McKean-Vlasov equations with singular coefficients - a review of recent results

TL;DR

This review paper summarizes recent advances in McKean-Vlasov stochastic differential equations with singular coefficients, highlighting new analytical tools and their applications to related non-linear Fokker-Planck equations.

Contribution

It provides a comprehensive overview of recent results, methods, and tools for analyzing McKean-Vlasov SDEs with various types of singularities in coefficients.

Findings

Different types of singularities are handled using advanced analytical tools.

Connections between McKean-Vlasov SDEs and non-linear Fokker-Planck equations are established.

Key tools like superposition principle and Zvonkin transformation are reviewed.

Abstract

This paper focuses on recent works on McKean-Vlasov stochastic differential equations (SDEs) involving singular coefficients. After recalling the classical framework, we review existing recent literature depending on the type of singularities of the coefficients: on the one hand they satisfy some integrability and measurability conditions only, while on the other hand the drift is allowed to be a generalised function. Different types of dependencies on the law of the unknown and different noises will also be considered. McKean-Vlasov SDEs are closely related to non-linear Fokker-Planck equations that are satisfied by the law (or its density) of the unknown. These connections are often established also in this singular setting and will be reviewed here. Important tools for dealing with singular coefficients are also included in the paper, such as Figalli-Trevisan superposition principle, Zvonkin transformation, Markov marginal uniqueness, and stochastic sewing lemma.