Exponential Ergodicity for Time-Periodic McKean-Vlasov SDEs

TL;DR

This paper establishes exponential ergodicity for time-periodic McKean-Vlasov SDEs across various dissipativity scenarios, extending known results from time homogeneous cases and applying to inhomogeneous granular media and reflecting SDEs.

Contribution

It extends exponential ergodicity results to time-periodic McKean-Vlasov SDEs in multiple dissipativity contexts, including non-dissipative cases, with new techniques and applications.

Findings

Proved exponential ergodicity in Wasserstein and entropy for dissipative cases.

Established ergodicity in cost-induced Wasserstein distances for partially dissipative cases.

Extended results to reflecting McKean-Vlasov SDEs in convex domains.

Abstract

As extensions to the corresponding results derived for time homogeneous McKean- Vlasov SDEs, the exponential ergodicity is proved for time-periodic distribution dependent SDEs in three different situations: 1) in the quadratic Wasserstein distance and relative entropy for the dissipative case; 2) in the Wasserstein distance induced by a cost function for the partially dissipative case; and 3) in the weighted Wasserstein distance induced by a cost function and a Lyapunov function for the fully non-dissipative case. The main results are illustrated by time inhomogeneous granular media equations, and are extended to reflecting McKean-Vlasov SDEs in a convex domain.