McKean-Vlasov SPDEs with H$\ddot{\text{o}}$lder continuous coefficients: existence, uniqueness, ergodicity, exponential mixing and limit theorems

The McKean-Vlasov stochastic differential equations (SDEs), also called mean-field SDEs, were initially elucidated by Kac [28] within the framework of the Boltzmann equation addressing particle density in sparse monatomic gases and subsequently was initiated by McKean [37]. These equations find extensive utility across a spectrum of disciplines, notably in physics, statistical mechanics, quantum mechanics, and quantum chemistry. For instance, as seen in the literature [38, 6], the utilization of McKean-Vlasov SDEs is demonstrated to characterize the asymptotic behavior of $N$-interacting particle systems with weak interactions manifesting chaotic propagation properties. Larsy and Lions [31, 32, 33], as well as Huang et al. [25, 24], respectively introduced mean-field strategies to investigate the deterministic behavior of large populations and stochastic differential strategies. Additionally, Michael [41] analyzed the propagation of chaos results for the mean-field limit of a model for a trimolecular chemical reaction called the ”Brusselator”. Given the widespread applications of mean-field equations, we encourage interested readers to refer to [43, 44, 20, 11, 10, 7, 27] and the references therein. This broad applicability stems from the recognition that the evolution of many stochastic systems is contingent not solely upon the state of the solution but also upon its macroscopic distribution and it is precisely, these broad applications that have piqued our interest in investigating the dynamic properties of McKean-Vlasov S(P)DEs.

In the present paper, we will consider the following McKean-Vlasov stochastic (partial) differential equations:

$\displaystyle\begin{cases}\text{d}u(t)=(A(u(t),\mathcal{L}_{u(t)})+f(u(t),\mathcal{L}_{u(t)}))\text{d}t+g(u(t),\mathcal{L}_{u(t)})\text{d}W(t),\\ u(0)=x\in U_{1},\end{cases}$

where $W(t)$ is a cylindrical Wiener process on a separable Hilbert space $(U_{2},\langle\cdot,\cdot\rangle_{U_{2}})$ with respect to a complete filtered probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$, and $\mathcal{L}_{u(t)}$ denotes the law of $u(t)$ taking values in $\mathcal{P}(U_{1})$. When the system coefficients are Lipschitz continuous and satisfy linear growth conditions, there have been several relevant studies on the aforementioned McKean-Vlasov stochastic systems. Wang [45] established the existence and uniqueness of solutions for McKean-Vlasov monotone SDEs, and delved into pertinent topics such as exponential ergodicity and Harnack-type inequalities. Buckdahn et al. [9] demonstrated the correlation between functionals in the form of $\mathbb{E}f(t,X(t),\mathcal{L}_{x(t)})$ and the associated second-order PDEs and obtained the unique classical solution of a nonlocal mean-field PDE. Cheng and Liu [12] demonstrates that, despite the coefficients of the mean-field system adhering to the global Lipschitz condition and fulfilling certain requisite regularity criteria, the solution typically does not exhibit the global homeomorphism property beyond the initial time point. Instead, it generally manifests the local diffeomorphism property. In addition, under locally Lipschitz coefficients, Liu and Ma [36] derived conditions ensuring the global existence of solutions, subject to Lyapunov conditions. Furthermore, they demonstrated the exponential convergence of invariant measures. Barbu and R$\ddot{\text{o}}$ckner [4] employed nonlinear Fokker-Planck equations to explore the existence of weak solutions to McKean-Vlasov SDEs. For further results on this subject, we refer to [3, 8, 50, 5, 46, 39, 21, 40, 22], and others.

While the Lipschitz continuity condition holds significant importance in the examination of dynamical properties within stochastic systems, it is commonly observed that the Lipschitz continuity condition is frequently violated by coefficients in numerous significant stochastic models. For instance, the Cox-Ingersoll-Ross model and the diffusion coefficients in the Ferrer branch diffusion are merely H$\ddot{\text{o}}$lder continuous, rather than Lipschitz continuous. Consequently, for many such models, the (local) Lipschitz condition imposes a considerable constraint. As a result, stochastic models featuring non-Lipschitz coefficients have garnered growing interest in recent years. For example, Fang, Zhang [19] and Wang et al. [49] investigated SDEs and stochastic functional differential equations (SFDEs) featuring non-Lipschitz coefficients, specifically examining the existence and uniqueness of strong solutions. Ding and Qiao [16] employed the stochastic McKean-Vlasov SDEs with non-Lipschitz coefficients as the basis of the study. The existence of a weak solution is established using the Euler-Maruyama approximation, and the pathwise uniqueness of the weak solution is demonstrated to ascertain the existence of a strong solution. We observe that the non-Lipschitz condition mentioned in the previous literature is, in fact, still stronger than H$\ddot{\text{o}}$lder continuity. In Reference [30], Kulik and Scheutzow utilized the method of generalized coupling to establish weak uniqueness, but not path uniqueness, for the weak solution of a specific class of SFDEs with H$\ddot{\text{o}}$lder continuous coefficients. Therefore, this paper aims to initially investigate the existence and uniqueness of (strong) solutions within the context of McKean-Vlasov S(P)DEs, where the coefficients of the system adhere solely to H$\ddot{\text{o}}$lder continuity and linear growth conditions.

