Coupling Methods and Applications on Path Dependent McKean-Vlasov SDEs

By using coupling by change of conditional probability measure, the log-Harnack inequality for path dependent McKean-Vlasov SDEs with distribution dependent diffusion coefficients is established, which together with the exponential contractivity in $L^2$-Wasserstein distance yields the exponential contractivity in entropy-cost under the uniformly dissipative condition. When the coefficients are only partially dissipative, the exponential contractivity in $L^1$-Wasserstein distance is also derived in the aid of asymptotic reflection coupling, which is new even in the distribution independent case. In addition, the uniform in time propagation of chaos in $L^1$- Wasserstein distance is also obtained for path dependent mean field interacting particle system.