Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables
Xing Huang, Feng-Yu Wang
TL;DR
This work develops a regularity theory for McKean-Vlasov SDEs with singularities across time, space and distribution by introducing a square-root Dini modulus $\alpha\in\mathscr A$ to quantify weak distributional continuity. It proves a dimension-free log-Harnack inequality and an entropy-cost bound under a drift decomposition and $\mathbb W_\alpha$-Lipschitz control, extending results beyond Lipschitz-in-distribution cases. The paper also establishes a Bismut-type formula for the intrinsic derivative of the MK-SDE semigroup, using a novel VP equation and fixed-point analysis to handle intrinsic/extrinsic derivatives in the distribution variable. Together, these results broaden the regime of singular coefficients for which quantitative regularity (in distribution) of the law flow can be analyzed and applied, improving the understanding of MK-SDEs with highly irregular drifts.
Abstract
The log-Harnack inequality and Bismut formula are established for McKean-Vlasov SDEs with singularities in all (time, space, distribution) variables, where the drift satisfies an integrability condition in time-space, and the continuity in distribution may be weaker than Dini. The main results considerably improve the existing ones for the case where the drift is $L$-differentiable and Lipschitz continuous in distribution with respect to the 2-Wasserstein distance.