A discrete log-Sobolev inequality under a Bakry-Emery type condition

TL;DR

This paper establishes a modified log-Sobolev inequality for probability measures on positive integers under a Bakry-Emery condition, leading to implications for measure concentration and hypercontractivity.

Contribution

It generalizes and strengthens existing log-Sobolev inequalities for discrete distributions using a Bakry-Emery framework, with potential extensions to higher dimensions.

Findings

Proves a new modified logarithmic Sobolev inequality for discrete measures.

Demonstrates implications for concentration of measure and hypercontractivity.

Discusses potential extensions to multidimensional settings.

Abstract

We consider probability mass functions $V$ supported on the positive integers using arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry--\'{E}mery condition for a Markov birth and death operator with invariant measure $V$. Under this condition, we prove a modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.