In the above literature, we discover that pathwise uniqueness plays a crucial role in determining the properties of (strong) solutions in stochastic differential systems. A classical finding reveals that under Lipschitz conditions, the influence of the initial value on the solution can be examined using Gronwall’s lemma. This leads to the establishment of pathwise uniqueness for the weak solution. However, when the regularity of the coefficients falls below the Lipschitz condition, employing Gronwall’s lemma becomes unfeasible. Consequently, one of the aims of our study in this paper is to address the aforementioned issues. Focusing on McKean-Vlasov S(P)DEs, we initially establish the existence and pathwise uniqueness of the weak solution for a class of finite-dimensional systems, where the coefficients solely adhere to the conditions of H$\ddot{\text{o}}$lder continuity and linear growth. We then proceed to establish the existence and uniqueness of the strong solution by leveraging the Yamada-Watanabe criterion. For detailed elaboration, refer to Theorem 3.1. In the context of infinite-dimensional McKean-Vlasov systems with coefficients satisfying the H$\ddot{\text{o}}$lder continuity, we employed the Galerkin projection technique. This method, combined with insights from finite-dimensional analysis, enables the proof of existence and uniqueness of variational solutions, as elaborated in Theorem 3.2.

Establishing limit theorems for Markov processes, particularly the strong law of large numbers (SLLN) and the central Limit Theorem (CLT), pivotal in elucidating the long-term dynamics of stochastic processes, constitutes a central pursuit in probability theory, see e.g. [1, 23, 14, 15, 18, 35]. Hence, another primary objective of this paper is to extend these theorems to McKean-Vlasov SPDEs with H$\ddot{\text{o}}$lder continuous coefficients. This endeavor necessitates an exploration of the exponential ergodicity. Previous investigations into the ergodic property of mean-field systems have primarily focused on scenarios characterized by monotonic or Lipschitz conditions, as evidenced by references [48, 34, 47], among others. Nonetheless, the exploration of lower regularity conditions, notably the H$\ddot{\text{o}}$lder continuous condition, remains relatively sparse in the existing literature. This study endeavors to refine the findings established in reference [36] and make extensions to McKean-Vlasov SPDEs with H$\ddot{\text{o}}$lder continuous coefficients. Drawing inspiration from references [36] and [13], we employ Lyapunov functions and the It$\hat{\text{o}}$ formula to unveil the exponential convergence of invariant measures within the framework of the Wasserstein quasi-metric. Furthermore, by reinforcing the condition to encompass integrable Lyapunov functions, we ascertain the exponential mixing property of invariant measures in the Wasserstein metric. Further elaboration on these advancements is provided in Theorem 4.2. This outcome is simple and interesting in its own right, as it imposes constraints on Lyapunov functions rather than solely on system coefficients, thereby broadening the method’s applicability.

As previously indicated, in the case of McKean-Vlasov SPDEs characterized by H$\ddot{\text{o}}$lder continuous coefficients, an invariant measure $\mu^{*}$ exhibiting uniform mixing properties is derived. Subsequently, leveraging both the exponential mixing property inherent in the invariant measure and the Markov property of the solution, we aim to demonstrate the applicability of the limit theorems delineated in [42] to the present system. Specifically, we establish the SLLN, the CLT, and their associated convergence rates, as detailed in Theorems 5.1 and 5.6:

1. 1.

   SLLN:

   $$\frac{1}{t}\int_{0}^{t}\Phi(u(s;0,x))\text{d}s\to\int_{U_{1}}\Phi(x)\mu^{*}(\text{d}x)\quad\text{as}\quad t\to\infty,\quad\mathbb{P}-a.s.,$$

   where $\Phi$ is the observation function;

2. 2.

   CLT:

   $$\frac{1}{\sqrt{t}}\int_{0}^{t}[\Phi(u(s;0,x))-\int_{\mathcal{H}}\Phi(x)\mu^{*}(\text{d}x)]\text{d}s\overset{W}{\to}\xi,$$

   where $\overset{W}{\to}$ means weak convergence and $\xi$ is a normal random variable.

The paper is structured as follows. The subsequent section delineates key definitions, notation, and lemmas. In Section 3, we first rigorously establish the existence and uniqueness of a strong solution for a specific class of finite-dimensional systems characterized by H$\ddot{\text{o}}$lder continuity. Subsequently, leveraging the sophisticated Galerkin projection technique and leveraging the insights garnered from the finite-dimensional context, we proceed to demonstrate the existence and uniqueness of solutions for the corresponding infinite-dimensional system. Section 4 elucidates properties of the solution including time homogeneity, the Markov and the Feller property. Based on these properties, this section also establishes the ergodicity and exponential mixing of invariant measure under Lyapunov conditions. Following this, in Section 5, we focus on establishing the SLLN, the CLT, and estimating the corresponding convergence rates for McKean-Vlasov SPDEs.

The $\left(\Omega,\mathscr{F},\mathbb{P}\right)$ be a certain complete probability space with a filtration $\{\mathscr{F}_{t}\}_{t\geq 0}$ satisfying the usual condition. If $K$ is a matrix or a vector, $K^{{}^{\prime}}$ is its transpose. For a matrix $K$, the norm is expressed as $\left\|K\right\|=\sqrt{\text{trace}(KK^{{}^{\prime}})}$ and denote by $\left\langle\cdot,\cdot\right\rangle$ the inner product of $\mathbb{R}^{n}$. $(U_{i},\left\|\cdot\right\|_{U_{i}})(i=1,2)$ are separable Hilbert spaces with inner product $\left\langle\cdot,\cdot\right\rangle_{U_{i}}$ and $(B,\left\|\cdot\right\|_{B})$ is a reflexive Banach space such that $B\subset U_{1}$. $U_{i}^{*}$, $B^{*}$ are the dual spaces of $U_{i}$, $B$, and

$\displaystyle B\subset U_{1}\subset B^{*},$

where the embedding $B\subset U_{1}$ is continuous and dense, hence we have $U_{1}^{*}\subset B^{*}$ continuously and densely. Let ${}_{B^{\ast}}\langle\cdot,\cdot\rangle_{B}$ denote the dualization between $B^{\ast}$ and $B$, which shows that for all $x\in U_{1}$, $y\in B$,

