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Nash Inequalities for Markov Processes in Dimension One

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Abstract

In this paper, we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product, we prove that for these processes, transience implies that the semigroups P(t) decay as

$$ {\left\| {P{\left( t \right)}} \right\|}_{{1 \to \infty }} \leqslant Ct^{{ - 1}} . $$

Sufficient conditions for general Markov chains are also obtained.

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References

  1. Chen M. F., Nash inequalities for general symmetric forms, Acta Mathematica Sinica, English Series, 1999, 15(3):353–370

    MATH  MathSciNet  Google Scholar 

  2. Wang F. Y., Sobolev type inequalities for general symmetric forms, Proc. Amer. Math. Soc., 1999

  3. Chen M. F., From Markov chains to non-equilibrium particle systems, Singapore: World Scientific, 1992

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Miclo L., An example of application of discrete Hardy’s inequalities, Markov Process and Related Fields, 1999, 5:319–330

    Google Scholar 

  6. Mao Y. H., Logarithmic Sobolev inequalities for birth-death process and diffusion process on the line, to appear, J. Appl. Prob. Stat., in Chinese, 2001

  7. Mao Y. H., On empty essential spectrum for Markov processes in dimension one, Preprint, 2000

  8. Carlen E. A., Kusuoka S., Stroock D. W., Upper bounds for symmetric Markov transition functions, Ann. Inst. Herri Poincaré, 1987, 2:245–287

    MathSciNet  Google Scholar 

  9. Varopoulos N., Hardy-Littlewood theory for semigroups, J. Funct. Anal., 1985, 63:240–260

    Article  MATH  MathSciNet  Google Scholar 

  10. Barky D., Coulhon T., Ledoux M., Saloff-Coste L., Sobolev inequalities in disguise, Indiana Univ. Math. J., 1995, 44(4):1033–1074

    MathSciNet  Google Scholar 

  11. Chen M. F., Analytic proof of dual variational formula for the first eigenvalue in dimension one, Science in China, 1999, 42(8):805–815

    Article  MATH  Google Scholar 

  12. Maz'ja V. G., Sobolev spaces, Berlin: Springer-Verlag, 1985

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Authors and Affiliations

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

    Young Hua Mao

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  1. Young Hua Mao
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Correspondence to Young Hua Mao.

Additional information

Research supported in part by RFDP (No 96002704), NSFC (No 19771008) and Fok Ying-Tung Youth Foundation

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Mao, Y.H. Nash Inequalities for Markov Processes in Dimension One. Acta Math Sinica 18, 147–156 (2002). https://doi.org/10.1007/s101140100128

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  • DOI: https://doi.org/10.1007/s101140100128

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