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2021 Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains
Frederic Weber
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Electron. J. Probab. 26: 1-31 (2021). DOI: 10.1214/21-EJP627

Abstract

In the setting of reversible continuous-time Markov chains, the CDΥ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the CDΥ(κ,F) condition, where the dimension term is expressed by a so called CD-function F. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

Funding Statement

The author is supported by a PhD-scholarship of the “Studienstiftung des deutschen Volkes”, Germany.

Citation

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Frederic Weber. "Entropy-information inequalities under curvature-dimension conditions for continuous-time Markov chains." Electron. J. Probab. 26 1 - 31, 2021. https://doi.org/10.1214/21-EJP627

Information

Received: 3 November 2020; Accepted: 3 April 2021; Published: 2021
First available in Project Euclid: 23 April 2021

Digital Object Identifier: 10.1214/21-EJP627

Subjects:
Primary: 60J27
Secondary: 39A12, 47D07

JOURNAL ARTICLE
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Vol.26 • 2021
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