Convexity and regularity properties for entropic interpolations

TL;DR

This paper establishes convexity and regularity properties of entropic interpolations and costs, deriving inequalities and contraction properties that connect to Wasserstein space results, advancing understanding of entropy in probabilistic and geometric contexts.

Contribution

It proves convexity and regularity properties of entropic interpolations and costs, linking these to Wasserstein space results and providing new inequalities and contraction properties.

Findings

Convexity of relative entropy along entropic interpolations

Regularity of entropic cost along heat flow

Dimensional EVI inequality and contraction property

Abstract

In this paper we prove a convexity property of the relative entropy along entropic interpolations (solutions of the Schr\"odinger problem), and a regularity property of the entropic cost along the heat flow. Then we derive a dimensional EVI inequality and a contraction property for the entropic cost along the heat flow. As a consequence, we recover the equivalent results in the Wasserstein space, proved by Erbar, Kuwada and Sturm.