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Weak logarithmic Sobolev inequalities and entropic convergence
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  • Published: 24 March 2007

Weak logarithmic Sobolev inequalities and entropic convergence

  • P. Cattiaux1,2,
  • I. Gentil3 &
  • A. Guillin4 

Probability Theory and Related Fields volume 139, pages 563–603 (2007)Cite this article

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  • 24 Citations

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Abstract

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.

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Author information

Authors and Affiliations

  1. Ecole Polytechnique, CMAP, 91128, Palaiseau Cedex, France

    P. Cattiaux

  2. Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMI, 200 avenue de la République, 92001, Nanterre Cedex, France

    P. Cattiaux

  3. Université Paris-Dauphine, CEREMADE, UMR CNRS 7534, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    I. Gentil

  4. Ecole Centrale Marseille et LATP UMR CNRS 6632, Centre de Mathematiques et Informatique Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453, Marseille Cedex 13, France

    A. Guillin

Authors
  1. P. Cattiaux
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  2. I. Gentil
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  3. A. Guillin
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Corresponding author

Correspondence to P. Cattiaux.

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Cattiaux, P., Gentil, I. & Guillin, A. Weak logarithmic Sobolev inequalities and entropic convergence. Probab. Theory Relat. Fields 139, 563–603 (2007). https://doi.org/10.1007/s00440-007-0054-5

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  • Received: 17 November 2005

  • Revised: 04 December 2006

  • Published: 24 March 2007

  • Issue Date: November 2007

  • DOI: https://doi.org/10.1007/s00440-007-0054-5

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Keywords

  • Logarithmic Sobolev inequalities
  • Concentration inequalities
  • Entropy

Mathematics Subject Classification (2000)

  • 26D10
  • 60E15
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