Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains

TL;DR

This paper extends curvature-dimension conditions for reversible continuous-time Markov chains to include a finite dimension term and positive curvature, deriving new entropy-information inequalities and related functional bounds.

Contribution

It introduces the $CD_\Upsilon(\kappa,F)$ condition combining finite dimension and positive curvature, and derives new entropy-information inequalities and applications.

Findings

Derived entropy-information inequalities relating entropy and Fisher information.

Established ultracontractivity bounds and exponential integrability results.

Provided finite diameter bounds and a modified Nash inequality.

Abstract

In the setting of reversible continuous-time Markov chains, the $CD_\Upsilon$ condition has been shown recently to be a consistent analogue to the Bakry-\'Emery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the $CD_\Upsilon(\kappa,F)$ condition, where the dimension term is expressed by a so called $CD$-function $F$. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.