Killed distribution dependent SDE for nonlinear Dirichlet problem (Q6107301)

The distribution dependent stochastic differential equation (DDSDE) is a crucial probability model characterizing the nonlinear Fokker-Planck equation. The author investigates killed DDSDE to characterize nonlinear Dirichlet problems in an open domain. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations: 1) monotone case with distribution dependent noise (possibly degenerate); 2) singular case with non-degenerate distribution dependent noise; 3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is \(C^2\) smooth such that the Lipschitz continuity in initial distributions is also derived, and in the last case the domain is arbitrary.