Log-Harnack Inequality and Bismut Formula for McKean-Vlasov SDEs with Singularities in all Variables

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.