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Dimension dependent hypercontractivity for Gaussian kernels
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  • Published: 16 September 2011

Dimension dependent hypercontractivity for Gaussian kernels

  • Dominique Bakry1,
  • François Bolley2 &
  • Ivan Gentil3 

Probability Theory and Related Fields volume 154, pages 845–874 (2012)Cite this article

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  • 16 Citations

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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 Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, Université Paul-Sabatier, Université de Toulouse, 31062, Toulouse, France

    Dominique Bakry

  2. Ceremade, Université Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75116, Paris, France

    François Bolley

  3. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne cedex, France

    Ivan Gentil

Authors
  1. Dominique Bakry
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  2. François Bolley
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  3. Ivan Gentil
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Corresponding author

Correspondence to François Bolley.

Additional information

Dominique Bakry is member of the Institut Universitaire de France.

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Cite this article

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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  • Received: 26 May 2010

  • Revised: 13 July 2011

  • Published: 16 September 2011

  • Issue Date: December 2012

  • DOI: https://doi.org/10.1007/s00440-011-0387-y

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Keywords

  • Hypercontractive bound
  • Diffusion semigroup
  • Logarithmic Sobolev inequality
  • Curvature-dimension criterion
  • Transportation inequality

Mathematics Subject Classification (2000)

  • 60J
  • 58J
  • 60H
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