Probability Distance Estimates Between Diffusion Processes and Applications to Singular McKean-Vlasov SDEs

The $L^k$-Wasserstein distance ${\mathbb{W}}_k(k\ge 1)$ and the probability distance ${\mathbb{W}}_{\psi }$ induced by a concave function $\psi$, are estimated between different diffusion processes with singular coefficients. As applications, the well-posedness, probability distance estimates and the log-Harnack inequality are derived for McKean-Vlasov SDEs with multiplicative distribution dependent noise, where the coefficients are singular in time-space variables and $({\mathbb{W}}_k+{\mathbb{W}}_{\psi })$-Lipschitz continuous in the distribution variable. This improves existing results derived in the literature under the ${\mathbb{W}}_k$-Lipschitz or derivative conditions in the distribution variable.