Skip to main content
SpringerLink
Log in

A Stein deficit for the logarithmic Sobolev inequality

  • Articles
  • Invited Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We provide some lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality in terms of the so-called Stein characterization of the Gaussian distribution. The techniques are based on the representation of the relative Fisher information along the Ornstein-Uhlenbeck semigroup by the Minimum Mean-Square Error from information theory.

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. Arstein-Avidan S, Klartag B, Schütt C, et al. Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J Funct Anal, 2012, 262: 4181–4204

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakry D. L’hypercontractivité et son utilisation en théorie des semigroupes. Berlin-Heidelberg: Springer, 1994

    MATH  Google Scholar 

  3. Bakry D, Émery M. Diffusions hypercontractives. In: Séminaire de Probabilités XIX. Lecture Notes in Mathematics, vol. 1123. Berlin-Heidelberg: Springer, 1985, 177–206

    Google Scholar 

  4. Bakry D, Gentil I, Ledoux M. Analysis and Geometry of Markov Diffusion Operators. New York: Springer, 2014

    Book  MATH  Google Scholar 

  5. Bakry D, Ledoux M. A logarithmic Sobolev form of the Li-Yau parabolic inequality. Rev Mat Iberoam, 2006, 22: 683–702

    Article  MathSciNet  MATH  Google Scholar 

  6. Barchiesi M, Brancolini A, Julin V. Sharp dimension free quantitative estimates for the Gaussian isoperimetric inequality. Ann Probab, in press, arXiv:1409.2106, 2014

    Google Scholar 

  7. Bobkov S, Gozlan N, Roberto C, et al. Bounds on the deficit in the logarithmic Sobolev inequality. J Funct Anal, 2014, 267: 4110–4138

    Article  MathSciNet  MATH  Google Scholar 

  8. Bucur D, Fragalà I. Lower bounds for the Prékopa-Leindler deficit by some distances modulo translations. J Convex Anal, 2014, 21: 289–305

    MathSciNet  MATH  Google Scholar 

  9. Caglar U, Fradelizi M, Guédon O, et al. Functional versions of Lp-affine surface area and entropy inequalities. Int Math Res Not IMRN, 2016, 2016: 1223–1250

    Article  MATH  Google Scholar 

  10. Carlen E. Some integral identities and inequalities for entire functions and their application to the coherent state transform. J Funct Anal, 1991, 97: 231–249

    Article  MathSciNet  MATH  Google Scholar 

  11. Carlen E. Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J Funct Anal, 1991, 101: 194–211

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen L H Y, Goldstein L, Shao Q-M. Normal Approximation by Stein’s Method. New York: Springer, 2011

    Book  MATH  Google Scholar 

  13. Cianchi A, Fusco N, Maggi F, et al. The sharp Sobolev inequality in quantitative form. J Eur Math Soc, 2009, 11: 1105–1139

    Article  MathSciNet  MATH  Google Scholar 

  14. Cordero-Erausquin D. Transport inequalities for log-concave measures, quantitative forms and applications. Canad J Math, in press, arXiv:1504.06147, 2015

    Google Scholar 

  15. Eldan R. A two-sided estimate for the Gaussian noise stability deficit. Invent Math, 2015, 201: 561–624

    Article  MathSciNet  MATH  Google Scholar 

  16. Fathi M, Indrei E, Ledoux M. Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete Contin Dyn Syst, 2016, 36: 6835–6853

    Article  MathSciNet  MATH  Google Scholar 

  17. Feo F, Indrei E, Posteraro M R, et al. Some remarks on the stability of the Log-Sobolev inequality for the Gaussian measure. Potential Anal, 2016, doi:10.1007/s11118-016-9607-5

    Google Scholar 

  18. Figalli A. Stability in geometric and functional inequalities. In: European Congress of Mathematics. Zürich: Eur Math Soc, 2013, 585–599

