\leq t}\left|\int
h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\cdot\right)\left(d\mathcal{L}_{\bar{U}^{N}_{s}}-d\mathcal{L}_{U_{s}}\right)\right|^{2}\right]+ATN^{-\frac{1}{2}}\right\\}$

_Proof of (i)._ As $h$ is Lipschitz continuous, the rate of convergence is
determined by the empirical measure convergence of independent and identically
distributed diffusion processes. However, obtaining the standard convergence
rate is not straightforward since we require uniform convergence in time. If
we only suppose that Assumption (A.1) and (A.2) holds, we obtain that

$\frac{1}{m^{2}}\mathbb{E}\left[\sup_{s\leq t}\left|\int
h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\cdot\right)\left(d\mathcal{L}_{\bar{U}^{N}_{s}}-d\mathcal{L}_{U_{s}}\right)\right|^{2}\right]\leq\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{s\leq
t}\mathcal{W}_{1}\left(\mathcal{L}_{\bar{U}^{N}_{s}},\mathcal{L}_{U_{s}}\right)^{2}\right]$

In view of the proof of [1][Theorem 3.3 Part (i)] , we have

$\mathbb{E}\left[\sup_{s\leq
t}\mathcal{W}_{1}\left(\mathcal{L}_{\bar{U}^{N}_{s}},\mathcal{L}_{U_{s}}\right)^{2}\right]\leq
CN^{-\frac{1}{2}}$

So, we have there exists a constant $C>0$

$\displaystyle\mathbb{E}\left[\sup_{r\leq
t}\left|X^{j}_{r}-\bar{X}^{j}_{r}\right|^{2}\right]\leq
Ce^{A}\left(C\frac{M^{2}}{m^{2}}+AT\right)N^{-\frac{1}{2}}\leq
CN^{-\frac{1}{2}}$ (4.12)

_Proof of (ii)._ We need to estimate

$\frac{1}{m^{2}}\mathbb{E}\left[\sup_{s\leq t}\left|\int
h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\cdot\right)\left(d\mathcal{L}_{\bar{U}^{N}_{s}}-d\mathcal{L}_{U_{s}}\right)\right|^{2}\right].$

Let $\psi$ is the Radon-Nikodym derivative of
$\bar{G}_{0}\left(\mu_{s}\right)$, $V^{i}$ is a semimartingale
$s\longmapsto\bar{G}_{0}(\mathcal{L}_{\bar{U}^{i}_{s}})+\bar{U}^{i}_{s}$

$\displaystyle R_{N}(s)$ $\displaystyle:=\int
h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\cdot\right)\left(d\mathcal{L}_{\bar{U}^{N}_{s}}-d\mathcal{L}_{U_{s}}\right)$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\bar{U}_{s}^{i}\right)-\mathbb{E}\left[h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+U_{s}\right)\right]$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\left\\{h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\bar{U}_{s}^{i}\right)-\mathbb{E}\left[h\left(\bar{G}_{0}\left(\mathcal{L}_{U_{s}}\right)+\bar{U}_{s}^{i}\right)\right]\right\\}$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\left\\{h\left(V_{s}^{i}\right)-\mathbb{E}\left[h\left(V_{s}^{i}\right)\right]\right\\}$

Apply Itô’s lemma, we have

$h(V^{i}_{s})=h(V^{i}_{0})+\int_{0}^{s}h^{\prime}(V^{i}_{r})\psi_{r}+\bar{\mathcal{A}}_{\bar{X}_{r},\mathcal{L}_{\bar{X}_{r}}}h(V^{i}_{r})dr+\int_{0}^{s}h^{\prime}(V^{i}_{r})\sigma\left(\bar{X}^{i}_{r},\mathcal{L}_{\bar{X}^{i}_{r}}\right)dB^{i}_{r}$

so that

$\displaystyle\mathbb{E}\left[h\left(V_{s}^{i}\right)\right]$
$\displaystyle=\mathbb{E}\left[h\left(V_{0}^{i}\right)\right]+\int_{0}^{s}\mathbb{E}\left[h^{\prime}\left(V_{r}^{i}\right)\psi_{r}+\bar{\mathcal{A}}_{\bar{X}_{r},\mathcal{L}_{\bar{X}_{r}}}h\left(V_{r}^{i}\right)\right]dr$
$\displaystyle=0+\int_{0}^{s}\mathbb{E}\left[h^{\prime}\left(V_{r}^{i}\right)\psi_{r}+\bar{\mathcal{A}}_{\bar{X}_{r},\mathcal{L}_{\bar{X}_{r}}}h\left(V_{r}^{i}\right)\right]dr.$

Then, in view of the proof of [1][Theorem 3.3 Part (ii)] we have there exists
a constant $C>0$ such that

$\mathbb{E}\left[\sup_{t\leq
T}\left|X^{j}_{t}-\tilde{X}^{j}_{t}\right|^{2}\right]\leq CN^{-1}.$

##  5 Asymptotic analysis

###  5.1 Stability analysis

In this subsection, We study stability properties of (1.1) with respect to
initial condition, coefficients and driven process.

Firstly, we establish the stability of (1.1) with respect to small
perturbation of the initial point $\xi$. For simplicity, We denote by
$X^{\xi_{n}}_{t}$ as the unique solution of following equation:

$\left\\{\begin{array}[]{l}X^{\xi_{n}}_{t}=\xi_{n}+\int_{0}^{t}b\left(X^{\xi_{n}}_{s},\mathcal{L}_{X^{\xi_{n}}_{s}}\right)ds+\int_{0}^{t}\sigma\left(X^{\xi_{n}}_{s},\mathcal{L}_{X^{\xi_{n}}_{s}}\right)dB_{s}+K_{t}^{\xi_{n}},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X^{\xi_{n}}_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X^{\xi_{n}}_{s}\right)\right]dK_{s}^{\xi_{n}}=0,\quad
t\geq 0\end{array}\right..$

######  Theorem 5.1 (Stability with respect to initial condition)

Suppose that Assumption (A.1) and (A.2) hold. The mapping

$\Xi:\mathbb{R}\rightarrow
L^{2}\left(\Omega,\mathcal{C}([0,T],\mathbb{R})\right)$

defined by $\Xi(\xi)\rightarrow X_{t}^{\xi}$ is continuous.

###### Proof 7

Let $(\xi_{n})$ be a sequence in $\mathbb{R}$ and
$\lim_{n\rightarrow\infty}\xi_{n}=\xi$. We will show that

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\sup_{t\leq
T}\left|X^{\xi_{n}}_{t}-X^{\xi}_{t}\right|^{2}\right]=0.$

That means the mapping $\Xi$ is continuous.

Apply the Lipschitz condition of $b$ and $\sigma$, and Cauchy-Schwarz
inequality, we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left|X^{\xi_{n}}_{t}-X^{\xi}_{t}\right|^{2}\right]$ $\displaystyle\leq
4\left|\xi_{n}-\xi\right|^{2}+4\mathbb{E}\left[\sup_{t\leq
T}\int_{0}^{t}\left|b\left(X_{s}^{\xi_{n}},\mathcal{L}_{X_{s}^{\xi_{n}}}\right)-b\left(X_{s}^{\xi},\mathcal{L}_{X_{s}^{\xi}}\right)\right|ds\right]^{2}$
(5.1) $\displaystyle+4\mathbb{E}\left[\sup_{t\leq
T}\int_{0}^{t}\left|\sigma\left(X_{s}^{\xi_{n}},\mathcal{L}_{X_{s}^{\xi_{n}}}\right)-\sigma\left(X_{s}^{\xi},\mathcal{L}_{X_{s}^{\xi}}\right)\right|dB_{s}\right]^{2}+4\sup_{t\leq
T}\left|K^{\xi_{n}}_{t}-K^{\xi}_{t}\right|^{2}$ $\displaystyle\leq
4\left|\xi_{n}-\xi\right|^{2}+4T\mathbb{E}\left[\int_{0}^{t}\left|b\left(X_{s}^{\xi_{n}},\mathcal{L}_{X_{s}^{\xi_{n}}}\right)-b\left(X_{s}^{\xi},\mathcal{L}_{X_{s}^{\xi}}\right)\right|^{2}ds\right]$
$\displaystyle+4C\mathbb{E}\left[\int_{0}^{t}\left|\sigma\left(X_{s}^{\xi_{n}},\mathcal{L}_{X_{s}^{\xi_{n}}}\right)-\sigma\left(X_{s}^{\xi},\mathcal{L}_{X_{s}^{\xi}}\right)\right|^{2}ds\right]+4\sup_{t\leq
T}\left|K^{\xi_{n}}_{t}-K^{\xi}_{t}\right|^{2}$ $\displaystyle\leq
4|\xi_{n}-\xi|+4TC\mathbb{E}\left[\int_{0}^{t}\left|X^{\xi_{n}}_{s}-X^{\xi}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\xi_{n}}_{s}},\mathcal{L}_{X^{\xi}_{s}}\right)^{2}ds\right]$
$\displaystyle+4C\mathbb{E}\left[\int_{0}^{t}\left|X^{\xi_{n}}_{s}-X^{\xi}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\xi_{n}}_{s}},\mathcal{L}_{X^{\xi}_{s}}\right)^{2}ds\right]+4\sup_{t\leq
T}\left|K^{\xi_{n}}_{t}-K^{\xi}_{t}\right|^{2}$ $\displaystyle\sup_{t\leq
T}\left|K^{\xi_{n}}_{t}-K^{\xi}_{t}\right|^{2}$
$\displaystyle\leq\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{r\in[0,t]}\left|U^{\xi_{n}}_{r}-U_{r}^{\xi}\right|^{2}\right]$
(5.2)
$\displaystyle\leq\frac{M^{2}}{m^{2}}\left\\{4|\xi_{n}-\xi|+(4T+1)C\mathbb{E}\left[\int_{0}^{t}\left|X^{\xi_{n}}_{s}-X^{\xi}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\xi_{n}}_{s}},\mathcal{L}_{X^{\xi}_{s}}\right)^{2}ds\right]\right\\}$

Apply Grownall’s lemma, we have

$\begin{aligned} \mathbb{E}\left[\sup_{t\leq
T}\left|X^{\xi_{n}}_{t}-X^{\xi}_{t}\right|^{2}\right]\leq
4\exp\left\\{C(1+\frac{M^{2}}{m^{2}})\right\\}\end{aligned}|\xi_{n}-\xi|^{2}$
(5.3)

Letting $n$ goes to infinity, we complete this proof.

