Well-posedness of McKean-Vlasov SDEs with density-dependent drift

TL;DR

This work studies the well-posedness of density-dependent McKean–Vlasov SDEs with drift depending pointwise on the marginal density and satisfying a local time–space integrability condition. The authors formulate a mollifier-based approximation scheme and prove strong existence, along with weak and strong uniqueness when $p=1$, the drift is bounded, and the diffusion is distribution-free. They establish a robust analytical framework combining optimal transport, heat-kernel estimates, and Krylov–Khasminskii-type bounds to control densities and marginal laws. The results yield a well-posedness theory for density-dependent SDEs under relatively flexible integrability and continuity assumptions, with explicit density bounds and Hölder-type continuity for the marginal laws, contributing to the mathematical understanding of mean-field-type dynamics and their associated Fokker–Planck equations.

Abstract

In this paper, we study well-posedness of McKean-Vlasov stochastic differential equations (SDE) whose drift depends pointwisely on marginal density and satisfies a local integrability condition in time-space variables. The drift and noise coefficients are assumed to be Lipschitz continuous in distribution variable with respect to Wasserstein metric $W_p$. Our approach is by approximation with mollifiers. We prove strong existence of a solution. Weak and strong uniqueness are obtained when $p=1$, the drift coefficient is bounded, and the diffusion coefficient is distribution free.

Theorems & Definitions (26)

• definition 1
• theorem 1: Existence
• theorem 2: Uniqueness
• lemma 1
• lemma 2
• remark 1
• lemma 3: see e.g. xia2020lq
• theorem 3
• proof
• lemma 4