${}_{B^{\ast}}\langle x,y\rangle_{B}=\langle x,y\rangle_{U_{1}},$

and ($B,U_{1},B^{\ast}$) is called a Gelfand triple. For any $q\geq 1$ and the Banach space $(Y,\left\|\cdot\right\|_{Y})$, $\mathfrak{L}^{q}(\Omega,Y)$ denotes the Banach space of all $Y$-value random variables:

$\displaystyle\mathfrak{L}^{q}(\Omega,Y)=\left\{y:\Omega\to Y:\mathbb{E}\left\|y\right\|^{q}=\int_{\Omega}\left\|y\right\|^{p}_{Y}\text{d}\mathbb{P}<\infty\right\}.$

Let $\mathcal{B}(U_{1})$ denote the $\sigma$-algebra generated by space $U_{1}$ and $\mathcal{P}(U_{1})$ be the family of all probability measures defined on $\mathcal{B}(U_{1})$. We introduce the following Banach space $\mathcal{P}_{2}(U_{1})$ of signed measures $\mu$ on $\mathcal{B}(U_{1})$ satisfying

$\displaystyle\left\|\mu\right\|_{U_{1}}^{2}=\int_{U_{1}}(1+\left\|x\right\|_{U_{1}})^{2}\left|\mu\right|(\text{d}x)<\infty,$

where $\left|\mu\right|=\mu^{+}+\mu^{-}$ and $\mu=\mu^{+}-\mu^{-}$ is the Jordan decomposition of $\mu$. Let $\mathcal{P}^{*}(U_{1})=\mathcal{P}(U_{1})\cap\mathcal{P}_{2}(U_{1})$ be the family of all probability measures on $(U_{1},\mathcal{B}(U_{1}))$ with the following metric:

$\displaystyle d_{U_{1}}(\mu,\nu):=\sup\{\left|\int\Phi\text{d}\mu-\int\Phi\text{d}\nu\right|:\left\|\Phi\right\|_{BL}\leq 1\},$

where $\left\|\Phi\right\|_{BL}:=\left\|\Phi\right\|_{\infty}+Lip(\Phi)$ and $\left\|\Phi\right\|_{\infty}=\sup_{x\in U_{1}}\frac{|\Phi(x)|}{(1+\left\|x\right\|_{U_{1}})^{2}}$, $Lip(\Phi)=\sup_{x\neq y}\frac{\left|\Phi(x)-\Phi(y)\right|}{\left\|x-y\right\|_{U_{1}}}$. Then it is not difficult to verify that the space $(\mathcal{P}^{*}(U_{1}),d_{U_{1}})$ is a complete metric space; see [17] for more details on $(\mathcal{P}^{*}(U_{1}),d_{U_{1}})$ and its properties.

Consider the following McKean-Vlasov stochastic partial differential equations(SPDEs):

$\displaystyle\begin{cases}\text{d}u(t)=(A(u(t),\mathcal{L}_{u(t)})+f(u(t),\mathcal{L}_{u(t)}))\text{d}t+g(u(t),\mathcal{L}_{u(t)})\text{d}W(t),\\ u(0)=x\in U_{1},\end{cases}$

(2.1)

where $W(t)$ is a cylindrical Wiener processes on a separable Hilbert space $(U_{2},\langle\cdot,\cdot\rangle_{U_{2}})$ with respect to a complete filtered probability space $\left(\Omega,\mathscr{F},\mathbb{P}\right)$, and $\mathcal{L}_{u(t)}$ denotes the law of $u(t)$ taking values in $\mathcal{P}(U_{1})$. These are the measurable maps:

$$A:B\times\mathcal{P}^{*}(U_{1})\to B^{*},\quad f:U_{1}\times\mathcal{P}^{*}(U_{1})\to U_{1},\quad g:U_{1}\times\mathcal{P}^{*}(U_{1})\to\mathscr{L}(U_{2},U_{1}),$$

where $\mathscr{L}(U_{2},U_{1})$ is the space of all Hilbert-Schmidt operators from $U_{2}$ into $U_{1}$. The definition of variational solution to system (2.1) is given as follows.  

Definition 2.1.[22] We call a continuous $U_{1}$-valued $\{\mathscr{F}_{t}\}_{t\geq 0}$-adapted process $\{u(t)\}_{t\in[0,T]}$ is a solution of the system (2.1), if for its $\text{d}t\times\mathbb{P}$-equivalent class $\{\hat{u}(t)\}_{t\in[0,T]}$ satisfying $\hat{u}(t)\in\mathfrak{L}^{p}([0,T]\times\Omega,B)\times\mathfrak{L}^{2}([0,T]\times\Omega,U_{1})$ and $\mathbb{P}$-a.s.,

$\displaystyle\text{d}u(t)=x+\int_{o}^{t}(A(\bar{u}(s),\mathcal{L}_{\bar{u}(s)})+f(\bar{u}(s),\mathcal{L}_{\bar{u}(s)}))\text{d}s+\int_{0}^{t}g(\bar{u}(s),\mathcal{L}_{\bar{u}(s)})\text{d}W(s),$

where $\bar{u}$ is a $B$-valued progressively measurable $\text{d}t\times\mathbb{P}$ of $\hat{u}$.  

The transition probability of the Markov process defined on $U_{1}$ is a function $p:\Delta\times U_{1}\times\mathcal{B}(U_{1})\to\mathbb{R}^{+}$, where $\Delta=\{(t,s):t\geq s,t,s\in\mathbb{R}\}$ for $u(t)\in U_{1}$ and $\Gamma\in\mathcal{B}(U_{1})$ with the following properties:

1. 1.

