Dimension dependent hypercontractivity for Gaussian kernels

Dominique Bakry (1, 2) , François Bolley (3) , Ivan Gentil (4)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IUF - Institut universitaire de France
3 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
4 ICJ - Institut Camille Jordan

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.

Mots clés: Transportation inequality; Hypercontractive bound; Diffusion semigroup; Logarithmic Sobolev inequality; Curvature-dimension criterion; Transportation inequality.

Domaines: Probabilités [math.PR]

Dominique Bakry, François Bolley, Ivan Gentil. Dimension dependent hypercontractivity for Gaussian kernels. Probability Theory and Related Fields, 2012, 154 (3-4), pp.845-874. ⟨10.1007/s00440-011-0387-y⟩. ⟨hal-00465879⟩