Exponential Contractivity and Propagation of Chaos for Langevin Dynamics of McKean-Vlasov Type with Lévy Noises

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-03-15 DOI:10.1007/s11118-024-10130-y

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引用次数: 0

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.

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带列维噪声的麦金-弗拉索夫型兰万动力学的指数收缩性和混沌传播

摘要通过概率耦合方法（该方法将新的精炼基本耦合与莱维过程的同步耦合结合在一起），我们为 $\mathbb R^{2d}$ 上 McKean-Vlasov 类型的以下朗格文动态 $(X_t,Y_t)_{t\ge 0}$ 得到了以标准 $L^1$ -Wasserstein 距离表示的明确指数收缩率： $$\begin{aligned}\dX_t=Y_t\,dt,dY_t=left( b(X_t)+\displaystyle int _{\mathbb R^d}\tilde{b}(X_t,z)\,\mu ^X_t(dz)-{\gamma }Y_t\right) \,dt+dL_t,\quad \mu ^X_t=\textrm{Law}(X_t), \end{array}.\right.\end{aligned}$$ where ${\gamma }>0$ , $b:\mathbb R^d\rightarrow \mathbb R^d$ and$\tilde{b}：\是两个全局李普齐兹连续函数，并且（(L_t)_{t\ge 0}）是一个（\mathbb R^d\）-值的纯跳跃李维过程。证明还基于一个新颖的距离函数，该函数是根据与所构建的耦合过程相关的边际的距离设计的。此外，通过应用上述耦合技术并进行一些修改，我们还提供了在标准 \(L^1$ -Wasserstein 距离下，具有莱维噪声的相应均场相互作用粒子系统在时间上均匀的混沌传播以及显式边界。

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.