Secondly, we study the stability of (1.1) with respect to small perturbation
of the cofficients $b$ and $\sigma$. Let $X_{t}^{n}$ is the unique solution of
following equation:

$\left\\{\begin{array}[]{l}X^{n}_{t}=\xi+\int_{0}^{t}b^{n}\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)ds+\int_{0}^{t}\sigma^{n}\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)dB_{s}+K_{t}^{n},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X^{n}_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X^{n}_{s}\right)\right]dK_{s}^{n}=0,\quad
t\geq 0\end{array}\right..$

######  Theorem 5.2 (Stability with respect to coefficients)

Assume that the functions $b(x,\mu)$, $b^{n}(x,\mu)$, $\sigma^{n}(x,\mu)$ and
$\sigma(x,\mu)$ satisfy Assumption (A.1) and (A.2), Further suppose that for
each $T>0$, and each compact set $K$ there exists $C>0$ such that

$\lim_{n\rightarrow\infty}\sup_{t\leq T}\sup_{x\in
K}\sup_{\mu\in\mathcal{P}_{2}(\mathbb{R})}\left|b^{n}(x,\mu)-b(x,\mu)\right|+\left|\sigma^{n}(x,\mu)-\sigma(x,\mu)\right|=0$

then

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\sup_{t\leq
T}\left|X^{n}_{t}-X_{t}\right|^{2}\right]=0$

###### Proof 8

Apply the Lipschitz condition of $b^{n}$ and $\sigma^{n}$, and Cauchy-Schwarz
inequality ,we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left|X^{n}_{t}-X_{t}\right|^{2}\right]$ $\displaystyle\leq
4\mathbb{E}\left[\int_{0}^{t}b^{n}\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)-b^{n}\left(X_{s},\mathcal{L}_{X_{s}}\right)ds\right]^{2}$
(5.4)
$\displaystyle+4\mathbb{E}\left[\int_{0}^{t}b^{n}\left(X_{s},\mathcal{L}_{X_{s}}\right)-b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds\right]^{2}$
$\displaystyle+4\mathbb{E}\left\\{\sup_{t\leq
T}\left[\int_{0}^{t}\sigma^{n}\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)-\sigma^{n}\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}\right]^{2}\right\\}$
$\displaystyle+4\mathbb{E}\left\\{\sup_{t\leq
T}\left[\int_{0}^{t}\sigma^{n}\left(X_{s},\mathcal{L}_{X_{s}}\right)-\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dB_{s}\right]^{2}\right\\}$
$\displaystyle+4\sup_{t\leq T}\left|K^{n}_{t}-K_{t}\right|^{2}$
$\displaystyle\leq
4(CT+C)\int_{0}^{t}\mathbb{E}\left[|X^{n}_{s}-X_{s}|^{2}\right]+\mathcal{W}_{2}\left(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X_{s}}\right)^{2}ds+C\lambda_{n}+4\sup_{t\leq
T}\left|K^{n}_{t}-K_{t}\right|^{2},$

where

$\lambda_{n}=\mathbb{E}\left[\int_{0}^{t}\left|b^{n}(X_{s},\mathcal{L}_{X_{s}})-b(X_{s},\mathcal{L}_{X_{s}})\right|^{2}+\left|\sigma^{n}(X_{s},\mathcal{L}_{X_{s}})-\sigma(X_{s},\mathcal{L}_{X_{s}})\right|^{2}ds\right]$

$\displaystyle\sup_{t\leq T}\left|K^{n}_{t}-K_{t}\right|^{2}$
$\displaystyle\leq\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{r\in[0,t]}\left|U^{n}_{r}-U_{r}\right|^{2}\right]$
(5.5)
$\displaystyle\leq\frac{M^{2}}{m^{2}}4(CT+C)\int_{0}^{t}\mathbb{E}\left[|X^{n}_{s}-X_{s}|^{2}\right]+\mathcal{W}_{2}\left(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X_{s}}\right)^{2}ds+C\lambda_{n}$

Apply Grownall’s lemma, we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq
T}\left|X^{n}_{t}-X_{t}\right|^{2}\right]\leq
C(1+\frac{M^{2}}{m^{2}})\exp\left\\{8C(1+T)(1+\frac{M^{2}}{m^{2}})\right\\}\lambda_{n}$
(5.6)

Letting $n$ goes to infinity, we complete this proof.

Thirdly, we study the stability with respect to the driving processes. We
consider the MR-MVSDE driving by continuous semimartingales of the following
form:

$\left\\{\begin{array}[]{l}X_{t}=\xi+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)dA_{s}+\int_{0}^{t}\sigma\left(X_{s},\mathcal{L}_{X_{s}}\right)dM_{s}+K_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}=0,\quad
t\geq 0\end{array}\right.$ (5.7)

where $A_{t}$ is an adapted continuous process of bounded variation and
$M_{t}$ is a continuous local martingale.

Let us consider the following sequence
$\left\\{X^{n}_{t}\right\\}_{n=1}^{\infty}$ of MR-MVSDE:

$\left\\{\begin{array}[]{l}X^{n}_{t}=\xi+\int_{0}^{t}b\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)dA^{n}_{s}+\int_{0}^{t}\sigma\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)dM^{n}_{s}+K_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X^{n}_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X^{n}_{s}\right)\right]dK_{s}=0,\quad
t\geq 0\end{array}\right.$ (5.8)

where $\left(A^{n}_{t}\right)$ is a sequence of $\mathcal{F}_{t}$-adapted
continuous process of bounded variation and $M^{n}_{t}$ is continuous
$\left(\mathcal{F}_{t},\mathbb{P}\right)$-local martingales.

######  Theorem 5.3 (Stability with respect to driving processes)

Assume that the functions $b(x,\mu)$, $\sigma(x,\mu)$ satisfy Assumption (A.1)
and (A.2), Further suppose that $(A_{t},A_{t}^{n},M_{t},M_{t}^{n})$ satisfy
that

  1. 1.

The family $\left(A_{t},A_{t}^{n},M_{t},M_{t}^{n}\right)$ is bounded in
$\mathcal{C}([0,T])^{\otimes 4}$

  2. 2.

$\left(M^{n}_{t}-M_{t}\right)$ converges to 0 in probability in
$\mathcal{C}([0,T])$

  3. 3.

The total variation $\left(A^{n}_{t}-A_{t}\right)$ converges to 0 in
probability.

Then

$\lim_{n\rightarrow\infty}\mathbb{E}\left[\sup_{t<T}\left|X_{t}^{n}-X_{t}\right|^{2}\right]=0$

where $(X^{n}_{t})$ and $(X_{t})$ are, respectively, solutions of (5.8) and
(5.7).