   $p(t,s,x,\Gamma)=\mathbb{P}(\omega:u(t;s,x)\in\Gamma|u_{s})$;

2. 2.

   $p(t,s,\cdot,\Gamma)$ is $\mathcal{B}(U_{1})$-measurable for every $t\geq s$ and $\Gamma\in\mathcal{B}(U_{1})$;

3. 3.

   $p(t,s,x,\cdot)$ is a probability measure on $\mathcal{B}(U_{1})$ for every $t\geq s$ and $x\in U_{1}$;

4. 4.

   The Chapman-Kolmogorov equation:

   $\displaystyle p(t,s,x,\Gamma)=\underset{U_{1}}{\int}p(t,\tau,y,\Gamma)p(\tau,s,x,\text{d}y)$

   holds for any $s\leq\tau\leq t$, $x\in U_{1}$ and $\Gamma\in\mathcal{B}(U_{1})$.

We further define a map $\hat{p}(t,s):\mathcal{P}(U_{1})\to\mathcal{P}(U_{1})$ for any $\mu\in\mathcal{P}(U_{1})$ and $\Gamma\in\mathcal{B}(U_{1})$ by

$\displaystyle\hat{p}(t,s)\mu(\Gamma)=\underset{U_{1}}{\int}p(t,s,y,\Gamma)\mu(\text{d}y).$

(2.2)

The Markov process $u(t)$ is said to be time-homogeneous with the transition function $p(t,s,x,\Gamma)$, if

$\displaystyle p(t,s,x,\Gamma)=p(t+\theta,s+\theta,x,\Gamma),$

for all $t\geq s\geq 0$, $\theta\geq 0$, $x\in U_{1}$ and $\Gamma\in\mathcal{B}(U_{1})$.

To study the existence and uniqueness of solutions for (2.1) with H$\ddot{\text{o}}$lder continuous coefficients, we assume that the initial value $x\in U_{1}$ is independent of $W(t)$. We first assume that the coefficients in (2.1) satisfy the following hypotheses:  

(H1) (Continuity) For all $x,y\in B$ and $\mu\in\mathcal{P}^{*}(U_{1})$, the map

$\displaystyle B\times\mathcal{P}^{*}(U_{1})\ni(x,\mu)\to_{B^{\ast}}\langle A(x,\mu),y\rangle_{B}$

is continuous.
(H2) (Coercivity) There exist constant $\lambda_{1}\in\mathbb{R}$, $\lambda_{2}>0$, $p\geq 2$ and $M>0$ such that for all $x\in B$, $\mu\in\mathcal{P}^{*}(U_{1})$

${}_{B^{\ast}}\langle A(x,\mu),x\rangle_{B}\leq\lambda_{1}(\left\|x\right\|_{U_{1}}^{2}+\left\|\mu\right\|_{U_{1}}^{2})-\lambda_{2}\left\|x\right\|_{B}^{p}+M.$

(H3) (Growth) For $A$, there exists constant $\lambda_{1}$ for all $x\in B$, $\mu\in\mathcal{P}^{*}(U_{1})$ such that

$\displaystyle\left\|A(x,\mu)\right\|_{B^{*}}^{\frac{p}{p-1}}\leq\lambda_{2}(\left\|x\right\|_{B}^{p}+\left\|\mu\right\|_{U_{1}}^{2})+M,$

and for the continuous functions $f$, $g$, there exist constant $\lambda_{1}$ for all $x\in U_{1}$, $\mu\in\mathcal{P}^{*}(U_{1})$ such that

$\displaystyle\left\|f(x,\mu)\right\|_{U_{1}}\vee\left\|g(x,\mu)\right\|_{\mathscr{L}(U_{2},U_{1})}\leq\lambda_{2}(\left\|x\right\|_{U_{1}}+\left\|\mu\right\|_{U_{1}})+M.$

(H4) (H$\ddot{\text{o}}$lder continuity) Let $\alpha\in(0,1]$ and $\beta\in(\frac{1}{2},1]$. The map $A$ satisfies, for all $x,y\in B$, $\mu,\nu\in\mathcal{P}^{*}(U_{1})$,

$\displaystyle 2_{B^{\ast}}\langle A(x,\mu)-A(y,\nu),x-y\rangle_{B}\leq\lambda_{1}(\left\|x-y\right\|_{U_{1}}^{\alpha+1}+d_{U_{1}}^{\alpha+1}(\mu,\nu)),$

and the functions $f$, $g$ satisfy, for all $x,y\in U_{1}$ and $\mu,\nu\in\mathcal{P}^{*}(U_{1})$ with $\left\|x-y\right\|_{U_{1}}\vee d_{U_{1}}(\mu,\nu)\leq 1$,

$\displaystyle\left\langle f(x,\mu)-f(y,\nu),x-y\right\rangle_{U_{1}}\leq\lambda_{1}(\left\|x-y\right\|_{U_{1}}^{2\beta}+d_{U_{1}}^{2\beta}(\mu,\nu)),$

$\displaystyle\left\|g(x,\mu)-g(y,\nu)\right\|_{\mathscr{L}(U_{2},U_{1})}\leq\lambda_{2}(\left\|x-y\right\|_{U_{1}}^{\beta}+d_{U_{1}}^{\beta}(\mu,\nu)).$

In this paper, we write $L_{M}$ to mean some positive constants which depend on $M$. If it does not cause confusion, we always assume that the constants $\lambda_{1}\in\mathbb{R}$, $\lambda_{2},M>0$ and these constants may change from line to line. Now we discuss the existence and uniqueness of solutions for (2.1) whose coefficients are assumed to be only H$\ddot{\text{o}}$lder continuous.  

