Modified logarithmic Sobolev inequalities and transportation inequalities

Gentil, Ivan; Guillin, Arnaud; Miclo, Laurent

Gentil, Ivan; Guillin, Arnaud; Miclo, Laurent (2005), Modified logarithmic Sobolev inequalities and transportation inequalities, Probability Theory and Related Fields, 133, 3, p. 409-436. http://dx.doi.org/10.1007/s00440-005-0432-9

Type

Article accepté pour publication ou publié

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http://hal.archives-ouvertes.fr/hal-00001609/en/

Date

2005

Nom de la revue

Probability Theory and Related Fields

Volume

133

Numéro

Pages

409-436

Résumé (EN)

We present a new class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type $\exp(-|x|^\al)$ or $\exp(-|x|^\al\log^\beta(2+|x|))$ ($\al\in]1,2[$ and $\be\in\dR$) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.

Mots-clés

Logarithmic Sobolev inequality; Poincaré inequality; Hardy inequality