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

Ansgar Jüngel, Wen Yue

Published 2015-11-19Version 1

Discrete Beckner inequalities, which interpolate between the modified logarithmic Sobolev inequality and the Poincar\'e inequality, are derived for time-continuous Markov chains on countable state spaces. The 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. The abstract result is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace models, and random transportation models, and to a finite-volume discretization of a one-dimensional Fokker-Planck equation, applying results by Mielke.