Theorem 3.1. Consider (2.1). Suppose that the assumptions (H1)$-$(H4) hold. For any initial values $x\in U_{1}$, system (2.1) has a unique solution $u(t)$ in the sense of Definition 2.1.  

Analyzing system (2.1), we will use the technique of Galerkin type approximation to get the existence and uniqueness of solutions. For ease of description, we first study the following McKean-Vlasov stochastic differential equations(MVSDEs) with H$\ddot{\text{o}}$lder continuous coefficients:

$\displaystyle\text{d}x(t)=F(x(t),\mathcal{L}_{x(t)})\text{d}t+G(x(t),\mathcal{L}_{x(t)})\text{d}B(t),$

(3.1)

with the initial data $x(0)=x_{0}\in\mathbb{R}^{n}$, where $B(t)$ is an $m$-dimensional Wiener process and $F:\mathbb{R}^{n}\times\mathcal{P}^{*}(\mathbb{R}^{n})\to\mathbb{R}^{n}$, $G:\mathbb{R}^{n}\times\mathcal{P}^{*}(\mathbb{R}^{n})\to\mathbb{R}^{n\times m}$ are two continuous maps. To ensure the existence and uniqueness of solutions for (3.1), we make the following assumptions:
(h1) The functions $F$ and $G$ are continuous in $(x,\mu)$ and satisfy, for all $x\in\mathbb{R}^{n}$, $\mu\in\mathcal{P}^{*}(\mathbb{R}^{n})$

$\displaystyle\left\langle F(x,\mu),x\right\rangle\vee\left\|G(x,\mu)\right\|^{2}\leq\lambda_{1}(\left\|x\right\|^{2}+\left\|\mu\right\|_{\mathbb{R}^{n}}^{2})+M.$

(h2) The functions $F,G$ satisfy, for all $x,y\in\mathbb{R}^{n}$ and $\mu,\nu\in\mathcal{P}^{*}(\mathbb{R}^{n})$ with $\left\|x-y\right\|\leq 1$ and $d_{\mathbb{R}^{n}}(\mu,\nu)\leq 1$,

$\displaystyle 2\left\langle F(x,\mu)-F(y,\nu),x-y\right\rangle\vee\left\|G(x,\mu)-G(y,\nu)\right\|^{2}\leq\lambda_{1}(\left\|x-y\right\|^{2\beta}+d^{2\beta}_{\mathbb{R}^{n}}(\mu,\nu)).$

Theorem 3.2. Consider (3.1). Suppose that the assumptions (h1)$-$(h2) hold. Then the following statement holds: for any $x_{0}\in\mathbb{R}^{n}$, there exist a unique strong solution $x(t)$ to (3.1) with $x(0)=x_{0}$.  

proof: Based on the above analysis, we divide the proof of Theorem 3.2 into the following two steps:
step 1: Existence of the weak solutions. We choose a family of finite-dimensional projections $\{\Gamma_{n}\}_{n\geq 1}$ in $\mathcal{P}^{*}(\mathbb{R}^{n})$ that satisfy the following property:

$\displaystyle\Gamma_{n}\to I(n\to\infty),$

where $I$ is the identity transformation on $\mathcal{P}^{*}(\mathbb{R}^{n})$. Let $F^{n}(x,\mu)=F(x,\Gamma_{n}(\mu))$ and $G^{n}(x,\mu)=G(x,\Gamma_{n}(\mu))$, which implies that $F^{n},G^{n}$ satisfy the following properties:

1. 1.

   $F^{n}\to F$, $G^{n}\to G$ as $n\to\infty$ uniformly on each compact subset of $\mathbb{R}^{n}\times\mathcal{P}^{*}(\mathbb{R}^{n})$;

1. 1.

   $F^{n},G^{n}$ satisfy the conditions (h1) and (h2), that is, the coefficients are independent of $n$.

We then convolve the $(F^{n},G^{n})$ with the finite-dimensional approximation $\delta$-function to obtain the $(F^{n,\delta},G^{n,\delta})$, hence it’s not hard to get that the functions $F^{n,\delta}$ and $G^{n,\delta}$ are Lipschitz continuous on each bounded subset of $\mathbb{R}^{n}\times\mathcal{P}^{*}(\mathbb{R}^{n})$. Fix $n\geq 1$ arbitrarily and we further consider the following equation:

$\displaystyle\text{d}x^{n}(t)=F^{n,\delta}(x^{n}(t),\mathcal{L}_{x^{n}(t)})\text{d}t+G^{n,\delta}(x^{n}(t),\mathcal{L}_{x^{n}(t)})\text{d}B(t).$

(3.2)

From the above analysis, (3.2) has a unique strong solution $x^{n}(t)$ with the initial condition $x^{n}(0)=x_{0}$. Then for any fixed $q\geq 2$, applying It$\hat{\text{o}}$ formula formula to $\|x^{n}(t)\|^{q}$, we have

$\displaystyle\begin{split}\|x^{n}(t)\|^{q}&=\left\|x_{0}\right\|^{q}+\int_{0}^{t}[q\left\|x^{n}(s)\right\|^{q-2}\left\langle F^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)}),x^{n}(s)\right\rangle\\ &~{}~{}~{}+\frac{q(q-1)}{2}\left\|x^{n}(s)\right\|^{q-2}\left\|G^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)})\right\|^{2}]\text{d}s\\ &~{}~{}~{}+\int_{0}^{t}q\left\|x^{n}(s)\right\|^{q-2}[x^{n}(s)]^{{}^{\prime}}G^{n,\delta}(x^{n}(t),\mathcal{L}_{x^{n}(t)})\text{d}B(s).\end{split}$

By Young inequality and assumptions (h1) and (h2), there exist some constants $L_{q,\lambda_{1}}$ such that

