Existence for low-regularity McKean-Vlasov dynamics via emergence of regularity

TL;DR

This paper proves the existence of solutions for low-regularity McKean-Vlasov equations with common noise, using a two-step approximation and a novel regularity emergence property to handle discontinuous coefficients.

Contribution

It introduces a new approach that leverages the emergence of regularity to solve McKean-Vlasov problems with discontinuous coefficients, especially in the presence of singular interactions.

Findings

Established existence of solutions for low-regularity coefficients.

Developed a two-step approximation method involving particle systems.

Proved regularity emergence leads to strong convergence and solution existence.

Abstract

We establish the existence of solutions to common noise McKean-Vlasov martingale problems for coefficients with low regularity. Our approach is able to handle the key challenge posed by drift coefficients that are discontinuous with respect to the narrow convergence of measures. This case arises for e.g. singular interactions. Our proof strategy proceeds via a two-step approximation using smoothed McKean-Vlasov $n$-particle systems: We first pass to the large system limit by taking $n\to \infty$, and subsequently remove the smoothing. A novel aspect of our work is the use of a crucial emergence of regularity property. It ensures that after the first limit, we obtain a process of measures that are absolutely continuous with respect to the Lebesgue measure and provides quantitative integrability bounds on their densities. We use this regularity to establish a tightness result in a stronger topology than is typically considered. In this way we obtain a sufficiently strong mode of convergence that lets us subsequently remove the smoothing and solve the McKean-Vlasov martingale problem via the particle system approximations.