Skip to main content

Advertisement

SpringerLink
Log in

Logarithmic Sobolev Inequalities of Diffusions for the \(L^2\) Metric

  • Published:
Potential Analysis Aims and scope Submit manuscript

An Erratum to this article was published on 10 February 2007

An Erratum to this article was published on 10 February 2007

Abstract

Under the Bakry–Emery's \(\Gamma_{2}\)-minoration condition, we establish the logarithmic Sobolev inequality for the Brownian motion with drift in the metric \(L^2\) instead of the usual Cameron–Martin metric. The involved constant is sharp and does not explode for large time. This inequality with respect to the \(L^2\)-metric provides us the gaussian concentration inequalities for the large time behavior of the diffusion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aida, S.: Gradient estimates of harmonic functions and the asymptotics of spectral gaps on path spaces, Interdiscip. Inform. Sci. 2(1) (1996), 75–84.

    Article  MATH  MathSciNet  Google Scholar 

  2. Aida, S., Masuda, T. and Shigekawa, I.: Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126(1) (1994), 83–101.

    Article  MATH  MathSciNet  Google Scholar 

  3. Aida, S. and Stroock, D.: Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1(1) (1994), 75–86.

    MATH  MathSciNet  Google Scholar 

  4. Bakry, D.: L'hypercontractivité et son utilisation en théorie des semigroupes, Ecole d'Eté de Probabilités de Saint-Flour (1992), Lecture Notes in Math, 1581, Springer, Berlin, 1994.

  5. Bakry, D. and Emery, M.: Diffusions hypercontractives. Seminaire de probabilités XIX, in Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 177–206.

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Capitaine, M., Hsu, E.P. and Ledoux, M.: Martingale representation and a simple proof of logarithmic Sobolev inequality on path spaces, Elect. Comm. Probab. 2: paper 7, 1997.

    Google Scholar 

  9. Djellout, H.: A. Guillin, Moderate deviation for Markov chains with atom, Stoch. Process. Appl. 95 (2001), 203–217.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Djellout, H., Guillin, A. and Wu, L.: Moderate deviations for non-linear functionals of moving average processes. Preprint 2004, to appear in Ann. Inst. H. Poincaré.

  12. Fang, S.: Inégalité du type de Poincaré sur l'espace des chemins riemanniens, C.R. Acad. Sci. 318 (1994), 257–260.

    MATH  Google Scholar 

  13. Gentil, Inégalités de Sobolev logarithmiques et hypercontractivité en mécanique statistique et en EDP, Thése de Doctorat 2001, Université de Toulouse III.

  14. Hsu, E.P.: Stochastic Analysis on Manifolds, Grad. Stud. Math. 38, 2001.

  15. Hsu, E.P.: Analysis on Path and Loop Spaces in Probability theory and applications, Princeton, New Jersey, 1996, 277–347.

  16. Gross, L.: Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083.

    Article  MathSciNet  Google Scholar 

  17. Ledoux, M.: Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités XXXIII, Lecture Notes in Math. 1709, Springer, Berlin, 1999, pp. 120–216.

  18. Ikeda, N. and Watanabe. S.: Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam, 1989.

  19. Nualart, D.: The Malliavin Calculus and Related Topics, Springer, Berlin, 1995.

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Villani, C.: Topics in Optimal Transportation, Grad. Stud. Math. 58, Amer. Math. Soc., Providence, 2003.

  22. Wang, F.Y.: Transportation cost inequalities on path spaces over Riemannian manifolds, Illinois J. Math. 46 (2002), 1197–1206.

    MATH  MathSciNet  Google Scholar 

  23. Wu, L.: A deviation inequality for non reversible Markov processes, Ann. Inst. Henri Poincaré, Probabilités et Statistiques 36(4) (2000), 435–445.

    Article  MATH  Google Scholar 

  24. Wu, L. and Zhang, Z.L.: Talagrand's \(T_2\)-transportation inequality w.r.t. a uniform metric for diffusions, Acta Math. Appl. Sinica (English Ser.) 20(3) (2004), 357–364.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177, Aubiére, France

    Mathieu Gourcy & Liming Wu

  2. Department of Mathematics, Wuhan University, 430072, Wuhan, China

    Liming Wu

Authors
  1. Mathieu Gourcy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Liming Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mathieu Gourcy.

Additional information

An erratum to this article can be found at http://dx.doi.org/10.1007/s11118-007-9037-5

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gourcy, M., Wu, L. Logarithmic Sobolev Inequalities of Diffusions for the \(L^2\) Metric. Potential Anal 25, 77–102 (2006). https://doi.org/10.1007/s11118-006-9009-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-006-9009-1

Mathematics Subject Classifications (2000)

Key words

Search

Navigation