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2004 Entropies, convexity, and functional inequalities, On $\Phi $-entropies and $\Phi $-Sobolev inequalities
Djalil Chafaï
J. Math. Kyoto Univ. 44(2): 325-363 (2004). DOI: 10.1215/kjm/1250283556


Our aim is to provide a short and self contained synthesis which generalise and unify various related and unrelated works involving what we call $\Phi$-Sobolev functional inequalities. Such inequalities related to $\Phi$-entropies can be seen in particular as an inclusive interpolation between Poincaré and Gross logarithmic Sobolev inequalities. In addition to the known material, extensions are provided and improvements are given for some aspects. Stability by tensor products, convolution, and bounded perturbations are addressed. We show that under simple convexity assumptions on $\Phi$, such inequalities hold in a lot of situations, including hyper-contractive diffusions, uniformly strictly log-concave measures, Wiener measure (paths space of Brownian Motion on Riemannian Manifolds) and generic Poisson space (includes paths space of some pure jumps Lévy processes and related infinitely divisible laws). Proofs are simple and relies essentially on convexity. We end up by a short parallel inspired by the analogy with Boltzmann-Shannon entropy appearing in Kinetic Gases and Information Theories.


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Djalil Chafaï. "Entropies, convexity, and functional inequalities, On $\Phi $-entropies and $\Phi $-Sobolev inequalities." J. Math. Kyoto Univ. 44 (2) 325 - 363, 2004.


Published: 2004
First available in Project Euclid: 14 August 2009

zbMATH: 1079.26009
MathSciNet: MR2081075
Digital Object Identifier: 10.1215/kjm/1250283556

Primary: 60J35
Secondary: 46E35, 47D07, 60J60, 94A15, 94A17

Rights: Copyright © 2004 Kyoto University


Vol.44 • No. 2 • 2004
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