Transition densities of reflecting diffusions

Abstract

Using a parametrix method, localization procedure and probabilistic arguments we construct the transition density function for a nondegenerate diffusion process in a smooth domain, with oblique reflection at the boundary and with "killing" terms in the interior and on the boundary; (our method obtains the density also on the boundary). In the case of half space, a Gaussian type upper bound and minimality are established for the transition density. We also prove some auxiliary results concerning Green function and Poisson kernel for a second order parabolic operator with mixed boundary conditions in certain domains.