A large deviation perspective on exponential decay of entropy and lower bounds on the Ricci-curvature

TL;DR

This paper introduces a new perspective on entropy decay and Ricci-curvature bounds using large deviation principles, connecting Markov process limits with geometric and functional inequalities.

Contribution

It defines the entropy-information inequality (EII) and entropy-convexity inequality (ECI), generalizing existing inequalities and linking them to large deviation rate functions.

Findings

EII is equivalent to exponential decay of the rate function.

ECI serves as an analogue of Ricci-curvature lower bounds.

Provides a unified framework connecting large deviations, entropy, and curvature.

Abstract

We offer a new point of view on the (Modified) Log-Sobolev inequality and lower bounds on the Ricci-curvature in the setting where the dynamics are obtained as the limit of Markov processes. In this setting, the large deviation rate function of the stationary measures of the Markov processes, plays the role of entropy. We define an entropy-information inequality (EII) that generalizes the (MLSI) and is equivalent to exponential decay of the rate function along the flow, and define an entropy-convexity inequality (ECI) that serves as an analogue of a lower bound on the Ricci-curvature in this setting.