Killed Distribution Dependent SDE for Nonlinear Dirichlet Problem

TL;DR

This paper studies killed distribution dependent SDEs to characterize nonlinear Dirichlet problems, establishing well-posedness under various conditions using coupling, Zvonkin/Girsanov transforms, and domain assumptions.

Contribution

It proves well-posedness of killed distribution dependent SDEs for nonlinear Dirichlet problems in different scenarios, including monotone and singular cases with various noise types.

Findings

Well-posedness established for three cases of distribution dependent SDEs.

Lipschitz continuity in initial distributions derived for smooth domains.

Results applicable to arbitrary domains in certain cases.

Abstract

To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. 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, and 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.