###### Proof 9

Apply the Lipschitz condition of $b$ and $\sigma$ and Cauchy-Schwarz
inequality, we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}|X^{n}_{t}-X_{t}|^{2}\right]$
$\displaystyle\leq 4\mathbb{E}\left\\{\sup_{t\leq
T}\left[\int_{0}^{t}\left|b\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)-b(X_{s},\mathcal{L}_{X_{s}})\right|dA^{n}_{s}\right]^{2}\right\\}$
(5.9) $\displaystyle+4\mathbb{E}\left\\{\sup_{t\leq
T}\left[\int_{0}^{t}\left|\sigma\left(X^{n}_{s},\mathcal{L}_{X^{n}_{s}}\right)-\sigma(X_{s},\mathcal{L}_{X_{s}})\right|dM^{n}_{s}\right]^{2}\right\\}$
$\displaystyle+4\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}|b(X_{s},\mathcal{L}_{X_{s}})|d|A^{n}_{s}-A_{s}|\right)^{2}\right]$
$\displaystyle+4\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}|\sigma(X_{s},\mathcal{L}_{X_{s}})|d|M^{n}_{s}-M_{s}|\right)^{2}\right]+4\sup_{t\leq
T}|K^{n}_{t}-K_{t}|^{2}$ $\displaystyle\leq
C\int_{0}^{T}\mathbb{E}\left[\sup_{s\leq
t}|X^{n}_{s}-X_{s}|^{2}\right]+\mathcal{W}_{2}\left(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X_{s}}\right)^{2}\left(dA^{n}_{s}+d\left\langle
M^{n}\right\rangle_{s}\right)+e_{n}$ $\displaystyle+4\sup_{t\leq
T}|K^{n}_{t}-K_{t}|^{2}$

Where

$e_{n}=4\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}|b(X_{s},\mathcal{L}_{X_{s}})|d|A^{n}_{s}-A_{s}|\right)^{2}\right]++4\mathbb{E}\left[\sup_{t\leq
T}\left(\int_{0}^{t}|\sigma(X_{s},\mathcal{L}_{X_{s}})|d|M^{n}_{s}-M_{s}|\right)^{2}\right]$
$\displaystyle\sup_{t\leq T}\left|K^{n}_{t}-K_{t}\right|^{2}$
$\displaystyle\leq\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{r\in[0,t]}\left|U^{n}_{r}-U_{r}\right|^{2}\right]$
(5.10)
$\displaystyle\leq\frac{M^{2}}{m^{2}}C\int_{0}^{t}\mathbb{E}\left[|X^{n}_{s}-X_{s}|^{2}\right]+\mathcal{W}_{2}\left(\mathcal{L}_{X^{n}_{s}},\mathcal{L}_{X_{s}}\right)^{2}ds+Ce_{n}$

Combing (5.9) and (5.10), we have

$\displaystyle\mathbb{E}\left[\sup_{t\leq T}|X^{n}_{t}-X_{t}|^{2}\right]\leq
C\int_{0}^{T}\mathbb{E}\left[\sup_{t\leq
T}|X^{n}_{t}-X_{t}|^{2}\right]\left(dA^{n}_{s}+d\left\langle
M^{n}\right\rangle_{s}\right)+Ce_{n}$ (5.11)

Since $A^{n}_{s}+\left\langle M^{n}\right\rangle_{s}$ is an increasing
process, then according to the Stochastic Gronwall lemma, we obtain

$\mathbb{E}\left[\sup_{t\leq T}|X^{n}_{t}-X_{t}|^{2}\right]\leq
Ce_{n}\mathbb{E}\left[A^{n}_{T}+\left\langle
M^{n}\right\rangle_{T}\right]<\infty,$

using the assumptions of $(A_{t},A_{t}^{n},M_{t},M_{t}^{n})$, we have

$\lim_{n\rightarrow\infty}e_{n}=0$

Therefore,

$\lim\limits_{n\rightarrow\infty}\mathbb{E}\left[\sup_{t\leq
T}|X^{n}_{t}-X_{t}|^{2}\right]=0$

###  5.2 Large Deviation Principle

For any $\varepsilon\in(0,1]$, let
$X^{\varepsilon}=\left\\{X_{t}^{\varepsilon},t\in[0,T]\right\\}$ be the unique
strong solution to the following MR-MVSDE:

$\left\\{\begin{array}[]{l}X^{\varepsilon}_{t}=\xi+\int_{0}^{t}b\left(X_{s}^{\varepsilon},\mathcal{L}_{X_{s}^{\varepsilon}}\right)ds+\sqrt{\varepsilon}\int_{0}^{t}\sigma\left(X_{s}^{\varepsilon},\mathcal{L}_{X_{s}^{\varepsilon}}\right)dB_{s}+K_{t}^{\varepsilon},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X_{t}^{\varepsilon}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}^{\varepsilon}\right)\right]dK_{s}^{\varepsilon}=0,\quad
t\geq 0\end{array}\right.$ (5.12)

In this subsection, we mainly consider the LDP for $X^{\varepsilon}$ as
$\varepsilon$ tending to 0.

######  Defination 5.4 (Large deviation)

A family $\left\\{X^{\varepsilon}\right\\}_{\varepsilon>0}$ of
$\mathcal{E}$-valued random variable is said to satisfy the large deviation
principle on $\mathcal{E}$, with the good rate function $I$ and with the speed
function $\lambda(\varepsilon)$ which is a sequence of positive numbers
tending to $+\infty$ as $\varepsilon\rightarrow 0$, if the following
conditions hold:

  1. 1.

for each $M<\infty$, the level set $\\{x\in\mathcal{E}:I(x)\leq M\\}$ is a
compact subset of $E$;

  2. 2.

for each closed subset $F$ of $\mathcal{E},\limsup_{\varepsilon\rightarrow
0}\frac{1}{\lambda(\varepsilon)}\log\mathbb{P}\left(X^{\varepsilon}\in
F\right)\leq-\inf_{x\in F}I(x)$;

  3. 3.

for each open subset $G$ of $\mathcal{E},\liminf_{\varepsilon\rightarrow
0}\frac{1}{\lambda(\varepsilon)}\log\mathbb{P}\left(X^{\varepsilon}\in
G\right)\geq-\inf_{x\in G}I(x)$.

From Theorem 3.2, we know that there exists a unique solution for the
following oblique reflected ODE

$\left\\{\begin{array}[]{l}\psi_{t}=\xi+\int_{0}^{t}b(\psi_{s},\delta_{\psi_{s}})ds+K^{\psi}_{t},\quad
t\geq 0\\\ h(\psi_{t})\geq
0\quad\int_{0}^{t}h(\psi_{s})dK^{\psi}_{s}=0\end{array}\right.,$ (5.13)

where,

$K^{\psi}_{s}=\sup_{r\leq s}\inf\left\\{x\geq 0,h(x+U_{s}^{\psi_{s}})\geq
0\right\\},$

and $\left(U^{\psi}\right)_{0\leq t\leq T}$ is the process defined by

$\displaystyle
U^{\psi}_{t}=\xi+\int_{0}^{t}b\left(\psi_{s},\delta_{\psi_{s}}\right)ds$
(5.14)

We denote $\psi$ is the solution of (5.13). For any $h\in
L^{2}\left([0,T];\mathbb{R}\right)$, consider the so called skeleton equation:

$\displaystyle\left\\{\begin{array}[]{l}Y^{h}_{t}=\xi+\int_{0}^{t}b\left(Y^{h}_{s},\delta_{\psi_{s}}\right)ds+\int_{0}^{t}\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)h(s)ds+K^{h}_{t}\\\
h(Y^{h}_{t})\geq
0,\quad\int_{0}^{t}h(Y^{h}_{s})dK^{h}_{s}=0\end{array}\right.$ (5.15)

###### Proposition 5.5

Under Assumption (A.1) and (A.2), there exists a unique strong solution to
(5.15).

###### Proof 10

Let
$\tilde{b}(y)=b\left(t,y,\delta_{\psi_{t}}\right)+\sigma\left(t,y,\delta_{\psi_{t}}\right)h(t)$,
it is easy show $\tilde{b}(y)$ satisfies Assumption (A.1). Then, apply Theorem
3.2, we complete this prove.

######  Theorem 5.6 (Large deviation principle)

Let Assumption (A.1) and (A.2) hold and $X^{\varepsilon}$ be the unique strong
solution to (5.12). Then the family of
$\left\\{X^{\varepsilon}\right\\}_{\varepsilon>0}$ satisfies a LDP on the
space $\mathcal{X}$ with rate function

$I(\phi):=\inf_{\left\\{h\in
L^{2}\left([0,T],\mathbb{R}\right):\phi=Y^{h}\right\\}}\frac{1}{2}\int_{0}^{T}|h(t)|^{2}\mathrm{~{}d}t,\phi\in\mathcal{X},$

with the convention $\inf\\{\emptyset\\}=+\infty$, here $Y^{h}\in\mathcal{X}$
solves equation (5.15).

###### Proof 11

According to Proposition 5.5 , there exists a measurable map
$\mathcal{G}^{0}$:

$\mathcal{G}^{0}:C\left([0,T];\mathbb{R}\right)\rightarrow\mathcal{X}$

such that for any $h\in L^{2}([0,T],\mathbb{R})$

$Y^{h}=\mathcal{G}^{0}\left(\int_{0}^{\cdot}h(s)ds\right).$

Let

$\mathcal{H}^{M}:=\left\\{h:[0,T]\rightarrow\mathbb{R}^{m},\int_{0}^{T}\|h(s)\|^{2}\mathrm{~{}d}s\leq
M\right\\},$

and

$\tilde{\mathcal{H}}^{M}:=\left\\{h:h\right.\text{is }\mathbb{R}\text{-valued
}\mathcal{F}_{t}\text{-predictable process such that
}h(\omega)\in\mathcal{H}^{M},\mathbb{P}-a.s\\}.$

$\mathcal{H}^{M}$ is endowed with the weak topology on
$L^{2}\left([0,T],\mathbb{R}\right)$. Then $\mathcal{H}^{M}$ is a Polish
space. For every $\varepsilon>0$, there exists a measurable mapping
$\mathcal{G}^{\varepsilon}(\cdot):C\left([0,T];\mathbb{R}\right)\rightarrow\mathcal{X}$
such that

