Skip to main content
SpringerLink
Log in

Coupling, convergence rates of Markov processes and weak Poincaré inequalities

  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

Some analytic and probabilistic properties of the weak Poincaré inequality are obtained. In particular, for strong Feller Markov processes the existence of this inequality is equivalent to each of the following: (i) the Liouville property (or the irreducibility); (ii) the existence of successful couplings (or shift-couplings); (iii) the convergence of the Markov process in total variation norm; (iv) the triviality of the tail (or the invariant) σ-field; (v) the convergence of the density. Estimates of the convergence rate in total variation norm of Markov processes are obtained using the weak Poincaré inequality

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Lindvall, T., Lectures on the Coupling Method, New York: Wiley, 1992.

    MATH  Google Scholar 

  2. Kendall, W. S., Nonnegative Ricci curvature and the Brownian coupling property, Stochastics, 1986, 19: 111–129.

    MATH  MathSciNet  Google Scholar 

  3. Kusuoka, S., Analysis on Wiener spaces II, J. Funct. Anal., 1992, 103: 229–427.

    Article  MATH  MathSciNet  Google Scholar 

  4. Aida, S., Uniformly positivity improving property, Sobolev inequalities and spectral gaps, J. Funct. Anal., 1998, 158: 152–185.

    Article  MATH  MathSciNet  Google Scholar 

  5. Hino, H., Exponential decay of positivity preserving semigroups on Lp, Osaka J. Math., 2000, 37: 603–624.

    MATH  MathSciNet  Google Scholar 

  6. Mathieu, P., Quand l’unégalite log-Sobolev implique l’inégalite de trou spectral, Séminaire de Probabilités XXXII, 30-35, Lecture Notes in Math. 1686, Berlin: Springer-Verlag, 1998.

    Google Scholar 

  7. Cranston, M., Greven, A., Coupling and harmonic functions in the case of continuous time Markov processes, Stoch. Proc. Appl., 1995, 60: 261–286.

    Article  MATH  MathSciNet  Google Scholar 

  8. Bogachev, V. I., Krylov, N. V., Röckner, M., Elliptic regularity and essential selfadjointness of Dirichlet operators on ℝn, Ann. Scuola Norm. Sup. Pisa XI. Sci., 1997, 24: 451–461.

    MATH  Google Scholar 

  9. Thorisson, H., Shift-coupling in continuous time, Probability Theory Relat. Fields, 1997, 109: 417–424.

    Article  Google Scholar 

  10. Aldous, D., Thorisson, H., Shift-coupling, Stoch. Proc. Appl., 1993, 44: 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  11. Chen, M.-F., Equivalence of exponential ergodicity and L2-exponential convergence for Markov chains, Stoch. Proc. Appl., 2000, 87: 281–297.

    Article  MATH  Google Scholar 

  12. Cranston, M., Wang, F.-Y., A condition for the equivalence of coupling and shift coupling, Ann. of Probab., 2000, 28: 1666–1679.

    Article  MATH  MathSciNet  Google Scholar 

  13. Wang, F.-Y., Functional inequalities, semigroup properties and spectrum estimates, Infinite Dimens. Anal. Quant. Probab. Rel. Topics, 2000, 3: 263–295.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Beijing Normal University, 100875, Beijing, China

    Fengyu Wang

Authors
  1. Fengyu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fengyu Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, F. Coupling, convergence rates of Markov processes and weak Poincaré inequalities. Sci. China Ser. A-Math. 45, 975–983 (2002). https://doi.org/10.1007/BF02879980

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02879980

Keywords

Search

Navigation