Dimension dependent hypercontractivity for Gaussian kernels

Bakry, Dominique; Bolley, François; Gentil, Ivan

Bakry, Dominique; Bolley, François; Gentil, Ivan (2012), Dimension dependent hypercontractivity for Gaussian kernels, Probability Theory and Related Fields, 154, 3-4, p. 845-874. http://dx.doi.org/10.1007/s00440-011-0387-y

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00465879/fr/

Date

2012

Journal name

Probability Theory and Related Fields

Volume

154

Number

3-4

Publisher

Springer

Pages

845-874

Abstract (EN)

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein-Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton-Jacobi equation. Hypercontractive bounds on the Ornstein-Uhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

Subjects / Keywords

Curvature-dimension criterion; Transportation inequality; Hypercontractive bound; Diffusion semigroups; Logarithmic Sobolev inequality