A discrete log-Sobolev inequality under a Bakry-Émery type condition

Oliver Johnson

doi:10.1214/16-AIHP778

A discrete log-Sobolev inequality under a Bakry-Émery type condition

Oliver Johnson

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

We consider probability mass functions V supported on the positive integers using
arguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–Émery condition for a Markov birth and death operator with invariant measure V . Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

Original language	English
Pages (from-to)	1952-1970
Number of pages	19
Journal	Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques
Volume	53
Issue number	4
Early online date	27 Nov 2017
DOIs	https://doi.org/10.1214/16-AIHP778
Publication status	Published - Nov 2017

Keywords

Bakry–Émery condition
Birth and death chain
Concentration of measure
Discrete probability measure
Log-concavity
Log-Sobolev inequality

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10.1214/16-AIHP778

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Cite this

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title = "A discrete log-Sobolev inequality under a Bakry-{\'E}mery type condition",

abstract = "We consider probability mass functions V supported on the positive integers usingarguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–{\'E}mery condition for a Markov birth and death operator with invariant measure V . Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.",

keywords = "Bakry–{\'E}mery condition, Birth and death chain, Concentration of measure, Discrete probability measure, Log-concavity, Log-Sobolev inequality",

author = "Oliver Johnson",

year = "2017",

month = nov,

doi = "10.1214/16-AIHP778",

language = "English",

volume = "53",

pages = "1952--1970",

journal = "Annales de l'Institut Henri Poincar{\'e} (B) Probabilit{\'e}s et Statistiques",

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number = "4",

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TY - JOUR

T1 - A discrete log-Sobolev inequality under a Bakry-Émery type condition

AU - Johnson, Oliver

PY - 2017/11

Y1 - 2017/11

N2 - We consider probability mass functions V supported on the positive integers usingarguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–Émery condition for a Markov birth and death operator with invariant measure V . Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

AB - We consider probability mass functions V supported on the positive integers usingarguments introduced by Caputo, Dai Pra and Posta, based on a Bakry–Émery condition for a Markov birth and death operator with invariant measure V . Under this condition, we prove a new modified logarithmic Sobolev inequality, generalizing and strengthening results of Wu, Bobkov and Ledoux, and Caputo, Dai Pra and Posta. We show how this inequality implies results including concentration of measure and hypercontractivity, and discuss how it may extend to higher dimensions.

KW - Bakry–Émery condition

KW - Birth and death chain

KW - Concentration of measure

KW - Discrete probability measure

KW - Log-concavity

KW - Log-Sobolev inequality

UR - http://arxiv.org/abs/1507.06268

U2 - 10.1214/16-AIHP778

DO - 10.1214/16-AIHP778

M3 - Article (Academic Journal)

VL - 53

SP - 1952

EP - 1970

JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -

A discrete log-Sobolev inequality under a Bakry-Émery type condition

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Professor Oliver T Johnson

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A discrete log-Sobolev inequality under a Bakry-Émery type condition

Abstract

Keywords

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Professor Oliver T Johnson

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