Moment estimates derived from Poincaré and logarithmic Sobolev inequalities
type: Item
instance of: scholarly article
title: Moment estimates derived from Poincaré and logarithmic Sobolev inequalities
zbMATH Open document ID: 0862.60064
author: Shigeki Aida
author: Daniel W. Stroock
published in: Mathematical Research Letters
publication date: 1997-05-25T00:00:00Z
review text: Assume that the space on which the probability measure \(\mu\) is defined possesses a natural gradient operation \(\nabla\). Then a test for mixing properties of the measure \(\mu\) is to examine what (if any) a priori estimates \(\mu\) satisfy relative to \(\nabla\). For example, such estimates are expressed by the Poincaré inequality \[ |f-\mathbb{E}^\mu [f] |^2_{L^2(\mu)}\leq P|\nabla f|^2_{L^2(\mu)} \] or by the logarithmic Sobolev inequality \[ \int f^2\log{f^2\over |f|^2_{L^2(\mu)}} d\mu\leq S|\nabla f|^2_{L^2(\mu)}. \] The aim of the paper is to show that each of these a priori estimates allows to obtain bounds of \(f\) in terms of bounds on \(|\nabla f|\).
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 794214
zbMATH Keywords: probability space
zbMATH Keywords: Markov's transition probability function
zbMATH Keywords: a priori estimates
zbMATH Keywords: moment estimates
reviewed by: Bohumír Opic
MaRDI profile type: Publication
Wikidata QID: Q98839630
DOI: 10.4310/MRL.1994.V1.N1.A9
description: scientific article; zbMATH DE number 794214