$X^{\varepsilon}=\mathcal{G}^{\varepsilon}(B(\cdot))$

and for any $M>0$ and $h^{\varepsilon}\in\tilde{\mathcal{H}}^{M}$ ,

$Z^{\varepsilon}:=\mathcal{G}^{\varepsilon}\left(B(\cdot)+\frac{1}{\sqrt{\varepsilon}}\int_{0}^{\cdot}h^{\varepsilon}(s)\mathrm{d}s\right).$

From Girsanov’s Theorem, we know $Z^{\varepsilon}$ is the solution of the
following MR-MVSDE:

$\displaystyle\left\\{\begin{array}[]{l}Z^{\varepsilon}_{t}=\xi+\int_{0}^{t}b\left(Z^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)ds+\int_{0}^{t}\sigma\left(Y^{h}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)h^{\varepsilon}(s)ds+K^{Z^{\varepsilon}}_{t}\\\
\mathbb{E}\left[h(Z^{\varepsilon}_{t})\right]\geq
0,\quad\int_{0}^{t}h(Z^{\varepsilon}_{s})dK^{Z^{\varepsilon}}_{s}=0\end{array}\right.$
(5.16)

Following [39], we only need to verify the two claims:

  * (LDP1)

For every $M<+\infty$ and any family
$\left\\{h^{n};n\in\mathbb{N}\right\\}\subset\mathcal{H}^{M}$ converging to
some element $h\in\mathcal{H}^{M}$ as $n\rightarrow\infty$

$\lim\limits_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\mathcal{G}^{0}\left(\int_{0}^{\cdot}h^{n}(s)ds\right)-\mathcal{G}^{0}\left(\int_{0}^{\cdot}h(s)ds\right)\right|=0$

  * (LDP2)

For every $M<+\infty$ and any family
$\left\\{h^{\varepsilon};\varepsilon>0\right\\}\subset\tilde{\mathcal{H}}^{M}$
and any $\theta>0$,

$\lim_{\varepsilon\rightarrow
0}\mathbb{P}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{t}-Y^{h^{\varepsilon}}_{t}\right|\geq\theta\right]=0$

where $Z^{\varepsilon}$ is the unique solution to (5.16) and
$Y^{h^{\varepsilon}}=\mathcal{G}^{0}\left(\int_{0}^{\cdot}h^{\varepsilon}(s)\mathrm{d}s\right)$.

The verification of (LDP1) will be given in Proposition 5.7. (LDP2) will be
established in Proposition 5.9.

####  5.2.1 Proof of (LDP1)

###### Proposition 5.7

Let Assumption (A.1) and (A.2) hold. For any $M<+\infty$, family
$\\{h^{n}\\}_{n\geq 1}\in\mathcal{H}^{M}$ and $h\in\mathcal{H}^{M}$, suppose
that $h^{n}\rightarrow h$ as $n\rightarrow\infty$ in the weak topology of
$L^{2}([0,T],\mathbb{R})$. Then

$\lim\limits_{n\rightarrow\infty}\sup_{t\in[0,T]}\left|\mathcal{G}^{0}\left(\int_{0}^{\cdot}h^{n}(s)ds\right)-\mathcal{G}^{0}\left(\int_{0}^{\cdot}h(s)ds\right)\right|=0$

###### Proof 12

For simplicity, we denote
$Y^{h_{n}}:=\mathcal{G}^{0}\left(\int_{0}^{\cdot}h^{n}(s)ds\right)$,
$Y^{h}=\mathcal{G}^{0}\left(\int_{0}^{\cdot}h(s)ds\right)$.

$\displaystyle Y^{h_{n}}-Y^{h}$
$\displaystyle=\int_{0}^{t}b\left(Y^{h^{n}}_{s},\delta_{\psi_{s}}\right)-b\left(Y^{h}_{s},\delta_{\psi_{s}}\right)ds+\int_{0}^{t}\sigma\left(Y^{h^{n}}_{s},\delta_{\psi_{s}}\right)h^{n}(s)-\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)h(s)ds$
(5.17) $\displaystyle+K^{h^{n}}_{t}-K^{h}_{t}$
$\displaystyle=\int_{0}^{t}b\left(Y^{h^{n}}_{s},\delta_{\psi_{s}}\right)-b\left(Y^{h}_{s},\delta_{\psi_{s}}\right)ds+\int_{0}^{t}\left[\sigma\left(Y^{h^{n}}_{s},\delta_{\psi_{s}}\right)-\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)\right]h^{n}(s)ds$
$\displaystyle+\int_{0}^{t}\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)\left[h^{n}(s)-h(s)\right]ds+K^{h^{n}}_{t}-K^{h}_{t}$
$\displaystyle:=I^{n}_{1}(t)+I^{n}_{2}(t)+I^{n}_{3}(t)+I^{n}_{4}(t).$

Let $\kappa^{n}(t):=\sup_{r\in[0,t]}\left|Y^{h^{n}}(r)-Y^{h}(r)\right|$. By
the Lipschitz condition of $b$, we have

$\displaystyle\left|I^{n}_{1}(t)\right|\leq\int_{0}^{t}\left|b\left(Y^{h^{n}}_{s},\delta_{\psi_{s}}\right)-b\left(Y^{h}_{s},\delta_{\psi_{s}}\right)\right|ds\leq
K\int_{0}^{t}\kappa^{n}(s)ds.$ (5.18)

By the Lipschitz condition of $\sigma$, we have

$\displaystyle\left|I^{n}_{2}(t)\right|\leq
K\int_{0}^{t}|h^{n}(s)|\kappa_{n}(s)ds$ (5.19)

By the linear growth of $\sigma$, we have

$\displaystyle\sup_{t\in[0,T]}\left|\sigma(x,\mu)\right|^{2}\leq C(1+K^{2})$
(5.20)

Since $h^{n}(t)\rightarrow h(t)$ weakly in
$L^{2}\left([0,T],\mathbb{R}\right)$ as $n\rightarrow\infty$, we have

$\displaystyle\int_{0}^{T}\left|h^{n}(s)\right|^{2}ds\leq
C(h)\quad\int_{0}^{T}\left|h(s)\right|^{2}ds\leq C(h).$ (5.21)

Then, by the Cauchy-Schwarz inequality, we have

$\displaystyle\sup_{t\in[0,T]}\left|I^{n}_{3}(t)\right|$
$\displaystyle\leq\int_{0}^{T}\left|\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)\left[h^{n}(s)-h(s)\right]\right|ds$
(5.22)
$\displaystyle\leq\left(\int_{0}^{T}\left|\sigma\left(Y^{h}_{s},\psi_{s}\right)\right|^{2}ds\right)^{\frac{1}{2}}\left(\int_{0}^{T}\left|h^{n}(s)-h(s)\right|^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq\left(T\sup_{t\in[0,T]}\left|\sigma\left(Y^{h}_{s},\psi_{s}\right)\right|^{2}\right)^{\frac{1}{2}}\left(2\int_{0}^{T}\left(|h^{n}(s)|^{2}+|h(s)|^{2}\right)ds\right)^{\frac{1}{2}}$
$\displaystyle\leq(CT(1+K^{2})C(h))^{\frac{1}{2}}<\infty.$

Similarly, we have, for any $0\leq t_{1}\leq t_{2}\leq T$,

$\left|I^{n}_{3}(t_{2})-I^{n}_{3}(t_{1})\right|\leq\left[C(t_{2}-t_{1})(1+K^{2})C(h)\right]^{\frac{1}{2}},$

which means that the sequence $\left\\{I^{n}_{3}:n\geq 1\right\\}$ is equi-
continuous. By the Arzela-Ascoli theorem, we deduce that
$\left\\{I^{n}_{3}:n\geq 1\right\\}$ is relatively compact in
$\mathcal{C}\left([0,T],\mathbb{R}\right)$.

We can deduce that for any $t\in[0,T]$,

$\displaystyle
I^{n}_{3}(t)=\int_{0}^{t}\left(h^{n}(s)-h(s)\right)\sigma\left(Y^{h}_{s},\delta_{\psi_{s}}\right)ds\rightarrow
0\quad\text{as }n\rightarrow\infty.$ (5.23)

Together with the relative compactness of $\left\\{I^{n}_{3}:n\geq 1\right\\}$
in $\mathcal{C}\left([0,T],\mathbb{R}\right)$, we can get

$\displaystyle\lim\limits_{n\rightarrow\infty}\sup_{t\in[0,T]}|I^{n}_{3}(t)|=0$
(5.24)

By lemma 2.4 , we have

$\displaystyle
K^{h^{n}}_{t}-K^{h}_{t}\leq\frac{M}{m}\sup_{r\in[0,t]}\left|U^{h^{n}}_{r}-U^{h}_{r}\right|=\frac{M}{m}\sup_{r\in[0,t]}\left[|I^{n}_{1}|(r)+|I^{n}_{2}|(r)\right]$
(5.25)

Combing (5.17), (5.18), (5.19), (5.22) and (5.25) together, we have

$\displaystyle\kappa^{n}(t)$ $\displaystyle\leq
C\left\\{\int_{0}^{t}\kappa^{n}(s)ds+\int_{0}^{t}\kappa^{n}(s)\left|h^{n}(s)\right|ds\right\\}+2\sup_{t\in[0,T]}\left|I^{n}_{3}(t)\right|.$
(5.26)

Then, by (5.23) and Gronwall’s lemma, we have

$\displaystyle\kappa^{n}(T)$ $\displaystyle\leq
2\sup_{t\in[0,T]}|I^{n}_{3}(t)|\exp\left\\{C\int_{0}^{T}|h^{n}(s)|ds\right\\}$
(5.27) $\displaystyle\leq C\sup_{t\in[0,T]}|I^{n}_{3}(t)|\rightarrow
0\quad\text{as }n\rightarrow\infty.$

The proof is complete.

