Regularity of the Schr{\"o}dinger cost

TL;DR

This paper investigates the regularity properties of the Schr{"o}dinger cost, focusing on its continuity with respect to probability measure marginals and its time derivative along measure-valued curves.

Contribution

It provides new insights into the regularity of the Schr{"o}dinger cost, including continuity and differentiability properties, which were previously not well-understood.

Findings

Established continuity of the Schr{"o}dinger cost with respect to marginals

Derived the time derivative of the Schr{"o}dinger cost along measure curves

Enhanced understanding of the regularity properties of entropy-based costs

Abstract

The Schr{\"o}dinger problem is an entropy minimisation problem on the space of probability measures. Its optimal value is a cost between two probability measures. In this article we investigate some regularity properties of this cost: continuity with respect to the marginals and time derivative of the cost along probability measures valued curves.