The Annals of Applied Probability

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

Ansgar Jüngel and Wen Yue

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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.

Article information

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2238-2269.

Received: November 2015
Revised: August 2016
First available in Project Euclid: 30 August 2017

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Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Time-continuous Markov chain functional inequality entropy decay discrete Beckner inequality stochastic particle systems


Jüngel, Ansgar; Yue, Wen. Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains. Ann. Appl. Probab. 27 (2017), no. 4, 2238--2269. doi:10.1214/16-AAP1258.

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