# Regularity of Schr\"odinger's functional equation in the weak topology and moment measures

## TL;DR

This paper investigates the continuity and measurability of solutions to Schrödinger's functional equation under weak topology, and applies these results to construct convex functions with prescribed moment measures via stochastic optimal transportation.

## Contribution

It extends previous work by analyzing the problem under weak topology and introduces a method to construct convex functions with specific moment measures using zero noise limits.

## Key findings

- Established continuity and measurability of solutions in weak topology.
- Constructed convex functions with given moment measures.
- Linked Schrödinger's equation solutions to stochastic optimal transportation.

## Abstract

We study the continuity and the measurability of the solution to Schr\"odinger's functional equation, with respect to space, kernel and marginals, provided the space of all Borel probability measures is endowed with the weak topology. This is a continuation of our previous result where the space of all Borel probability measures was endowed with the strong topology. As an application, we construct a convex function of which the moment measure is a given probability measure, by the zero noise limit of a class of stochastic optimal transportation problems.