Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Transportation-information inequalities for Markov processes
Download PDF
Download PDF
  • Published: 15 July 2008

Transportation-information inequalities for Markov processes

  • Arnaud Guillin1,
  • Christian Léonard2,
  • Liming Wu3,4 &
  • …
  • Nian Yao4 

Probability Theory and Related Fields volume 144, pages 669–695 (2009)Cite this article

  • 589 Accesses

  • 49 Citations

  • Metrics details

Abstract

In this paper, one investigates the transportation-information T c I inequalities: α(T c (ν, μ)) ≤ I (ν|μ) for all probability measures ν on a metric space \({(\mathcal{X}, d)}\), where μ is a given probability measure, T c (ν, μ) is the transportation cost from ν to μ with respect to the cost function c(x, y) on \({\mathcal{X}^2}\), I(ν|μ) is the Fisher–Donsker–Varadhan information of ν with respect to μ and α : [0, ∞) → [0, ∞] is a left continuous increasing function. Using large deviation techniques, it is shown that T c I is equivalent to some concentration inequality for the occupation measure of a μ-reversible ergodic Markov process related to I(·|μ). The tensorization property of T c I and comparisons of T c I with Poincaré and log-Sobolev inequalities are investigated. Several easy-to-check sufficient conditions are provided for special important cases of T c I and several examples are worked out.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. In: Ecole d’Eté de Probabilités de Saint-Flour (1992). Lecture Notes in Mathematics, vol. 1581. Springer, New York (1994)

  2. Bakry D., Cattiaux P., Guillin A.: Rates of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254(3), 727–759 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. 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 

  4. 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 

  5. Bogachev V., Kolesnikov A.: Integrability of absolutely continuous transformations of measures and applications to optimal mass transportation. Probab. Theory Appl. 50(3), 3–25 (2005)

    MathSciNet  Google Scholar 

  6. 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 

  7. Carlen E.A.: Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101(1), 194–211 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cattiaux P., Guillin A.: Deviation bounds for additive functionals of Markov process. ESAIM P S 12, 12–29 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cattiaux P., Guillin A.: On quadratic transportation cost inequalities. J. Math. Pures Appl. 86(9), 341–361 (2006)

    MathSciNet  Google Scholar 

  10. Cattiaux, P., Guillin, A., Wu, L., Wang, F.Y.: Preprint, available on Arxiv (2007)

  11. Chen, M.F.: Eigenvalues, inequalities, and ergodic theory. In: Probability and its Applications. Springer, New York (2005)

  12. Deuschel, J.-D., Stroock, D.W.: Large Deviations, vol. 137 of Pure and Applied Mathematics. Academic Press, London (1989)

  13. 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 

  14. Djellout, H., Wu, L.: Lipschitzian spectral gap for one dimensional diffusions. In preparation (2008)

  15. Donsker, M.D., Varadhan, S.R.S.: Asymptotic evaluations of certain Markov process expectations for large time, I, III, IV. Commun. Pure Appl. Math. 28:1–47 (1975), 29:389–461 (1976), 36:183–212 (1983)

  16. Douc, R., Fort, G., Guillin, A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Preprint, available on Arxiv (2006)

  17. Fernique, X.: Extension du théorème de Cameron-Martin aux translations aléatoires. II. Intégrabilité des densités. In: High Dimensional Probability III (Sandjberg 2002), Progresses in Probability, vol. 55, 95-102. Birkhäuser, Basel (2003)

  18. Feyel D., Ustunel A.S.: The Monge-Kantorovitch problem and Monge-Ampère equation on Wiener space. Probab. Theor. Relat. Fields 128(3), 347–385 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  19. Gao, F.Q., Wu, L.: Transportation-information inequalities for Gibbs measures. Preprint (2007)

  20. Gibbs A., Su F.: On choosing and bounding probability metrics. Int. Stat. Rev. 70(3), 419–435 (2002)

    Article  MATH  Google Scholar 

  21. Gozlan, N.: Characterization of Talagrand’s like transportation cost inequalities on the real line. To appear in J. Funct. Anal. (2006)