    Google Scholar 

  19. Figalli A, Jerison D. Quantitative stability for the Brunn-Minkowski inequality. J Eur Math Soc, 2014, in press

    Google Scholar 

  20. Figalli A, Maggi F, Pratelli A. A refined Brunn-Minkowski inequality for convex sets. Ann Inst H Poincaré Anal Non-Linéaire, 2009, 26: 2511–2519

    Article  MathSciNet  MATH  Google Scholar 

  21. Figalli A, Maggi F, Pratelli A. A mass transportation approach to quantitative isoperimetric inequalities. Invent Math, 2010, 182: 167–211

    Article  MathSciNet  MATH  Google Scholar 

  22. Figalli A, Maggi F, Pratelli A. Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation. Adv Math, 2013, 242: 80–101

    Article  MathSciNet  MATH  Google Scholar 

  23. Fusco N, Maggi F, Pratelli A. The sharp quantitative isoperimetric inequality. Ann of Math, 2008, 168: 941–980

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Guo D, Shamai S, Verd´u S. Mutual information and minimum mean-square error in Gaussian channels. IEEE Trans Inform Theory, 2005, 51: 1261–1282

    Article  MathSciNet  MATH  Google Scholar 

  26. Guo D, Wu Y, Shamai S, et al. Estimation in Gaussian noise: Properties of the minimum mean-square error. IEEE Trans Inform Theory, 2011, 57: 2371–2385

    Article  MathSciNet  Google Scholar 

  27. Ledoux M. On an integral criterion for hypercontractivity of diffusion semigroups and extremal functions. J Funct Anal, 1992, 105: 444–465

    Article  MathSciNet  MATH  Google Scholar 

  28. Ledoux M. Heat flow derivatives and Minimun Mean-Square Error in Gaussian noise. IEEE Trans Inform Theory, 2016, 62: 3401–3409

    Article  MathSciNet  MATH  Google Scholar 

  29. Ledoux M, Nourdin I, Peccati G. Stein’s method, logarithmic Sobolev and transport inequalities. Geom Funct Anal, 2015, 25: 256–306

    Article  MathSciNet  MATH  Google Scholar 

  30. Mossel E, Neeman J. Robust dimension free isoperimetry in Gaussian space. Ann Probab, 2015, 43: 971–991

    Article  MathSciNet  MATH  Google Scholar 

  31. Nourdin I, Peccati G. Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge: Cambridge University Press, 2012

    Book  MATH  Google Scholar 

  32. Nourdin I, Peccati G, Swan Y. Integration by parts and representation of information functionals. In: Proceedings of the 2014 IEEE International Symposium on Information Theory. New York: IEEE, 2014, 2217–2221

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Stein C. Approximate Computation of Expectations. Hayward: Institute of Mathematical Statistics, 1986

    MATH  Google Scholar 

  35. Talagrand M. Transportation cost for Gaussian and other product measures. Geom Funct Anal, 1996, 6: 587–600

    Article  MathSciNet  MATH  Google Scholar 

  36. Villani C. Optimal Transport: Old and New. Berlin: Springer, 2009

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was partially supported by Grants Nos. F1R-MTH-PUL-15CONF and F1R-MTH-PUL-15STAR at Luxembourg University.

Author information

Authors and Affiliations

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

    Michel Ledoux

  2. Institut Universitaire de France, Paris, France

    Michel Ledoux

  3. UR en Mathématiques-FSTC, Luxembourg University, Luxembourg City, L-1359, Luxembourg

    Ivan Nourdin & Giovanni Peccati

Authors
  1. Michel Ledoux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ivan Nourdin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Giovanni Peccati
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Peccati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ledoux, M., Nourdin, I. & Peccati, G. A Stein deficit for the logarithmic Sobolev inequality. Sci. China Math. 60, 1163–1180 (2017). https://doi.org/10.1007/s11425-016-0134-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-016-0134-7

Keywords

MSC(2010)

Search

Navigation