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December 2019 Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities
André Schlichting, Martin Slowik
Ann. Appl. Probab. 29(6): 3438-3488 (December 2019). DOI: 10.1214/19-AAP1484

Abstract

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz’ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie–Weiss model, where metastability and the additional regularity assumptions are verifiable.

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André Schlichting. Martin Slowik. "Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities." Ann. Appl. Probab. 29 (6) 3438 - 3488, December 2019. https://doi.org/10.1214/19-AAP1484

Information

Received: 1 May 2017; Revised: 1 January 2019; Published: December 2019
First available in Project Euclid: 7 January 2020

zbMATH: 07172339
MathSciNet: MR4047985
Digital Object Identifier: 10.1214/19-AAP1484

Subjects:
Primary: 60J10
Secondary: 34L15, 49J40, 60J45, 82C26

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 6 • December 2019
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