Existence and Uniqueness for McKean-Vlasov Equations with Singular Interactions

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-06-04 DOI:10.1007/s11118-024-10148-2

Guohuan Zhao

引用次数: 0

Abstract

We investigate the well-posedness of following McKean-Vlasov equation in $\mathbb {R}^d$:

$$\textrm{d} X_t=\sigma (t,X_t, \mu _{X_t})\textrm{d} W_t+b(t, X_t, \mu _{X_t}) \textrm{d} t,$$

where $\mu _{X_t}$ is the law of $X_t$. The existence of solutions is demonstrated when $\sigma $ satisfies certain non-degeneracy and continuity assumptions, and when b meets some integrability conditions, and continuity requirements in the (generalized) total variation distance. Furthermore, uniqueness is established under additional continuity assumptions of a Lipschitz type.

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具有奇异相互作用的麦金-弗拉索夫方程的存在性和唯一性

我们研究了以下麦金-弗拉索夫方程在 $\mathbb {R}^d$ 中的良好提出性：$$\textrm{d}X_t=sigma (t,X_t, \mu _{X_t})\textrm{d}W_t+b(t, X_t, \mu _{X_t}) \textrm{d} t,$$其中 $\mu _{X_t}$ 是 $X_t$的规律。当 $\sigma $ 满足某些非退化性和连续性假设时，当 b 满足某些可整性条件和（广义）总变化距离的连续性要求时，解的存在性就得到了证明。此外，在利普希兹类型的额外连续性假设下，唯一性也得以确立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.