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arXiv · 1003.5072

Dimension dependent hypercontractivity for Gaussian kernels

Dominique Bakry·François Bolley·Ivan Gentil
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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.

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Dominique Bakry, François Bolley, Ivan Gentil. 2010-03-26. Dimension dependent hypercontractivity for Gaussian kernels. https://doi.org/10.1007/s00440-011-0387-y

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