Exponential Ergodicity in $\W_1$ for SDEs with Distribution Dependent Noise and Partially Dissipative Drifts

TL;DR

This paper proves exponential ergodicity in the Wasserstein-1 distance for a broad class of McKean--Vlasov SDEs with distribution-dependent noise and partially dissipative drifts, extending previous results.

Contribution

It establishes a general exponential ergodicity result for McKean--Vlasov SDEs with distribution-dependent coefficients and applies it to various noise types, including stable noise.

Findings

Proves exponential ergodicity in Wasserstein-1 distance for complex SDEs.

Extends ergodicity results to distribution-dependent coefficients and various noise types.

Improves upon existing results by relaxing assumptions on coefficients.

Abstract

Being concerned with ergodicity of McKean--Vlasov SDEs, we establish a general result on exponential ergodicity in the $L^1$-Wasserstein distance. The result is successfully applied to non-degenerate and multiplicative Brownian motion cases, degenerate second order systems, and even the additive $\alpha$-stable noise, where the coefficients before the noise are allowed to be distribution dependent and the drifts are only assumed to be partially dissipative. Our results considerably improve existing ones whose coefficients before the noise are distribution-free.