####  5.2.2 Proof of (LDP2)

###### Lemma 5.8

Suppose Assumption (A.1) and (A.2) hold, let $X^{\varepsilon}_{t}$ and
$\psi_{t}$ satisfy (5.12) and (5.13), respectively. Then

$\lim_{\varepsilon\rightarrow
0}\sup_{t\in[0,T]}\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{t}},\delta_{\psi_{t}}\right)^{2}=0$

###### Proof 13

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|X^{\varepsilon}_{t}-\psi_{t}\right|^{2}\right]$
$\displaystyle\leq
3\left\\{\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}b\left(X^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)-b\left(\psi_{s},\delta_{\psi_{s}}\right)ds\right]^{2}\right.$
(5.28)
$\displaystyle\left.+\varepsilon\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\sigma\left(X^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)dB_{s}\right]^{2}+\sup_{t\in[0,T]}\left|K^{\varepsilon}_{t}-K^{\psi}_{t}\right|^{2}\right\\}$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

By the Lipschitz condition of $b$ and $\sigma$, we have

$\displaystyle I_{1}\leq
TK\int_{0}^{T}\left\\{\mathbb{E}\left[\left|X^{\varepsilon}_{s}-\psi_{s}\right|^{2}\right]+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s}},\delta_{\psi_{s}}\right)^{2}\right\\}ds$
(5.29) $\displaystyle
I_{2}\leq\mathbb{E}\int_{0}^{T}\sigma\left(X^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)^{2}ds\leq\mathbb{E}\int_{0}^{T}C(1+K^{2})ds=C(1+K^{2})T<\infty$
(5.30) $\displaystyle
I_{3}=\sup_{t\in[0,T]}\left|K^{\varepsilon}_{t}-K^{\psi}_{t}\right|^{2}$
$\displaystyle\leq\frac{M^{2}}{m^{2}}\mathbb{E}\left[\sup_{r\in[0,t]}\left|U^{\varepsilon}_{r}-U^{\psi}_{r}\right|^{2}\right]\leq
3\frac{M^{2}}{m^{2}}(I_{1}+I_{2})$ (5.31)

Putting (5.29), (5.30) and (5.31) to (5.28) and applying the Gronwall’s lemma,
we have

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|X^{\varepsilon}_{t}-\psi_{t}\right|^{2}\right]\leq
e^{C}\left\\{T\int_{0}^{T}\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s}},\delta_{\psi_{s}}\right)^{2}ds+\varepsilon\right\\}.$
(5.32)

By the definition of
$\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{t}},\delta_{\psi_{t}}\right)$,
we have following estimation

$\displaystyle\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{t}},\delta_{\psi_{t}}\right)^{2}\leq\mathbb{E}\left[\sup_{t\in[0,T]}\left|X^{\varepsilon}_{t}-\psi_{t}\right|^{2}\right]\leq
e^{C}\left\\{T\int_{0}^{T}\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s}},\delta_{\psi_{s}}\right)^{2}ds+\varepsilon\right\\}$
(5.33)

Thanking to Gronwall’s lemma, we have

$\lim_{\varepsilon\rightarrow
0}\sup_{t\in[0,T]}\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{t}},\delta_{\psi_{t}}\right)^{2}=0$

###### Proposition 5.9

Let Assumption (A.1) and (A.2) hold. Then for any $\theta>0$,

$\lim_{\varepsilon\rightarrow
0}\mathbb{P}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{t}-Y^{h^{\varepsilon}}_{t}\right|\geq\theta\right]=0$

###### Proof 14

By Chebyshev’s inequality, we only need to show that

$\lim_{\varepsilon\rightarrow
0}\mathbb{E}\left[\sup_{t\in[0,T]}|Z_{t}^{\varepsilon}-Y_{t}^{h^{\varepsilon}}|^{2}\right]=0$
$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{t}-Y^{h^{\varepsilon}_{t}}\right|^{2}\right]$
$\displaystyle\leq
T\mathbb{E}\left\\{\sup_{t\in[0,T]}\int_{0}^{T}\left|b\left(Z^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)-b\left(Y^{h^{\varepsilon}}_{s},\delta_{\psi_{s}}\right)\right|^{2}ds\right\\}$
(5.34)
$\displaystyle+\mathbb{E}\left\\{\sup_{t\in[0,T]}\left[\int_{0}^{T}h^{\varepsilon}(s)\left[\sigma\left(Z^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)-\sigma\left(Y^{h^{\varepsilon}_{s},\delta_{\psi_{s}}}\right)\right]ds\right]^{2}\right\\}$
$\displaystyle+\varepsilon
C_{p}\int_{0}^{T}\sigma\left(Z^{\varepsilon}_{s},\mathcal{L}_{X^{\varepsilon}_{s}}\right)^{2}ds+\sup_{t\in[0,T]}\left|K^{Z^{\varepsilon}}_{t}-K^{h^{\varepsilon}}_{t}\right|^{2}$
$\displaystyle\leq
KT\sup_{t\in[0,T]}\mathbb{E}\int_{0}^{T}\left[\left|Z^{\varepsilon}_{s}-Y^{h^{\varepsilon}}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s}},\delta_{\psi_{s}}\right)^{2}\right]ds$
$\displaystyle+N\sup_{t\in[0,T]}\mathbb{E}\int_{0}^{T}\left[\left|Z^{\varepsilon}_{s}-Y^{h^{\varepsilon}}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s},\delta_{\psi_{s}}}\right)^{2}\right]ds$
$\displaystyle+\sup_{t\in[0,T]}\left|K^{Z^{\varepsilon}}_{t}-K^{h^{\varepsilon}}_{t}\right|^{2}$
$\displaystyle\sup_{t\in[0,T]}\left|K^{Z^{\varepsilon}}_{t}-K^{h^{\varepsilon}}_{t}\right|^{2}$
$\displaystyle\leq\frac{M^{2}}{m^{2}}\sup_{r\in[0,t]}\left|U^{Z^{\varepsilon}}_{r}-U^{h^{\varepsilon}}_{r}\right|^{2}$
(5.35)
$\displaystyle\leq\frac{M^{2}}{m^{2}}KT\int_{0}^{T}\mathbb{E}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{s}-Y^{h^{\varepsilon}}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s}},\delta_{\psi_{s}}\right)^{2}\right]ds$
$\displaystyle+\frac{M^{2}}{m^{2}}N\int_{0}^{T}\mathbb{E}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{s}-Y^{h^{\varepsilon}}_{s}\right|^{2}+\mathcal{W}_{2}\left(\mathcal{L}_{X^{\varepsilon}_{s},\delta_{\psi_{s}}}\right)^{2}\right]ds$

By (5.34) and (5.35) and apply Gronwall’s lemma, we deduce that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\left|Z^{\varepsilon}_{t}-Y^{h^{\varepsilon}_{t}}\right|^{2}\right]$
$\displaystyle\leq\left(CT+N\right)(1+\frac{M^{2}}{m^{2}})\int_{0}^{t}\mathbb{E}\sup_{t\in[0,s]}\left|Z^{\varepsilon}_{s}-Y^{h^{\varepsilon}}_{s}\right|^{2}ds$
(5.36)
$\displaystyle+\left(CT+N\right)(1+\frac{M^{2}}{m^{2}})T\sup_{t\in[0,T]}\mathcal{W}\left(\mathcal{L}_{X^{\varepsilon}_{t}},\delta_{\psi_{t}}\right)+C(1+\frac{M^{2}}{m^{2}})\varepsilon$

Apply Gronwall’s lemma, Lemma 5.8 and letting $\varepsilon\rightarrow\infty$,
we get

$\lim_{\varepsilon\rightarrow
0}\mathbb{E}\left[\sup_{t\in[0,T]}|Z_{t}^{\varepsilon}-Y_{t}^{h^{\varepsilon}}|^{2}\right]=0$

The proof is complete.

##  6 Invariant probability measure and Harnack inequality

###  6.1 Existence of invariant probability measure

For any $X_{s,t}^{x}$ is the solution of (1.1) with initial condition $x$ from
$s$ to $t$, let us define operator $P_{s,t}$ and its adjoin $P^{\star}_{s,t}$
by

$P_{s,t}f(x):=\mathbb{E}f\left(X_{s,t}^{x}\right)=\int_{\mathbb{R}}f(y)\left(P_{s,t}^{*}\delta_{x}\right)(\mathrm{d}y)$

$P_{s,t}$ is not a semigroup, but the dual $P_{s,t}^{*}$ is a non-linear
semigroup on $\mathcal{P}_{2}$.

We first prove the exponential convergence.

