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April 2022 A probabilistic approach to convex (ϕ)-entropy decay for Markov chains
Giovanni Conforti
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Ann. Appl. Probab. 32(2): 932-973 (April 2022). DOI: 10.1214/21-AAP1700

Abstract

We study the exponential dissipation of entropic functionals along the semigroup generated by a continuous time Markov chain and the associated convex Sobolev inequalities, including MLSI and Beckner inequalities. We propose a method that combines the Bakry–Émery approach and coupling arguments, which we use as a probabilistic alternative to the discrete Bochner identities. In particular, the validity of the method is not limited to the perturbative setting and we establish convex entropy decay for interacting random walks beyond the high temperature/weak interaction regime. In this framework, we show that the exponential contraction of the Wasserstein distance implies MLSI. We also revisit classical examples often obtaining new inequalities and sometimes improving on the best known constants. In particular, we analyse the zero range dynamics, hardcore and Bernoulli–Laplace models and the Glauber dynamics for the Curie–Weiss and Ising model.

Acknowledgments

The author wishes to thank Paolo Dai Pra and Matthias Erbar for providing insightful comments at an early stage of this work.

Citation

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Giovanni Conforti. "A probabilistic approach to convex (ϕ)-entropy decay for Markov chains." Ann. Appl. Probab. 32 (2) 932 - 973, April 2022. https://doi.org/10.1214/21-AAP1700

Information

Received: 1 May 2020; Revised: 1 April 2021; Published: April 2022
First available in Project Euclid: 28 April 2022

Digital Object Identifier: 10.1214/21-AAP1700

Subjects:
Primary: ‎39B62 , 60J80 , 60K35

Keywords: Beckner inequalities , convexity of the entropy , coupling , Glauber dynamics , hardcore models , Markov chains , Modified logarithmic Sobolev inequality

Rights: Copyright © 2022 Institute of Mathematical Statistics

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Vol.32 • No. 2 • April 2022
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