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A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality
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  • Published: 16 June 2009

A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality

  • Patrick Cattiaux1,
  • Arnaud Guillin2 &
  • Li-Ming Wu2 

Probability Theory and Related Fields volume 148, pages 285–304 (2010)Cite this article

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  • 36 Citations

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Abstract

We give by simple arguments sufficient conditions, so called Lyapunov conditions, for Talagrand’s transportation information inequality and for the logarithmic Sobolev inequality. Those sufficient conditions work even in the case where the Bakry–Emery curvature is not lower bounded. Several new examples are provided.

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References

  1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10. Société Mathématique de France, Paris (2000)

  2. Arnaudon M., Thalmaier A., Wang F.Y.: Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130, 223–233 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités, number 1123 in Lecture Notes in Mathematics. Springer (1985)

  4. Bakry D., Barthe F., Cattiaux P., Guillin A.: A simple proof of the Poincaré inequality for a large class of measures including the logconcave case. Electron. Commun. Probab. 13, 60–66 (2008)

    MATH  MathSciNet  Google Scholar 

  5. Bakry D., Cattiaux P., Guillin A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254, 727–759 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barthe F., Kolesnikov A.V.: Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18(4), 921–979 (2009)

    Article  MathSciNet  Google Scholar 

  7. Bobkov S.G., Götze F.: Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163, 1–28 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bolley F., Villani C.: Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Fac. Sci. Toulouse 14, 331–352 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Bobkov S.G., Gentil I., Ledoux M.: Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Cattiaux P.: Hypercontractivity for perturbed diffusion semi-groups. Ann. Fac. Sci. Toulouse 14(4), 609–628 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Cattiaux P., Guillin A.: On quadratic transportation cost inequalities. J. Math. Pures Appl. 86, 342–361 (2006)

    MATH  MathSciNet  Google Scholar 

  12. Cattiaux, P., Gozlan, N., Guillin, A., Roberto, C.: Functional inequalities for heavy tails distributions and application to isoperimetry. Available on arXiv:0807.3112 (2008)

    Google Scholar 

  13. Cattiaux P., Guillin A., Wang F.Y., Wu L.: Lyapunov conditions for logarithmic Sobolev and super Poincaré inequality. J. Func. Anal. 256, 1821–1841 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chavel I.: Isoperimetric Inequalities. Differential Geometric and Analytic Perspectives. Cambridge Tracts in Mathematics, vol. 145. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  15. Djellout H., Guillin A., Wu L.: Transportation cost-information inequalities for random dynamical systems and diffusions. Ann. Probab. 32(3B), 2702–2732 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Douc R., Fort G., Guillin A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stoch. Proc. Appl. 119(3), 897–923 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Down N., Meyn S.P., Tweedie R.L.: Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4), 1671–1691 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gao, F.Q., Wu, L.: Transportation-information inequalities for Gibbs measures (2007, in press)

  19. Gozlan N.: Characterization of Talagrand’s like transportation cost inequalities on the real line. J. Funct. Anal. 250(2), 400–425 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gozlan, N.: Poincaré inequalities for non euclidean metrics and transportation cost inequalities on \({\mathbb{R}^{d}}\) (2007, in press)

  21. Gozlan, N.: A characterization of dimension free concentration and transportation inequalities. Ann. Probab. (2008, to appear)

  22. Gozlan N., Léonard C.: A large deviation approach to some transportation cost inequalities. Probab. Theory Relat. Fields 139(1–2), 235–283 (2007)

    Article  MATH  Google Scholar 

  23. Guillin, A., Léonard, C., Wang, F.Y., Wu, L.-M.: Transportation information inequalities for Markov processes (II). Available on arXiv:0902.2101 (2008, preprint)

  24. Guillin A., Léonard C., Wu L.-M., Yao N.: Transportation information inequalities for Markov processes. Probab. Theory Relat. Fields 144(3–4), 669–696 (2009)

    Article  MATH  Google Scholar 

  25. Ledoux M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence (2001)

    Google Scholar 

  26. Marton K.: Bounding \({\bar{d}}\) -distance by informational divergence: A way to prove measure concentration. Ann. Probab. 24, 857–866 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  27. Marton K.: A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6, 556–571 (1997)

    Article  MathSciNet  Google Scholar 

  28. Meyn S.P., Tweedie R.L.: Markov chains and stochastic stability. Springer, London (1993)

    MATH  Google Scholar 

  29. Otto F., Villani C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Talagrand M.: Transportation cost for gaussian and other product measures. Geom. Funct. Anal. 6, 587–600 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Villani, C.: Saint-Flour Lecture Notes, 2005. Optimal transport, old and new. Available online via http://www.umpa.ens-lyon.fr/~cvillani/

  32. Villani C.: Topics in Optimal Transportation. Graduate Studies in Mathematics 58. American Mathematical Society, Providence (2003)

    Google Scholar 

  33. Wang F.Y.: Logarithmic Sobolev inequalities: Conditions and counterexamples. J. Oper. Theory 46, 183–197 (2001)

    Google Scholar 

  34. Wang, F.Y.: Logarithmic Sobolev inequalities: Different roles of Ric and Hess. Ann. Probab. (2008, to appear)

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Authors and Affiliations

  1. Laboratoire de Statistique et Probabilités, Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse cedex 09, France

    Patrick Cattiaux

  2. Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, Avenue des Landais, 63177, Aubière, France

    Arnaud Guillin & Li-Ming Wu

Authors
  1. Patrick Cattiaux
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  2. Arnaud Guillin
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  3. Li-Ming Wu
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Correspondence to Patrick Cattiaux.

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Cattiaux, P., Guillin, A. & Wu, LM. A note on Talagrand’s transportation inequality and logarithmic Sobolev inequality. Probab. Theory Relat. Fields 148, 285–304 (2010). https://doi.org/10.1007/s00440-009-0231-9

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  • Received: 27 October 2008

  • Revised: 14 May 2009

  • Published: 16 June 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0231-9

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Keywords

  • Lyapunov functions
  • Talagrand transportation information inequality
  • Logarithmic Sobolev inequality

Mathematics Subject Classification (2000)

  • 26D10
  • 47D07
  • 60G10
  • 60J60
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