Discrete Bochner inequalities via the Bochner-Bakry-Emery approach for Markov chains

TL;DR

This paper develops discrete Bochner inequalities using the Bakry-Emery approach to derive convex Sobolev and Beckner inequalities for various Markov chains, connecting these inequalities to entropy and functional analysis.

Contribution

It extends the Bakry-Emery method to discrete Markov chains, establishing new inequalities and applying them to diverse models and discretizations.

Findings

Derived convex Sobolev and Beckner inequalities for Markov chains.

Unified framework connecting inequalities with entropy methods.

Applied results to multiple Markov chain models and discretizations.

Abstract

Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincar\'e inequality. Their proof is based on the Bakry-Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace models, and random transposition models, and to a finite-volume discretization of a one-dimensional Fokker-Planck equation, applying results by Mielke.