Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity of Schroedinger's functional equation and mean field PDEs for
  h-path processes   E. Schroedinger proposed the equation to find the statistical property of a
quantum particle on a finite time interval. It is called "Schroedinger's
functional equation". Given probability distributions of a particle at initial
and terminal times, it determines the joint distribution of a quantum particle
at initial and terminal times so that a particle is Markovian. S. Bernstein
generalized Schroedinger's idea and introduced the so-called Bernstein
processes which are also called reciprocal processes or one-dimensional Markov
random fields. The theory of stochastic differential equation for
Schroedinger's functional equation was given by B. Jamison. The solution is
Doob's h-path process with given two end point marginals. We show that the
solution of Schroedinger'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 marginals
satisfy a class of mean field PDE systems of which the coefficients are
measurable function of space, time and marginal. We also show that
Schroedinger's functional equation is the Euler equation of a stochastic
optimal transportation problem.

15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex