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E. Schroedinger proposed the equation to find the statistical property of aquantum particle on a finite time interval. It is called "Schroedinger'sfunctional equation". Given probability distributions of a particle at initialand terminal times, it determines the joint distribution of a quantum particleat initial and terminal times so that a particle is Markovian. S. Bernsteingeneralized Schroedinger's idea and introduced the so-called Bernsteinprocesses which are also called reciprocal processes or one-dimensional Markovrandom fields. The theory of stochastic differential equation forSchroedinger's functional equation was given by B. Jamison. The solution isDoob's h-path process with given two end point marginals. We show that thesolution of Schroedinger's functional equation is measurable in space, kerneland marginals. As an application, we show that the drift vector of the h-pathprocess with given two end point marginals is a measurable function of space,time and marginal at each time. In particular, we show that the marginalssatisfy a class of mean field PDE systems of which the coefficients aremeasurable function of space, time and marginal. We also show thatSchroedinger's functional equation is the Euler equation of a stochasticoptimal transportation problem.

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