  22. Gozlan, N., Léonard, C.: A large deviation approach to some transportation cost inequalities. To appear in Probab. Theory Relat. Fields (2008)

  23. Guillin, A., Léonard, C., Wu, L.: Transportation cost inequalities. In preparation (2008)

  24. Joulin, A.: Concentration et fluctuations de processus stochastiques avec sauts. Ph.D. thesis, Université La Rochelle (2006)

  25. Joulin, A.: A new Poisson-type deviation inequality for Markov jump process with positive Wasserstein curvature. Preprint (2007)

  26. Klein T., Ma Y., Privault N.: Convex concentration inequalities and forward-backward stochastic calculus. Electron. J. Probab. 11, 486–512 (2006)

    MathSciNet  Google Scholar 

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

  28. Lezaud P.: Chernoff and Berry-Esséen inequalities for Markov processes. ESAIM Probab. Stat. 5, 183–201 (2001) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, W., Ma, Y.: Spectral gap and deviation inequalities for birth-death processes. Preprint 2006, contained in the Ph.D. thesis of Y. Ma at Université La Rochelle 2007. To appear in Ann. IHP Probab. Stat. (2008)

  30. Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. To appear in Ann. Math. (2008)

  31. Marton K.: Bounding d̄-distance by informational divergence: a way to prove measure concentration. Ann. Probab. 24, 857–866 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  33. Meyn S.P., Tweedie R.L.: Markov chains and stochastic stability. Communications and Control Engineering Series. Springer, New York (1993)

    Google Scholar 

  34. 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 

  35. Saloff-Coste, L.: Lectures on finite Markov chains. École d’été de Probabilités de Saint-Flour 1996, LNM, vol. 1685, pp. 301–413. Springer, New York (1997)

  36. Sturm K.-T.: On the geometry of metric measure spaces, I. Acta Math. 196, 65–131 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  37. Sturm K.-T.: On the geometry of metric measure spaces, II. Acta Math. 196, 133–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Villani, C.: Saint-Flour Lecture Notes. Optimal transport, old and new. http://www.umpa.ens-lyon.fr/~cvillani/ (2005)

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

  41. Wang F.Y.: Functional Inequalities, Markov Semigroup and Spectral Theory. Chinese Sciences Press, Beijing (2005)

    Google Scholar 

  42. Wu L.: A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri Poincaré (Sér. Probab. Stat.) 36, 435–445 (2000)

    Article  MATH  Google Scholar 

  43. Wu L.: Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172, 301–376 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  44. Wu L.: Large and moderate deviations for stochastic damping Hamiltonian systems. Stoch. Proc. Appl. 91, 205–238 (2001)

    Article  MATH  Google Scholar 

  45. Wu L.: Essential spectral radius for Markov semigroups (I) : discrete time case. Probab. Theory Relat. Fields 128, 255–321 (2004)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Centrale de Marseille et LATP, Centre de Mathématiques et Informatique, Technopôle de Château-Gombert, 13453, Marseille, France

    Arnaud Guillin

  2. Modal-X, Université Paris Ouest, Bât. G, 200 av. de la République, 92001, Nanterre, France

    Christian Léonard

  3. Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177, Aubière, France

    Liming Wu

  4. Department of Mathematics, Wuhan University, 430072, Hubei, China

    Liming Wu & Nian Yao

Authors
  1. Arnaud Guillin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Léonard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liming Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nian Yao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arnaud Guillin.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Guillin, A., Léonard, C., Wu, L. et al. Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields 144, 669–695 (2009). https://doi.org/10.1007/s00440-008-0159-5

Download citation

  • Received: 22 July 2007

  • Revised: 28 April 2008

  • Published: 15 July 2008

  • Issue Date: July 2009

  • DOI: https://doi.org/10.1007/s00440-008-0159-5

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 26D10
  • 47D07
  • 60E15
  • 60F10
  • 60G99
  • 60J25
  • 60J60
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature