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Modified logarithmic Sobolev inequalities and transportation inequalities

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
External document link
http://hal.archives-ouvertes.fr/hal-00001609/en/
Date
2005
Journal name
Probability Theory and Related Fields
Volume
133
Number
3
Pages
409-436
Publication identifier
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Author(s)
Gentil, Ivan
Guillin, Arnaud
Miclo, Laurent cc
Abstract (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.
Subjects / Keywords
Logarithmic Sobolev inequality; Poincaré inequality; Hardy inequality