Regularity of Schrödinger's functional equation and mean field PDEs for h-path processes

Abstract

We show that the solution of Schrödinger's functional equation is measurable in space, kernel and marginals. As an application, we show that the drift vector of the h-path process with given two end point marginals is a measurable function of space, time and marginal at each time. In particular, we show that the coefficients of mean field PDE systems which the marginals satisfy are measurable functions of space, time and marginal.