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Erschienen in:
Markov Semigroups Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

5. Logarithmic Sobolev Inequalities

verfasst von : Dominique Bakry, Ivan Gentil, Michel Ledoux

Erschienen in: Analysis and Geometry of Markov Diffusion Operators

Verlag: Springer International Publishing

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Abstract

After Poincaré inequalities, logarithmic Sobolev inequalities are amongst the most studied functional inequalities for semigroups. They contain much more information than Poincaré inequalities, and are at the same time sufficiently general to be available in numerous cases of interest, in particular in infinite dimension (as limits of Sobolev inequalities on finite-dimensional spaces). After the basic definition of a logarithmic Sobolev inequality together with its first properties, the first sections of this chapter present the exponential decay in entropy and the fundamental equivalence between the logarithmic Sobolev inequality and smoothing properties of the semigroup in the form of hypercontractivity. Next, integrability properties of eigenvectors and of Lipschitz functions under a logarithmic Sobolev inequality are discussed together with a criterion for measures on the real line to satisfy a logarithmic Sobolev inequality (for the usual gradient). The further sections deal with curvature conditions, first for the local logarithmic Sobolev inequalities for heat kernel measures, then for the invariant measure with an additional dimensional information. Local hypercontractivity and some applications of the local logarithmic Sobolev inequalities towards heat kernel bounds are further presented. Harnack-type inequalities under the infinite-dimensional curvature conditions, linked with reverse local logarithmic Sobolev inequalities complete the chapter.

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Vorheriges Kapitel Poincaré Inequalities
Nächstes Kapitel Sobolev Inequalities
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Metadaten
Titel
Logarithmic Sobolev Inequalities
verfasst von
Dominique Bakry
Ivan Gentil
Michel Ledoux
Copyright-Jahr
2014
Buch
Analysis and Geometry of Markov Diffusion Operators

Print ISBN: 978-3-319-00226-2

Electronic ISBN: 978-3-319-00227-9

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-00227-9_5