######  Lemma 6.1 (Exponential convergence)

Suppose Assumption (A.1), (A.2) and (A.5) hold, let $X_{t}$ and $Y_{t}$ be two
solution of (1.1) such that $\mathcal{L}_{\xi^{X_{0}}}=\mu_{0}$,
$\mathcal{L}_{\xi^{Y_{0}}}=\nu_{0}$ and
$\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}=\mathbb{E}\left[\left|X_{0}-Y_{0}\right|^{2}\right]$.
We have

$\mathcal{W}_{2}\left(P^{*}_{t}\mu_{0},P^{*}_{t}\nu_{0}\right)^{2}\leq\mathrm{e}^{-\left(C_{2}-C_{1}\right)t}\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}$

###### Proof 15

From the definition of $P^{*}_{t}$, we have $P^{*}_{t}\mu_{0}=\mu_{t}$,
$P^{*}_{t}\nu_{0}=\nu_{t}$.

Apply Itô’s lemma to $(X_{t}-Y_{t})^{2}$, we have

$\displaystyle\left(X_{t}-Y_{t}\right)^{2}$
$\displaystyle=(\xi^{X_{0}}-\xi^{Y_{0}})^{2}+\int_{0}^{t}2\left(X_{t}-Y_{t}\right)\left[b(X_{t},\mu_{t})-b(Y_{t},\nu_{t})\right]dt+\int_{0}^{t}\sigma(X_{t},\mu_{t})-\sigma(Y_{t},\nu_{t})dt$
(6.1)
$\displaystyle+\int_{0}^{t}2\left(X_{t}-Y_{t}\right)\left[\sigma(X_{t},\mu_{t})-\sigma(Y_{t},\nu_{t})\right]dB_{t}+\int_{0}^{t}2(X_{t}-Y_{t})d(K^{X}_{t}-K^{Y}_{t})$

Since $X_{t}$ and $Y_{t}$ is the solution of (1.1) and $h$ is bi-Lipschitz ,
we have $\mathbb{E}\left[X_{t}\right]\geq 0$ and
$\mathbb{E}\left[Y_{t}\right]\geq 0$. Combining $K^{X}_{t}$ and $K^{Y}_{t}$
are non-decrease processes, we have

$\int_{0}^{t}\mathbb{E}\left[X_{t}\right]dK^{Y}_{t}\geq
0\quad\int_{0}^{t}\mathbb{E}\left[Y_{t}\right]dK^{X}_{t}\geq 0$

By the definition of solution (1.1), we obtain

$\int_{0}^{t}\mathbb{E}\left[h(X_{t})\right]dK^{X}_{t}=\int_{0}^{t}\mathbb{E}\left[h(Y_{t})\right]dK^{Y}_{t}=0$

Taking expectation of (6.1) and applying stochastic Fubini’s Theorem, we have
following estimation

$\mathbb{E}\left(X_{t}-Y_{t}\right)^{2}\leq(\xi^{X_{0}}-\xi^{Y_{0}})^{2}+\mathbb{E}\int_{0}^{t}2\left(X_{t}-Y_{t}\right)\left[b(X_{t},\mu_{t})-b(Y_{t},\nu_{t})\right]dt+\mathbb{E}\int_{0}^{t}\sigma(X_{t},\mu_{t})-\sigma(Y_{t},\nu_{t})dt$

Apply Assumption 5 and Gronwall’s lemma, we obtain

$\mathcal{W}_{2}\left(P^{*}_{t}\mu_{0},P^{*}_{t}\nu_{0}\right)^{2}\leq\mathbb{E}\left[|X_{t}-Y_{t}|^{2}\right]\leq
e^{-(C_{2}-C_{1})t}\mathbb{E}\left[|\xi^{X_{0}}-\xi^{Y_{0}}|^{2}\right]\leq
e^{-(C_{2}-C_{1})t}\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}$

######  Theorem 6.2 (Existence of invariant probability measure)

Suppose Assumption (A.1), (A.2) and (A.5) hold. $P_{t}$ has a unique invariant
probability measure $\mu\in\mathcal{P}_{2}$ i.e. $P_{t}^{*}\mu=\mu,t>0$ such
that

$\mathcal{W}_{2}\left(P_{t}^{*}\nu_{0},\mu\right)^{2}\leq\mathcal{W}_{2}\left(\nu_{0},\mu\right)^{2}\mathrm{e}^{-\left(C_{2}-C_{1}\right)t},\quad
t\geq 0,\nu_{0}\in\mathcal{P}_{2}.$

###### Proof 16

We first show that for any $\mu\in\mathcal{P}_{2}(\mathbb{R})$ we have

$\displaystyle\lim_{t\rightarrow\infty}\mathcal{W}_{2}\left(P^{*}_{t}\delta_{0},\mu\right)=0$
(6.2)

To this end, it suffices to show that
$\left\\{P_{t}^{*}\delta_{0}\right\\}_{t\geq 0}$ is a $\mathcal{W}_{2}$-Cauchy
family when $t\rightarrow\infty$. Using the semigroup property of
$P_{t+s}^{*}$ and the exponential convergence, we have

$\displaystyle\mathcal{W}_{2}\left(P_{t+s}^{*}\delta_{0},P_{t}^{*}\delta_{0}\right)^{2}\leq\mathcal{W}_{2}\left(P_{s}^{*}\delta_{0},\delta_{0}\right)^{2}\mathrm{e}^{-\left(C_{2}-C_{1}\right)t}=\mathrm{e}^{-\left(C_{2}-C_{1}\right)t}\mathbb{E}\left|X_{s}^{0}\right|^{2},\quad
s,t\geq 0.$

where $X^{0}_{S}$ is a solution of MR-MVSDE which initial condition is
$\xi=0$. Then, we only need to show

$\sup_{s\geq 0}\mathbb{E}\left[|X^{0}_{s}|^{2}\right]\leq\infty$

It is true from Proposition 3.4. By (6.2), $P^{*}_{t+s}=P^{*}_{t}P^{*}_{s}$
and

$\mathcal{W}_{2}\left(P^{*}_{t}\mu_{1},P^{*}_{t}\mu_{2}\right)\leq
e^{-(C_{2}-C_{1})t/2}\mathcal{W}_{2}\left(\mu_{1},\mu_{2}\right),$

we obtain that

$\mathcal{W}_{2}\left(P^{*}_{t}\mu,\mu\right)=\lim_{s\rightarrow\infty}\mathcal{W}_{2}\left(P^{*}_{t}\mu,P^{*}_{t+s}\delta_{0}\right)\leq
e^{-(C_{2}-C_{1})t/2}\lim_{s\rightarrow\infty}\mathcal{W}_{2}\left(\mu,P^{*}_{s}\delta_{0}\right)=0$

That means $P^{*}_{t}\mu=\mu$.

###  6.2 Log-Harnack inequality and Shift Harnack inequality

In this subsection, we investigate the dimension-free log-Harnack inequality
and Shift Harnack inequality. Using the coupling by change of measure method,
we establish log-Harnack inequalities for $P_{t}f$, as detailed in [51, 57].
However, for this purpose, we need to assume that the noise term is
distribution-free and that $|\sigma(x)^{-1}|\leq\lambda$. This implies that we
consider a particular version of (1.1):

$\left\\{\begin{array}[]{l}X_{t}=\xi^{X_{0}}+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds+\int_{0}^{t}\sigma\left(X_{s}\right)dB_{s}+K_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}=0,\quad
t\geq 0\end{array}\right.$ (6.3)

For any $\mu_{0}\in\mathcal{P}_{2}(\mathbb{R})$ and $r\geq 0$, let
$B(\mu_{0},r)=\left\\{\nu\in\mathcal{P}_{2}(\mathbb{R}):\mathcal{W}_{2}(\mu_{0},\nu)\leq
r\right\\}$. Let

$\phi(s,t)=\lambda^{2}\left(\frac{C_{1}}{1-e^{-C_{1}(t-s)}}+\frac{tC_{2}^{2}\exp\left(2(C_{1}+C_{2})(t-s)\right)}{2}\right),\quad
0\leq s<t.$

Let $\xi_{t}=\frac{1}{C_{1}}(1-e^{C_{1}(t-T)})$, $\nu_{t}=P^{*}_{t}\nu_{0}$
and let $\xi^{Y_{0}}$ be $\mathcal{F}_{0}-$measurable with
$\mathcal{L}_{\xi^{Y_{0}}}=\nu_{0}$. Consider the following MR-MVSDE:

$\left\\{\begin{array}[]{l}Y_{t}=\xi^{Y_{0}}+\int_{0}^{t}b\left(Y_{s},\mathcal{L}_{Y_{s}}\right)ds+\int_{0}^{t}\frac{1}{\xi_{s}}\sigma(Y_{s})\sigma(X_{s})^{-1}(X_{s}-Y_{s})ds+\int_{0}^{t}\sigma\left(Y_{s}\right)dB_{s}+K^{Y}_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(Y_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(Y_{s}\right)\right]dK^{Y}_{s}=0,\quad
t\geq 0\end{array}\right.$ (6.4)

By Theorem 3.2, this MR-MVSDE has unique solution
$\left(Y_{t}\right)_{t\in[0,T]}$. Let

$\tau_{n}=T\wedge\inf\left\\{t\in[0,T):|X_{t}|+|Y_{t}|\geq n\right\\},$

we have $\tau_{n}\rightarrow T$ as $n\rightarrow\infty$. We first show

$\displaystyle
R_{t}:=\exp\left\\{\int_{0}^{t}\frac{1}{\xi_{s}}\sigma\left(X_{s}\right)^{-1}(Y_{s}-X_{s})dB_{s}-\frac{1}{2}\int_{0}^{t}\frac{\left|\sigma\left(X_{s}\right)^{-1}(Y_{s}-X_{s})\right|^{2}}{\xi_{s}^{2}}ds\right\\}$
(6.5)

is a uniformly integrable martingale for $t\in[0,T]$.

