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

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