Local convergence near equilibria for distribution dependent SDEs

TL;DR

This paper studies the local exponential convergence near stationary distributions for distribution dependent SDEs, providing criteria for stability and illustrating with concrete examples like the granular media equation.

Contribution

It introduces a linearization approach of the nonlinear Markov semigroup to establish local convergence criteria for distribution dependent SDEs.

Findings

Established criteria for local exponential stability of stationary distributions.

Connected convergence of nonlinear semigroups to linear operator semigroups.

Provided concrete examples demonstrating the applicability of the results.

Abstract

Owing to exhibiting phase transitions, we investigate the local convergence near a stationary distribution for distribution dependent stochastic differential equations. By linearizing the nonlinear Markov semigroup associated with the distribution dependent equation around the stationary distribution, the local exponential convergence of the solution is related to the exponential convergence of a semigroup of linear operators. Our result can be used as a criteria for the locally exponential stability of stationary distributions. Concrete examples, including the granular media equation with double-wells landscapes and quadratic interaction, are given to illustrate our main result.