###### Lemma 6.3

Suppose Assumption (A.1) and (A.2) hold, and $\sigma(x)^{-1}<\infty$. Let
$X_{0}$, $Y_{0}$ be two $\mathcal{F}_{0}-$measurable random variables such
that $\mathcal{L}_{\xi^{X_{0}}}=\mu_{0}$, $\mathcal{L}_{\xi^{Y_{0}}}=\nu_{0}$
and

$\displaystyle\mathbb{E}\left[\left|\xi^{X_{0}}-\xi^{Y_{0}}\right|^{2}\right]=\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}$
(6.6)

Then $\left(R_{t}\right)_{t\in[0,T]}$ is a uniformly intgrable martingale with

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[R_{t}\log
R_{t}\right]\leq\phi(0,T)\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}$
(6.7)

###### Proof 17

By Assumption (A.1), for any $n\geq 1$,
$\left(R_{t\wedge\tau_{n}}\right)_{t\in[0,T]}$ is a uniformly integrable
continuous martingale. Since $\tau_{n}\rightarrow T$ as $n\rightarrow\infty$,
by the martingale convergence Theorem, we only need to prove

$\displaystyle\sup_{t\in[0,T),n\geq 1}\mathbb{E}\left[R_{t\wedge\tau_{n}}\log
R_{t\wedge\tau_{n}}\right]\leq\phi(0,T)\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}$
(6.8)

We fix $t\in[0,T)$ and $n\geq 1$. By Girsanov’s Theorem,

$\tilde{B}_{s}:=B_{s}-\frac{1}{\xi_{s}}\sigma(X_{s})^{-1}(Y_{s}-X_{s}),s\in[0,t\wedge\tau_{n}]$

is a Brownian motion under the weighted probability
$\mathbb{Q}_{t,n}:=R_{t\wedge\tau_{n}}\mathbb{P}$.

Under $\mathbb{Q}_{t,n}$, reformulating (6.3) and (6.4)

$\left\\{\begin{array}[]{l}X_{t\wedge\tau_{n}}=\xi^{X_{0}}+\int_{0}^{t\wedge\tau_{n}}b\left(X_{s},\mathcal{L}_{X_{s}}\right)-\frac{X_{s}-Y_{s}}{\xi_{s}}ds+\int_{0}^{t\wedge\tau_{n}}\sigma\left(X_{s}\right)d\tilde{B}_{s}+\tilde{K}^{X}_{t\wedge\tau_{n}}\\\
\mathbb{E}^{\mathbb{Q}_{t,n}}\left[h\left(X_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}^{\mathbb{Q}_{t,n}}\left[h\left(X_{s}\right)\right]d\tilde{K}^{X}_{s}=0\end{array}\right.$
(6.9)

$\left\\{\begin{array}[]{l}Y_{t\wedge\tau_{n}}=\xi^{Y_{0}}+\int_{0}^{t\wedge\tau_{n}}b\left(Y_{s},\mathcal{L}_{Y_{s}}\right)ds+\int_{0}^{t\wedge\tau_{n}}\sigma\left(Y_{s}\right)d\tilde{B}_{s}+\tilde{K}^{Y}_{t\wedge\tau_{n}}\\\
\mathbb{E}\left[h\left(Y_{t\wedge\tau_{n}}\right)\right]\geq
0,\quad\int_{0}^{t\wedge\tau_{n}}\mathbb{E}^{\mathbb{Q}_{t,n}}\left[h\left(Y_{s}\right)\right]d\tilde{K}^{Y}_{s}=0\end{array}\right.$
(6.10)

Apply Itô’s Lemma, we have

$\displaystyle\left(X_{t\wedge\tau_{n}}-Y_{t\wedge\tau_{n}}\right)^{2}$
$\displaystyle\leq\int_{0}^{t\wedge\tau_{n}}C_{1}|X_{s}-Y_{s}|^{2}+C_{2}\mathcal{W}_{2}\left(\mu_{s},\nu_{s}\right)-\frac{2|X_{s}-Y_{s}|^{2}}{\xi_{s}}ds$
(6.11)
$\displaystyle+\int_{0}^{t\wedge\tau_{n}}\left(X_{s}-Y_{s}\right)\left(\sigma(X_{s})-\sigma(Y_{s})\right)d\tilde{B}_{s}+\int_{0}^{t\wedge\tau_{n}}(X_{s}-Y_{s})d\left(\tilde{K}^{X}_{s}-\tilde{K}^{Y}_{s}\right)$

Using the some argument of proving Lemma 6.1, we have

$\int_{0}^{t\wedge\tau_{n}}(X_{s}-Y_{s})d\left(\tilde{K}^{X}_{s}-\tilde{K}^{Y}_{s}\right)\leq
0$

Then, apply Young’s inequality, we have

$\displaystyle\frac{\left(X_{t\wedge\tau_{n}}-Y_{t\wedge\tau_{n}}\right)^{2}}{\xi_{t\wedge\tau_{n}}}$
$\displaystyle\leq\frac{C^{2}}{2}\int_{0}^{t\wedge\tau_{n}}\mathcal{W}_{2}\left(\mu_{s},\nu_{s}\right)^{2}ds-\int_{0}^{t\wedge\tau_{n}}\left(\frac{3}{2}-C_{1}\xi_{s}\right)\frac{|X_{s}-Y_{s}|^{2}}{\xi_{s}^{2}}ds$
(6.12)
$\displaystyle+\int_{0}^{t\wedge\tau_{n}}\frac{\left(X_{s}-Y_{s}\right)}{\xi_{s}}\left(\sigma(X_{s})-\sigma(Y_{s})\right)d\tilde{B}_{s}$

By the exponential convergence (Lemma 6.1), we have

$\mathcal{W}_{2}\left(\mu_{s},\nu_{s}\right)\leq\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)e^{(C_{1}+C_{2})s}$

Then, we have

$\displaystyle\mathbb{E}^{\mathbb{Q}_{t,n}}\int_{0}^{t\wedge\tau_{n}}\frac{|X_{s}-Y_{s}|^{2}}{\xi_{s}^{2}}ds$
$\displaystyle\leq\frac{2}{\xi_{0}}+\mathbb{E}^{\mathbb{Q}_{t,n}}\int_{0}^{t\wedge\tau_{n}}C_{2}^{2}\mathcal{W}_{2}\left(\mu_{s},\nu_{s}\right)^{2}ds$
(6.13)
$\displaystyle\leq\frac{2}{\xi_{0}}+TC_{2}^{2}e^{2(C_{1}+C_{2})T}\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}.$

Therefore, (6.8) holds since $t\in[0,T)$ and $n\geq 1$ are arbitrary.

######  Theorem 6.4 (Log-Harnack inequality)

Suppose Assumption (A.1) and (A.2) hold and $\sigma(x)^{-1}<\infty$. Let
$0\leq s<t\leq T$

  1. 1.

For any $\mu_{0},\nu_{0}\in\mathcal{P}_{2}(\mathbb{R})$,

$\left(P_{s,t}\log
f\right)\left(\nu_{0}\right)\leq\log\left(P_{s,t}f\right)\left(\mu_{0}\right)+\phi(s,t)\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2},\quad
f\in\mathcal{B}_{b}^{+}\left(\mathbb{R}\right).$

Consequently,

$\left|\nabla P_{s,t}f\right|^{2}\leq
2\phi(s,t)\left\\{P_{s,t}f^{2}-\left(P_{s,t}f\right)^{2}\right\\},\quad
f\in\mathcal{B}_{b}\left(\mathbb{R}\right)$

  2. 2.

