Entropy and Fisher information in non-convex domains: one chain to rule them all

TL;DR

This paper demonstrates that Fisher information acts as a strong Wasserstein upper gradient of entropy on non-convex Riemannian domains, bypassing the need for displacement convexity and providing new control and chain rule results.

Contribution

It establishes Fisher information as a strong Wasserstein upper gradient on non-convex domains and introduces novel short-time control and chain rule results for measure curves.

Findings

Fisher information is a strong Wasserstein upper gradient of entropy in non-convex domains.

New quantitative short-time control of Fisher information along Neumann heat flow.

Exact chain rule established under $AC_2$ assumptions for measure curves.

Abstract

We prove that the (square root) Fisher information functional is a strong Wasserstein upper gradient of the entropy on non-convex Riemannian domains. This fills a gap in the literature by allowing one to completely dispense from $\lambda$-displacement convexity arguments. Along the way we establish a novel quantitative short-time control of the Fisher information along the Neumann heat flow, and establish an exact chain rule under stronger $AC_2$ assumptions typically satisfied by curves of measures obtained as limits of JKO schemes.