Coupling Methods and Applications on Path Dependent McKean-Vlasov SDEs

TL;DR

This paper establishes new coupling methods for path dependent McKean-Vlasov SDEs, proving inequalities and contractivity properties, and demonstrates uniform propagation of chaos for related particle systems.

Contribution

It introduces novel coupling techniques for path dependent McKean-Vlasov SDEs and derives new exponential contractivity and propagation of chaos results.

Findings

Log-Harnack inequality established for these SDEs

Exponential contractivity in entropy-cost under dissipative conditions

Uniform propagation of chaos in Wasserstein distance for particle systems

Abstract

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.