For any different $\mu_{0},\nu_{0}\in\mathcal{P}_{2}(\mathbb{R})$ and
$f\in\mathcal{B}_{b}^{+}\left(\mathbb{R}\right)$,

$\displaystyle\frac{\left|\left(P_{s,t}f\right)\left(\mu_{0}\right)-\left(P_{s,t}f\right)\left(\nu_{0}\right)\right|^{2}}{\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}}\quad\leq
2\phi(s,t)\sup_{\nu\in
B\left(\mu_{0},\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)\right)}\left\\{\left(P_{s,t}f^{2}\right)(\nu)-\left(P_{s,t}f\right)^{2}(\nu)\right\\}.$

Consequently,

$\displaystyle\left\|P_{s,t}^{*}\mu_{0}-P_{s,t}^{*}v_{0}\right\|_{var}$
$\displaystyle:=2\sup_{A\in\mathcal{B}_{b}\left(\mathbb{R}\right)}\left|\left(P_{s,t}^{*}\mu_{0}\right)(A)-\left(P_{s,t}^{*}v_{0}\right)(A)\right|\leq\sqrt{2\phi(s,t)}\mathcal{W}_{2}\left(\mu_{0},v_{0}\right).$

###### Proof 18

By Lemma 6.3 and Girsanov’s Theorem, $d\mathbb{Q}:=R_{T}d\mathbb{P}$ is a
probability measure such that

$\tilde{B}_{t}:=B_{t}-\int_{0}^{t}\frac{\sigma(X_{s})^{-1}(Y_{s}-X_{s})}{\xi_{s}}ds\quad
t\in[0,T]$

is a Brownian motion. Then, (6.4) can be written to

$\left\\{\begin{array}[]{l}Y_{t}=\xi^{Y_{0}}+\int_{0}^{t}b\left(Y_{s},\mathcal{L}_{Y_{s}}\right)ds+\int_{0}^{t}\sigma\left(Y_{s}\right)d\tilde{B}_{s}+\tilde{K}^{Y}_{t}\\\
\mathbb{E}^{\mathbb{Q}}\left[h\left(Y_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}^{\mathbb{Q}}\left[h\left(Y_{s}\right)\right]d\tilde{K}^{Y}_{s}=0\end{array}\right.$
(6.14)

Consider MR-MVSDE under $\mathbb{Q}$ measure

$\left\\{\begin{array}[]{l}\tilde{X}_{t}=\xi^{Y_{0}}+\int_{0}^{t}b\left(\tilde{X}_{s},\mathcal{L}_{\tilde{X}_{s}}\right)ds+\int_{0}^{t}\sigma\left(\tilde{X}_{s}\right)d\tilde{B}_{s}+\tilde{K}^{\tilde{X}}_{t}\\\
\mathbb{E}^{\mathbb{Q}}\left[h\left(\tilde{X}_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}^{\mathbb{Q}}\left[h\left(\tilde{X}_{s}\right)\right]d\tilde{K}^{Y}_{s}=0\end{array}\right.$
(6.15)

By Theorem 3.2, we have $\tilde{X}_{t}=Y_{t}$ for $t\in[0,T]$. So we deduce
that

$\left(P_{T}\log
f\right)(\nu_{0})\leq\log\left(P_{T}f\right)(\mu_{0})+\phi(0,T)\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}.$

In particular,

$P_{T}\log f(x)\leq\left(\log P_{T}f\right)(y)+\phi(0,T)|x-y|^{2}.$

Following [64] We can get for any $f\in\mathcal{B}_{b}^{+}(\mathbb{R})$

$\left|\nabla P_{s,t}f\right|^{2}\leq
2\phi(s,t)\left[P_{s,t}f^{2}-\left(P_{s,t}f\right)^{2}\right]$

Repeating the proof of [46][Theorem 4.1 (ii)] we can obtain for any different
$\mu_{0}$ and $\nu_{0}\in\mathcal{P}_{2}(\mathbb{R})$ and any
$f\in\mathcal{B}_{b}^{+}(\mathbb{R})$,

$\displaystyle\frac{\left|\left(P_{s,t}f\right)\left(\mu_{0}\right)-\left(P_{s,t}f\right)\left(\nu_{0}\right)\right|^{2}}{\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)^{2}}$
(6.16) $\displaystyle\quad\leq 2\phi(s,t)\sup_{\nu\in
B\left(\mu_{0},\mathcal{W}_{2}\left(\mu_{0},\nu_{0}\right)\right)}\left\\{\left(P_{s,t}f^{2}\right)(\nu)-\left(P_{s,t}f\right)^{2}(\nu)\right\\}.$

and

$\displaystyle\left\|P_{s,t}^{*}\mu_{0}-P_{s,t}^{*}v_{0}\right\|_{var}$
$\displaystyle:=2\sup_{A\in\mathcal{B}_{b}\left(\mathbb{R}^{d}\right)}\left|\left(P_{s,t}^{*}\mu_{0}\right)(A)-\left(P_{s,t}^{*}v_{0}\right)(A)\right|$
(6.17)
$\displaystyle\leq\sqrt{2\phi(s,t)}\mathcal{W}_{2}\left(\mu_{0},v_{0}\right).$

Then we establish Shift Harnack inequality for $P_{t}$ introduced in [60] . We
should assume that $\sigma(x,\mu)$ does not depend on $x$. Then, MR-MVSDE
(1.1) becomes

$\left\\{\begin{array}[]{l}X_{t}=\xi+\int_{0}^{t}b\left(X_{s},\mathcal{L}_{X_{s}}\right)ds+\int_{0}^{t}\sigma\left(\mathcal{L}_{X_{s}}\right)dB_{s}+K_{t},\quad
t\geq 0\\\ \mathbb{E}\left[h\left(X_{t}\right)\right]\geq
0,\quad\int_{0}^{t}\mathbb{E}\left[h\left(X_{s}\right)\right]dK_{s}=0,\quad
t\geq 0\end{array}\right.$ (6.18)

######  Theorem 6.5 (Shift Harnack inequality)

Suppose Assumption (A.1) and (A.2) hold and $\sigma(\mu)$ is invertible with
$\sigma(\mu)+\sigma(\mu)^{-1}$ is bounded. For any $p>1$, $t\in[0,T]$,
$\mu\in\mathcal{P}_{2}(\mathbb{R})$, $\nu\in\mathbb{R}$ and
$f\in\mathcal{B}^{+}_{b}(\mathbb{R})$.

$\left(P_{t}f\right)^{p}(\mu_{0})\leq\left(P_{t}f(v+\cdot)^{p}\right)(\mu_{0})\exp{\frac{p\int_{0}^{t}\sigma(\mu_{s})^{-2}\left\\{|v|/t+s|v|/t\right\\}^{2}ds}{2(p-1)}}$

Moreover, for any $f\in\mathcal{B}_{b}^{+}(\mathbb{R})$ with $f>1$,

$\left(P_{t}\log
f\right)(\mu_{0})\leq\log\left(P_{t}f(v+\cdot)\right)(\mu_{0})+\frac{1}{2}\int_{0}^{t}\sigma(\mu_{s})^{-2}\left\\{|v|/t+s|v|/t\right\\}^{2}ds$

###### Proof 19

Let $Y_{t}=X_{t}+\frac{tv}{T}$, then

$dY_{t}=b(Y_{t},\mu_{t})dt+\sigma(\mu_{t})d\tilde{B}_{t}+dK_{t}$

where

$\displaystyle\tilde{B}_{t}$ $\displaystyle:=B_{t}+\int_{0}^{t}\eta_{s}ds$
$\displaystyle\eta_{t}$
$\displaystyle:=\sigma(\mu_{t})^{-1}\left\\{\frac{v}{T}+b(X_{t},\mu_{t})-b(X_{t}+\frac{tv}{T},\mu_{t})\right\\}.$

Let
$R_{T}=\exp\left\\{\int_{0}^{T}\eta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}|\eta_{s}|^{2}ds\right\\}$.
Since
$\mathbb{E}\left[\exp(\frac{1}{2}\int_{0}^{T}|\eta_{s}|^{2}ds)\right]<\infty$,
from Nokikov’s condition, we deduce that $R_{t}$ is a martingale and
$\mathbb{E}R_{T}=1$. By the Girsanov’s Theorem we can get

$\left(P_{T}f\right)(\mu_{0})=\mathbb{E}\left[R_{T}f(Y_{T})\right]=\mathbb{E}\left[R_{T}f(X_{T}+v)\right]\leq\left(P_{T}f(v+\cdot)^{p}\right)^{\frac{1}{p}}(\mu_{0})\left(\mathbb{E}R_{T}^{\frac{p}{p-1}}\right)^{\frac{p-1}{p}}$

apply Young’s inequality, we obtain

$\displaystyle\left(P_{T}\log f\right)(\mu_{0})$
$\displaystyle=\mathbb{E}\left[R_{T}\log
f(Y_{T})\right]=\mathbb{E}\left[R_{T}\log f\left(X_{T}+v\right)\right]$
$\displaystyle\leq\log P_{T}f(v+\cdot)(\mu_{0})+\mathbb{E}\left[R_{T}\log
R_{T}\right]$

Then we have

$\mathbb{E}R_{T}^{\frac{p}{p-1}}\leq\exp\left\\{\frac{p}{2(p-1)^{2}}\int_{0}^{T}|\eta_{s}|^{2}ds\right\\}\leq\exp\left\\{\frac{p\int_{0}^{T}\sigma(\mu_{t})^{-2}\left\\{|v|/T+t|v|/T\right\\}^{2}dt}{2(p-1)^{2}}\right\\}$

and

$\mathbb{E}\left[R_{T}\log R_{T}\right]=\mathbb{E}^{\mathbb{Q}}\log
R_{T}\leq\frac{1}{2}\mathbb{E}^{\mathbb{Q}}\int_{0}^{T}|\eta_{s}|^{2}ds\leq\frac{1}{2}\int_{0}^{T}\sigma(\mu_{t})^{-2}\left\\{|v|/T+t|v|/T\right\\}^{2}dt$

## Acknowledgements

The research is supported by NSFC (No. 71471075)

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