Dimension dependent hypercontractivity for Gaussian kernels
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2010
- DOI:
- 10.48550/arXiv.1003.5072
- arXiv:
- arXiv:1003.5072
- Bibcode:
- 2010arXiv1003.5072B
- Keywords:
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- Mathematics - Probability
- E-Print:
- 24 pages