Convexity and regularity properties for entropic interpolations

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.