Log-Sobolev Inequalities and Exponential Ergodicity for Non-degenerate and Degenerate McKean-Vlasov SDEs

TL;DR

This paper establishes new log-Sobolev inequalities and combines them with other inequalities to prove exponential ergodicity in $L^2$-Wasserstein distance and relative entropy for both non-degenerate and degenerate McKean-Vlasov SDEs.

Contribution

It introduces uniform log-Sobolev inequalities for invariant measures and extends exponential ergodicity results to degenerate cases using advanced inequalities.

Findings

Proved uniform log-Sobolev inequalities for invariant measures.

Derived exponential ergodicity in $L^2$-Wasserstein distance.

Extended results to degenerate diffusion cases.

Abstract

The exponential ergodicity of partially dissipative McKean-Vlasov SDEs in the \(L^1\)-Wasserstein distance has been extensively studied using asymptotic reflection coupling. However, the reflection coupling method is not applicable for the exponential ergodicity in $L^2$-Wasserstein distance and relative entropy. In this paper, we first establish uniform log-Sobolev inequalities (in the frozen measure variable with bounded second moments) for the invariant probability measure of the corresponding SDEs with frozen distribution. Second, for the McKean-Vlasov SDEs, we combine the log-Harnack inequality and Talagrand's inequality to derive exponential ergodicity in both $L^2$-Wasserstein distance and relative entropy. Furthermore, we extend these main results to the case of degenerate diffusion.