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 Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.
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Dominique Bakry is member of the Institut Universitaire de France.
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Bakry, D., Bolley, F. & Gentil, I. Dimension dependent hypercontractivity for Gaussian kernels. Probab. Theory Relat. Fields 154, 845–874 (2012). https://doi.org/10.1007/s00440-011-0387-y
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DOI: https://doi.org/10.1007/s00440-011-0387-y
Keywords
- Hypercontractive bound
- Diffusion semigroup
- Logarithmic Sobolev inequality
- Curvature-dimension criterion
- Transportation inequality