Dimension dependent hypercontractivity for Gaussian kernels
Probability
2013-09-19 v1
Abstract
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\'evy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.
Cite
@article{arxiv.1003.5072,
title = {Dimension dependent hypercontractivity for Gaussian kernels},
author = {Dominique Bakry and François Bolley and Ivan Gentil},
journal= {arXiv preprint arXiv:1003.5072},
year = {2013}
}
Comments
24 pages