	$\displaystyle\|x^{n}(t)\|^{q}$	$\displaystyle\leq\left\|x_{0}\right\|^{q}+L_{q,\lambda_{1}}\int_{0}^{t}(\left\|x^{n}(s)\right\|^{q}+\left\|\mathcal{L}_{x^{n}(s)}\right\|_{\mathbb{R}^{n}}^{q}+M)\text{d}s$
		$\displaystyle~{}~{}~{}+L_{q,\lambda_{1}}\int_{0}^{t}(\left\|x^{n}(s)\right\|^{q}+\left\|\mathcal{L}_{x^{n}(s)}\right\|_{\mathbb{R}^{n}}^{q}+M)\text{d}B(s),$

where $L_{q,\lambda_{1}}$ depends only on $q,\lambda_{1}$. By Cauchy’s inequality and Burkholder–Davis–Gundy inequality, we obtain

$\displaystyle\begin{split}\mathbb{E}\underset{z\in[0,t]}{\sup}\left\|x^{n}(t)\right\|^{2q}&\leq 3\left\|x_{0}\right\|^{q}+L_{q,\lambda_{1}}\mathbb{E}[\int_{0}^{t}(\left\|x^{n}(s)\right\|^{q}+\left\|\mathcal{L}_{x^{n}(s)}\right\|_{\mathbb{R}^{n}}^{q}+M)\text{d}s]^{2}\\ &~{}~{}~{}+L_{q,\lambda_{1}}\mathbb{E}(\underset{z\in[0,t]}{\sup}\int_{0}^{t}(\left\|x^{n}(s)\right\|^{q}+\left\|\mathcal{L}_{x^{n}(s)}\right\|_{\mathbb{R}^{n}}^{q}+M)\text{d}B(s))^{2}\\ &\leq 3\left\|x_{0}\right\|^{q}+L_{q,\lambda_{1}}\mathbb{E}\int_{0}^{t}(\left\|x^{n}(s)\right\|^{2q}+\left\|\mathcal{L}_{x^{n}(s)}\right\|_{\mathbb{R}^{n}}^{2q}+M^{2})\text{d}s\\ &\leq 3\left\|x_{0}\right\|^{q}+L_{q,\lambda_{1}}\int_{0}^{t}[\mathbb{E}\left\|x^{n}(s)\right\|^{2q}+L_{q}\mathbb{E}(1+\left\|x^{n}(s)\right\|^{2q})+M^{2}]\text{d}s\\ &\leq 3\left\|x_{0}\right\|^{2q}+L_{q,\lambda_{1},M}[\int_{0}^{t}\mathbb{E}\underset{z\in[0,s]}{\sup}\left\|x^{n}(z)\right\|^{2q}\text{d}s+t].\end{split}$

Then, applying the Gronwall inequality yields

$\displaystyle\begin{split}\mathbb{E}\underset{z\in[0,t]}{\sup}\left\|x^{n}(t)\right\|^{2q}\leq L_{q,\lambda_{1},M}(\left\|x_{0}\right\|^{2q}+t+e^{t})<\infty,\end{split}$

(3.3)

for any $T>0$ and $t\in[0,T]$. We can follow from condition (h1) that $F^{n,\delta}$ and $G^{n,\delta}$ are bounded on every bounded subset of $\mathbb{R}^{n}\times\mathcal{P}^{*}(\mathbb{R}^{n})$, further by (h1) and (3.3), there exists a constant $L_{q,\lambda_{1},M,T}>0$ independent of $n$ such that

$\displaystyle\left\|F^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)})\right\|\vee\left\|G^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)})\right\|\leq L_{q,\lambda_{1},M,T}.$

Hence for any $0\leq z,t\leq T<\infty$, we have

$\displaystyle\begin{split}&\underset{n\geq 1}{\sup}\mathbb{E}\left\|x^{n}(t)-x^{n}(z)\right\|^{2q}\\ &\leq L_{q,\lambda_{1}}\underset{n\geq 1}{\sup}\mathbb{E}\left\|\int_{z}^{t}F^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)})\text{d}s\right\|^{2q}+L_{q,\lambda_{1}}\mathbb{E}\underset{n\geq 1}{\sup}\left\|\int_{z}^{t}G^{n,\delta}(x^{n}(s),\mathcal{L}_{x^{n}(s)})\text{d}B(s)\right\|^{2q}\\ &\leq L_{q,\lambda_{1},M,T}\left\|t-z\right\|^{q},\end{split}$

(3.4)

which yields that the family of laws $\mathcal{L}_{x^{n}(t)}$ of $x^{n}(t)$ is weakly compact. Further, from property (a) we can obtain the weak limit point $x^{*}(t)$ of $x^{n}(t)(n\to\infty)$, being a weak solution to (3.1). This proof is again quite standard and see Appendix I for details.  

step 2: Pathwise uniqueness of the weak solutions. In the following, we will present the pathwise uniqueness for (3.1). Suppose that two stochastic processes $x(t),y(t)$ satisfy the following form:

$\displaystyle x(t)=x_{0}+\int_{0}^{t}F(x(s),\mathcal{L}_{x(s)})\text{d}s+\int_{0}^{t}G(x(s),\mathcal{L}_{x(s)})\text{d}B(s),$

and

$\displaystyle y(t)=y_{0}+\int_{0}^{t}F(y(s),\mathcal{L}_{y(s)})\text{d}s+\int_{0}^{t}G(y(s),\mathcal{L}_{y(s)})\text{d}B(s).$

