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Weak Poincaré Inequalities, Decay of Markov Semigroups and Concentration of Measures

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Abstract

For a reasonable class of weak Poincaré inequalities, the decay of the corresponding Markov semigroups obtained earlier by Röckner and the first named author is improved by removing an extra L 2–norm. Next, a concentration estimate of the reference measure is presented for the weak Poincaré inequality, which is sharp as illustrated by some examples of one–dimensional diffusion processes.

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References

  1. Röckner, M.,Wang, F. Y.: Weak Poincaré inequalities and L2–convergence rates of Markov Semigroups. J. Funct. Anal., 185, 564–603 (2001)

    Article  MathSciNet  Google Scholar 

  2. Wang, F. Y.: Functional inequalities for the decay of sub–Markov semigroups. Pot. Anal., 18, 1–23 (2003)

    Article  Google Scholar 

  3. Liggett, T. M.: L2–rates of convergence for attractive reversible nearest particle systems: the critical case. Ann. Probab., 19, 935–959 (1991)

    MathSciNet  Google Scholar 

  4. Wang, F. Y.: Coupling, convergence rates of Markov processes and weak Poincar´e inequalities. Science in China (A), 45(8), 975–983 (2002)

    Google Scholar 

  5. Wu, L.: Uniform positive improvingness of resolvent and spectral gap, Preprint, 2001

  6. Wang, F. Y.: Functional inequalities for empty essential spectrum. J. Funct. Anal., 170, 219–245 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Gross, L., Rothaus, O.: Herbst inequalities for supercontractive semigroups. J. Math. Kyoto Univ., 38, 295–318 (1998)

    MathSciNet  Google Scholar 

  9. Röckner, M., Wang, F. Y.: Supercontractivity and ultracontractivity for (non–symmetric) diffusion semigroups on manifolds. Forum Math., 15, 893–921 (2003)

    Article  MathSciNet  Google Scholar 

  10. Ledoux, M.: Concentration of measures and logarithmic Sobolev inequalities. Lecture Notes in Math., 1709, 120–216 (1999)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Chen, M. F.: Explicit bounds of the first eigenvalue. Science in China (A), 43, 1051–1059 (2000)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P. R. China

    Feng Yu Wang & Qi Zhi Zhang

Authors
  1. Feng Yu Wang
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  2. Qi Zhi Zhang
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Correspondence to Feng Yu Wang.

Additional information

Research supported in part by NNSFC (10121101, 10025105), TRAPOYT and the 973-Project

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Wang, F.Y., Zhang, Q.Z. Weak Poincaré Inequalities, Decay of Markov Semigroups and Concentration of Measures. Acta Math Sinica 21, 937–942 (2005). https://doi.org/10.1007/s10114-004-0435-y

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  • DOI: https://doi.org/10.1007/s10114-004-0435-y

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