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August 2017 Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains
Ansgar Jüngel, Wen Yue
Ann. Appl. Probab. 27(4): 2238-2269 (August 2017). DOI: 10.1214/16-AAP1258

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.

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Ansgar Jüngel. Wen Yue. "Discrete Beckner inequalities via the Bochner–Bakry–Emery approach for Markov chains." Ann. Appl. Probab. 27 (4) 2238 - 2269, August 2017. https://doi.org/10.1214/16-AAP1258

Information

Received: 1 November 2015; Revised: 1 August 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 1374.60144
MathSciNet: MR3693525
Digital Object Identifier: 10.1214/16-AAP1258

Subjects:
Primary: ‎39B62 , 60J27 , 60J80

Keywords: discrete Beckner inequality , Entropy decay , functional inequality , Stochastic particle systems , Time-continuous Markov chain

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 4 • August 2017
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