McKean-Vlasov SDEs with Singular Coefficients and Distribution Dependent Noise: Well-posedness and Regularity

TL;DR

This paper establishes well-posedness and regularity results for McKean-Vlasov SDEs with singular coefficients, allowing for distribution-dependent noise and providing new estimates under various Wasserstein distances.

Contribution

It introduces novel well-posedness and regularity results for McKean-Vlasov SDEs with singular coefficients and distribution-dependent noise, extending previous work with improved estimates.

Findings

Proved well-posedness for SDEs with singular coefficients.

Established regularity estimates under Wasserstein distances.

Improved upon previous regularity results in the literature.

Abstract

The well-posedness for SDEs with singularity in both space and distribution variables is derived, where the interacting drift term is bounded and Lipschitz continuous under total variation distance and the diffusion term is allowed to be Lipschitz continuous under $L^\eta$($\eta\in(0,1]$)-Wasserstein distance in the distribution variable. When the diffusion term is Lipschitz continuous under $L^k$-Wasserstein distance for some $k\geq 1$, the regularity estimate $$\|P_t^\ast\gamma^1-P_t^\ast\gamma^2\|_{var}\leq ct^{-\frac{1}{2}}\W_{k}(\gamma^1,\gamma^2),\ \ t\in(0,T]$$ is established. This improves the results in \cite[Theorem 1.3]{HRWJDE}.