Existence and non-uniqueness of stationary distributions for distribution dependent SDEs

TL;DR

This paper investigates the existence and non-uniqueness of stationary distributions for distribution dependent SDEs, employing ergodicity, fixed point theorems, and transformations to handle singular coefficients, with concrete examples illustrating non-uniqueness.

Contribution

It provides new results on the existence of stationary distributions and demonstrates non-uniqueness in certain classes of distribution dependent SDEs, including those with singular coefficients.

Findings

Existence of stationary distributions established using ergodicity and Schauder fixed point theorem.

Non-uniqueness of stationary distributions demonstrated for regular coefficient equations.

Concrete examples include McKean-Vlasov equations with quadratic and non-quadratic interactions.

Abstract

The existence of stationary distributions to distribution dependent stochastic differential equations are investigated by using the ergodicity of the associated decoupled equation and the Schauder fixed point theorem. By using Zvonkin's transformation, we also establish the existence result for equations with singular coefficients. Instead of the uniqueness, the non-uniqueness of stationary distributions are considered for equations with regular coefficients. Concrete examples including McKean-Vlasov stochastic equations with the quadratic interaction and the non-quadratic interaction, and equations with a bounded and discontinuous drift are presented to illustrate our non-uniqueness results.