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

TL;DR

This work advances exponential ergodicity results for McKean–Vlasov SDEs with distribution-dependent diffusion coefficients and partially dissipative drifts by establishing a general contraction criterion in the $\mathbb{W}_1$ metric. The authors prove a two-part framework: (i) a decoupled SDE with fixed law $\mu$ possesses a uniformly contractive ergodic behavior, and (ii) a bound linking different laws via a function $G(t)$ ensures global contraction, yielding a unique invariant measure $\mu^*$ and explicit exponential convergence rates. They then demonstrate broad applicability to Brownian and $\alpha$-stable noise scenarios, including non-degenerate, degenerate/kinetic, and second-order Langevin-type systems, under small coupling $\kappa$ and suitable dissipativity conditions. An appendix introduces a time-change technique to address challenges arising from Lévy noise in the coupling analysis, broadening the impact to distribution-dependent diffusion with jump noise.

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 $α$-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.

Theorems & Definitions (20)

• Theorem 2.1
• proof
• Theorem 3.1
• Remark 3.2
• Example 3.3
• Corollary 3.4
• proof
• Lemma 3.5
• proof : Proof of Lemma \ref{['lemma of sde']}
• proof : Proof of Theorem \ref{['main thm']}