Distribution Dependent Reflecting Stochastic Differential Equations

To characterize the Neumann problem for nonlinear Fokker-Planck equations, we investigate distribution dependent reflecting SDEs (DDRSDEs) in a domain. We first prove the well-posedness and establish functional inequalities for reflecting SDEs with singular drifts, then extend these results to DDRSDEs with singular or monotone coefficients, for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting SDEs is established. Moreover, three different types of exponential ergodicity are derived for DDRSDEs under dissipative, partially dissipative, and fully non-dissipative conditions respectively.