Without loss of generality, let’s assume $\left\|x_{0}-y_{0}\right\|<1$. Denote

$\displaystyle\tau_{N}=\underset{t\geq 0}{\inf}\{\left\|x(t)\right\|\vee\left\|y(t)\right\|>N\},\quad\tau_{\gamma}=\underset{t\geq 0}{\inf}\{\left\|x(t)-y(t)\right\|>\gamma\},$

where $N>\left\|x_{0}\right\|\vee\left\|y_{0}\right\|$ and $\gamma\in(\left\|x_{0}-y_{0}\right\|,1]$. In order to better prove the pathwise uniqueness of the weak solution, we first assume that the following property holds:

$\displaystyle\lim_{\left\|x_{0}-y_{0}\right\|\to 0}\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{N}\wedge\tau_{\gamma})-y(z\wedge\tau_{N}\wedge\tau_{\gamma})\right\|^{2})=0.$

(3.5)

Then by (3.3), we obtain that $\lim_{N\to\infty}\tau_{N}=\infty$ a.s., which together with Fatou’s lemma leads to

$$\lim_{\left\|x_{0}-y_{0}\right\|\to 0}\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{\gamma})-y(z\wedge\tau_{\gamma})\right\|^{2})=0.$$

In addition, by the definition of $\tau_{\gamma}$, we have $\left\|x(\tau_{\gamma})-y(\tau_{\gamma})\right\|=\gamma$, which implies

	$\displaystyle\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{\gamma})-y(z\wedge\tau_{\gamma})\right\|^{2})$	$\displaystyle=\mathbb{P}(t\geq\tau_{\gamma})\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{\gamma})-y(z\wedge\tau_{\gamma})\right\|^{2})$
		$\displaystyle~{}~{}~{}+\mathbb{P}(t<\tau_{\gamma})\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{\gamma})-y(z\wedge\tau_{\gamma})\right\|^{2})$
		$\displaystyle\geq\mathbb{P}(t\geq\tau_{\gamma})\gamma^{2},$

i.e., $\mathbb{P}(t\geq\tau_{\gamma})=0$ as $\left\|x_{0}-y_{0}\right\|\to 0$. Hence, by (h1) and (3.3), we have

	$\displaystyle\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z)-y(z)\right\|^{2})$	$\displaystyle=\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z)-y(z)\right\|^{2}\chi_{\{t>\tau_{\gamma}\}})+\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z)-y(z)\right\|^{2}\chi_{\{t\leq\tau_{\gamma}\}})$
		$\displaystyle\leq\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{\gamma})-y(z\wedge\tau_{\gamma})\right\|^{2})$
		$\displaystyle=0,$

which implies that as $\left\|x_{0}-y_{0}\right\|\to 0$,

$\displaystyle\mathbb{P}(\underset{z\in[0,t]}{\sup}\left\|x(z)-y(z)\right\|^{2}=0)=1,$

i.e., the pathwise uniqueness of weak solution holds, and then by Yamada–Watanabe principle, we obtain that there exists a unique global strong solution of (3.1).

From the above analysis, we know that the key condition for pathwise uniqueness is that (3.5) holds. So next, we’re going to apply the contradiction to prove that (3.5) is true. We assume that (3.5) does not hold, that is, there is a constant $a>0$, such that as $\left\|x_{0}-y_{0}\right\|\to 0$,

$$\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{N}\wedge\tau_{\gamma})-y(z\wedge\tau_{N}\wedge\tau_{\gamma})\right\|^{2})\geq a.$$

Then applying the It$\hat{\text{o}}$ formula to $\left\|x(t\wedge\tau_{N}\wedge\tau_{\gamma})-y(t\wedge\tau_{N}\wedge\tau_{\gamma})\right|^{2}$, Young inequality, BDG inequality and (h2) yield

		$\displaystyle\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{N}\wedge\tau_{\gamma})-y(z\wedge\tau_{N}\wedge\tau_{\gamma})\right\|^{2})$
		$\displaystyle=\left\|x_{0}-y_{0}\right\|^{2}+\mathbb{E}\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}[\left\|G(x(s),\mathcal{L}_{x(s)})-G(y(s),\mathcal{L}_{y(s)})\right\|^{2}$
		$\displaystyle~{}~{}~{}+2\left\langle F(x(s),\mathcal{L}_{x(s)})-F(y(s),\mathcal{L}_{y(s)}),x(s)-y(s)\right\rangle]\text{d}s$
		$\displaystyle~{}~{}~{}+2\mathbb{E}(\underset{z\in[0,t]}{\sup}\int_{0}^{z\wedge\tau_{N}\wedge\tau_{\gamma}}[x(s)-y(s)]^{\prime}(G(x(s),\mathcal{L}_{x(s)})-G(y(s),\mathcal{L}_{y(s)}))\text{d}B(s))$		(3.6)
		$\displaystyle\leq\left\|x_{0}-y_{0}\right\|^{2}+2\lambda_{1}\mathbb{E}\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}(\left\|x(s)-y(s)\right\|^{2\beta}+d_{\mathbb{R}^{n}}^{2\beta}(\mathcal{L}_{x(s)},\mathcal{L}_{y(s)}))\text{d}s$
		$\displaystyle~{}~{}~{}+12\mathbb{E}[\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}\left\|(x(s)-y(s))^{\prime}(G(x(s),\mathcal{L}_{x(s)})-G(y(s),\mathcal{L}_{y(s)}))\right\|^{2}\text{d}s]^{\frac{1}{2}}$
		$\displaystyle\leq\left\|x_{0}-y_{0}\right\|^{2}+2\lambda_{1}\mathbb{E}\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}(\left\|x(s)-y(s)\right\|^{2\beta}+d_{\mathbb{R}^{n}}^{2\beta}(\mathcal{L}_{x(s)},\mathcal{L}_{y(s)}))\text{d}s$
		$\displaystyle~{}~{}~{}+\frac{1}{2}\mathbb{E}(\underset{z\in[0,t\wedge\tau_{N}\wedge\tau_{\gamma}]}{\sup}\left\|x(z)-y(z)\right\|^{2})+72\lambda_{1}\mathbb{E}\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}(\left\|x(s)-y(s)\right\|^{2\beta}+d_{\mathbb{R}^{n}}^{2\beta}(\mathcal{L}_{x(s)},\mathcal{L}_{y(s)}))\text{d}s.$

