Abstract

By the probabilistic coupling approach which combines a new refined basic coupling with the synchronous coupling for Lévy processes, we obtain explicit exponential contraction rates in terms of the standard \(L^1\) -Wasserstein distance for the following Langevin dynamic \((X_t,Y_t)_{t\ge 0}\) of McKean-Vlasov type on \(\mathbb R^{2d}\) : $$\begin{aligned} \left\{ \begin{array}{l} dX_t=Y_t\,dt,\\ dY_t=\left( b(X_t)+\displaystyle \int _{\mathbb R^d}\tilde{b}(X_t,z)\,\mu ^X_t(dz)-{\gamma }Y_t\right) \,dt+dL_t,\quad \mu ^X_t=\textrm{Law}(X_t), \end{array} \right. \end{aligned}$$ where \({\gamma }>0\) , \(b:\mathbb R^d\rightarrow \mathbb R^d\) and \(\tilde{b}:\mathbb R^{2d}\rightarrow \mathbb R^d\) are two globally Lipschitz continuous functions, and \((L_t)_{t\ge 0}\) is an \(\mathbb R^d\) -valued pure jump Lévy process. The proof is also based on a novel distance function, which is designed according to the distance of the marginals associated with the constructed coupling process. Furthermore, by applying the coupling technique above with some modifications, we also provide the propagation of chaos uniformly in time for the corresponding mean-field interacting particle systems with Lévy noises in the standard \(L^1\) -Wasserstein distance as well as with explicit bounds.

中文翻译：

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具有Lévy噪声的McKean-Vlasov型Langevin动力学的指数收缩性和混沌传播

摘要

通过将新的改进的基本耦合与 Lévy 过程的同步耦合相结合的概率耦合方法，我们获得了以下 Langevin 动态\((X_t ) 的标准\(L^1\) -Wasserstein 距离的显式指数收缩率\(\mathbb R^{2d}\)上的 McKean-Vlasov 类型的,Y_t)_{t\ge 0}\) : $$\begin{aligned} \left\{ \begin{array}{l} dX_t =Y_t\,dt,\\ dY_t=\left( b(X_t)+\displaystyle \int _{\mathbb R^d}\tilde{b}(X_t,z)\,\mu ^X_t(dz)- {\gamma}Y_t\right) \,dt+dL_t,\quad \mu ^X_t=\textrm{Law}(X_t), \end{array} \right。\end{aligned}$$其中\({\gamma }>0\)、\(b:\mathbb R^d\rightarrow \mathbb R^d\)和\(\tilde{b}:\mathbb R^ {2d}\rightarrow \mathbb R^d\)是两个全局 Lipschitz 连续函数，而\((L_t)_{t\ge 0}\)是一个\(\mathbb R^d\)值的纯跳跃 Lévy过程。该证明还基于一种新颖的距离函数，该函数是根据与构造的耦合过程相关的边缘距离而设计的。此外，通过应用上述耦合技术并进行一些修改，我们还为具有标准\(L^1\) -Wasserstein 距离中的 Lévy 噪声的相应平均场相互作用粒子系统提供了时间上均匀的混沌传播，以及有明确的界限。