A system of Schr\"odinger's problems and functional equations

TL;DR

This paper introduces a generalized framework of Schr"odinger's problems and functional equations in probability theory, focusing on variational problems of relative entropies with fixed endpoint marginals, and establishes existence and uniqueness results.

Contribution

It extends Schr"odinger's problem and functional equations to a broader class with inductive definitions, providing new insights into stochastic optimal transport.

Findings

Proved existence and uniqueness of solutions to the generalized functional equations.

Established a variational approach for a stochastic optimal transport analog.

Connected the framework to the Knothe--Rosenblatt rearrangement.

Abstract

We propose and study a system of Schr\"odinger's problems and functional equations in probability theory. More precisely, we consider a system of variational problems of relative entropies for probability measures on a Euclidean space with given two endpoint marginals, which can be defined inductively. We also consider an inductively defined system of functional equations, which are Euler's equations for our variational problems. These are generalizations of Schr\"odinger's problem and functional equation. % in probability theory. We prove the existence and uniqueness of solutions to our functional equations, % up to a multiplicative function, from which we show the existence and uniqueness of a minimizer of our variational problem. Our problem gives an approach for a stochastic optimal transport analog of the Knothe--Rosenblatt rearrangement via a variational problem point of view.