By the definitions of $d_{\mathbb{R}^{n}}(\cdot,\cdot)$ ,

	$\displaystyle d_{\mathbb{R}^{n}}(\mathcal{L}_{x(s)},\mathcal{L}_{y(s)})$	$\displaystyle=\sup\{\left\|\int_{\mathbb{R}^{n}}\Phi\text{d}\mathcal{L}_{x(s)}-\int_{\mathbb{R}^{n}}\Phi\text{d}\mathcal{L}_{y(s)}\right\|:\left\|\Phi\right\|_{BL}\leq 1\}$
		$\displaystyle=\sup\{\left\|\int_{\mathbb{R}^{n}}\Phi(x)\mathbb{P}(\omega:x(s)\in\text{d}x)-\int_{\mathbb{R}^{n}}\Phi(x)\mathbb{P}(\omega:y(s)\in\text{d}x)\right\|:\left\|\Phi\right\|_{BL}\leq 1\}$
		$\displaystyle=\sup\{\left\|\mathbb{E}\Phi(x(s))-\mathbb{E}\Phi(y(s))\right\|:\left\|\Phi\right\|_{BL}\leq 1\}$		(3.7)
		$\displaystyle\leq\sup\{\left\|Lip(\Phi)\right\|\cdot\mathbb{E}\left\|x(s)-y(s)\right\|_{\mathbb{R}^{n}}:\left\|\Phi\right\|_{BL}\leq 1\}$
		$\displaystyle\leq\mathbb{E}\left\|x(s)-y(s)\right\|.$

Substituting (3) into (3), by H$\ddot{\text{o}}$lder inequality we have

		$\displaystyle\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{N}\wedge\tau_{\gamma})-y(z\wedge\tau_{N}\wedge\tau_{\gamma})\right\|^{2})$
		$\displaystyle\leq 2\left\|x_{0}-y_{0}\right\|^{2}+296\lambda_{1}\mathbb{E}\int_{0}^{t\wedge\tau_{N}\wedge\tau_{\gamma}}\left\|x(s)-y(s)\right\|^{2\beta}\text{d}s$		(3.8)
		$\displaystyle\leq 2\left\|x_{0}-y_{0}\right\|^{2}+296\lambda_{1}\int_{0}^{t}\mathbb{E}(\underset{z\in[0,s\wedge\tau_{N}\wedge\tau_{\gamma}]}{\sup}\left\|x(z)-y(z)\right\|^{2})^{\beta}\text{d}s$
		$\displaystyle:=\Lambda(t).$

Note that $\beta\in(\frac{1}{2},1]$ and let $H(t)=\frac{t^{1-2\beta}}{1-2\beta}$, then

1. 1.

   $H(t)$ is a monotonically increasing function;

1. 1.

   $\lim_{t\to 0^{+}}H(t)=-\infty$ ,

i.e., $H(t)$ satisfies $H(t)>-\infty$ for any $t>0$. Hence

$\displaystyle H(a)\leq H(\mathbb{E}(\underset{z\in[0,t]}{\sup}\left\|x(z\wedge\tau_{N}\wedge\tau_{\gamma})-y(z\wedge\tau_{N}\wedge\tau_{\gamma})\right\|^{2}))\leq H(\Lambda(t)),$

(3.9)

further

	$\displaystyle H(\Lambda(t))$	$\displaystyle=H(\Lambda(0))+\int_{0}^{t}H^{\prime}(\Lambda(s))\text{d}\Lambda(s)$
		$\displaystyle=H(2\left\|x_{0}-y_{0}\right\|^{2})$
		$\displaystyle~{}~{}~{}+\int_{0}^{t}\frac{1}{\Lambda^{\beta}(s)}\cdot\frac{296\mathbb{E}(\underset{z\in[0,s\wedge\tau_{N}\wedge\tau_{\gamma}]}{\sup}\left\|x(z)-y(z)\right\|^{2})^{\beta}}{\Lambda^{\beta}(s)}\text{d}s$		(3.10)
		$\displaystyle\leq H(2\left\|x_{0}-y_{0}\right\|^{2})+\int_{0}^{t}\frac{296}{\Lambda^{\beta}(s)}\text{d}s$
		$\displaystyle\leq H(2\left\|x_{0}-y_{0}\right\|^{2})+\frac{296}{a^{\beta}}t.$

By (3.9) and (3), we obtain

$\displaystyle H(2\left\|x_{0}-y_{0}\right\|^{2})\geq\frac{a^{1-2\beta}}{1-2\beta}-\frac{296}{a^{\beta}}t,$

which implies $\left\|x_{0}-y_{0}\right\|\neq 0$. This contradicts the condition in (3.5). Therefore we use the technique of contradiction to show that (3.5) holds. This completes the proof.  $\Box$  

Remark 3.3. For a stochastic differential equation(SDE) with low regularity coefficients, although considerable advance in the properties of its (weak) solutions, there are few work on whether there are strong solutions when the drift and diffusion coefficients of the system satisfy the H$\ddot{\text{o}}$lder continuity. Through the proof of Theorem 3.2, it is not difficult to find that for general SDEs, when its drift coefficient and diffusion coefficient only satisfy the H$\ddot{\text{o}}$lder continuity and linear growth conditions (i.e. similar to conditions and h1 and h2), we affirmatively prove the existence and uniqueness of the strong solution by Yamada–Watanabe principle.