Dimension dependent hypercontractivity for Gaussian kernels

• 成果类型: Article
• 署名作者: Bakry, Dominique; Bolley, Francois; Gentil, Ivan
• 署名单位: Universite PSL; Universite Paris-Dauphine; Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-011-0387-y
• 发表日期: 2012
• 页码: 845-874
• 关键词: logarithmic sobolev inequalities; transportation cost

摘要

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 L,vy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.