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

Shao-Qin Zhang

Published 2021-05-10Version 1

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. The non-uniqueness of stationary distribution are considered instead of the uniqueness for equations with regular coefficients. Concrete examples including stochastic equations with a bounded and discontinuous drift are presented to illustrate our non-uniqueness result.