Existence and Uniqueness for McKean-Vlasov Equations with Singular Interactions

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research 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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.