Exponential convergence in Wasserstein metric for distribution dependent SDEs

TL;DR

This paper proves exponential convergence in Wasserstein distance for distribution dependent SDEs, providing explicit rates and applicable to models like Curie-Weiss and granular media without uniform dissipativity.

Contribution

It introduces a twinned Talagrand inequality and establishes exponential convergence for a broad class of distribution dependent SDEs, including non-uniform dissipative cases.

Findings

Exponential convergence in Wasserstein distance is established.

Explicit convergence rates are derived.

Applicable to models like Curie-Weiss and granular media.

Abstract

The existence and uniqueness of stationary distributions and the exponential convergence in $L^p$-Wasserstein distance are derived for distribution dependent SDEs from associated decoupled equations. To establish the exponential convergence, we introduce a twinned Talagrand inequality of the original SDE and the associated decoupled equation, and explicit convergence rate is obtained. Our results can be applied to SDEs without uniformly dissipative drift and distribution dependent diffusion term, which cover the Curie-Weiss model and the granular media model in double-well landscape with quadratic interaction as examples.