McKean SDEs with singular coefficients

TL;DR

This paper studies McKean stochastic differential equations with singular, distribution-dependent drift, establishing existence and uniqueness through a singular martingale problem and analysis of the associated Fokker-Planck equation.

Contribution

It introduces a novel approach to analyze McKean SDEs with singular coefficients using a singular martingale problem framework and Fokker-Planck equation analysis.

Findings

Proved existence of solutions for McKean SDEs with distributional drift.

Established uniqueness of solutions under certain conditions.

Linked the SDE analysis to the properties of the associated Fokker-Planck equation.

Abstract

The paper investigates existence and uniqueness for a stochastic differential equation (SDE) with distributional drift depending on the law density of the solution. Those equations are known as McKean SDEs. The McKean SDE is interpreted in the sense of a suitable singular martingale problem. A key tool used in the investigation is the study of the corresponding Fokker-Planck equation.