Journal of Mathematics of Kyoto University

Entropies, convexity, and functional inequalities, On $\Phi $-entropies and $\Phi $-Sobolev inequalities

Djalil Chafaï

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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.

Article information

J. Math. Kyoto Univ., Volume 44, Number 2 (2004), 325-363.

First available in Project Euclid: 14 August 2009

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Zentralblatt MATH identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60J60: Diffusion processes [See also 58J65] 94A15: Information theory, general [See also 62B10, 81P94] 94A17: Measures of information, entropy


Chafaï, Djalil. Entropies, convexity, and functional inequalities, On $\Phi $-entropies and $\Phi $-Sobolev inequalities. J. Math. Kyoto Univ. 44 (2004), no. 2, 325--363. doi:10.1215/kjm/1250283556.

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