Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains

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 Poincaré 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.