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Symmetric Markov Processes with Tightness Property

  • Masayoshi TakedaEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 229)

Abstract

A symmetric Markov process X is said to be in Class (T) if it is irreducible, strong Feller and possesses a tightness property. We give some properties of X in Class (T) and of the semi-group \(p_t\) of X: the uniform large deviation principle of X, \(L^p\)-independence of growth bounds of \(p_t\), compactness of \(p_t\) as an operator in \(L^2\), and boundedness of every eigenfunction of \(p_t\).

Keywords

Symmetric Markov process Dirichlet form Compactness of semigroup 

2010 Mathematics Subject Classification

60J45 60G52 31C25 

Notes

Acknowledgements

The author would like to thank the referee for helpful comments on this paper.

References

  1. 1.
    Chen, Z.-Q.: Gaugeability and conditional gaugeability. Trans. Am. Math. Soc. 354, 4639–4679 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chung, K.L., Zhao, Z.X.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995)CrossRefGoogle Scholar
  3. 3.
    Deuschel, J.-D., Stroock, D.W.: Large Deviations. American Mathematical Society, Providence (2001)CrossRefGoogle Scholar
  4. 4.
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987)Google Scholar
  5. 5.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, de Gruyter, 2nd rev. and ext. ed. (2011)Google Scholar
  6. 6.
    Itô, K.: Essentials of Stochastic Processes. American Mathematical Society, Providence (2006)CrossRefGoogle Scholar
  7. 7.
    Kaleta, K., Kulczycki, T.: Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians. Potential Anal. 33, 313–339 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lenz, D., Stollmann, P., Wingert, D.: Compactness of Schrödinger semigroups. Math. Nachr. 283, 94–103 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schilling, R. L.: Measures, Integrals and Martingales. Cambridge University Press, Cambridge (2005)Google Scholar
  10. 10.
    Simon, B.: Schrödinger operators with purely discrete spectra. Methods Funct. Anal. Topol. 15, 61–66 (2009)zbMATHGoogle Scholar
  11. 11.
    Simon, B.: Operator Theory, A Comprehensive Course in Analysis, Part 4. American Mathematical Society (2015)Google Scholar
  12. 12.
    Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5, 109–138 (1996)Google Scholar
  13. 13.
    Takeda, M.: Lp-independence of spectral bounds of Schrödinger type semigroups. J. Funct. Anal. 252, 550–565 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Takeda, M.: A tightness property of a symmetric Markov process and the uniform large deviation principle. Proc. Am. Math. Soc. 141, 4371–4383 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Takeda, M.: A variational formula for Dirichlet forms and existence of ground states. J. Funct. Anal. 266, 660–675 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Takeda, M.: Criticality and subcriticality of generalized Schrödinger forms. Illinois J. Math. 58, 251–277 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Takeda, M., Tawara, Y., Tsuchida, K.: Compactness of Markov and Schrödinger semi-groups: a probabilistic approach. Osaka J. Math. 54, 517–532 (2017)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Wu, L.: Some notes on large deviations of Markov processes, Acta Math. Sin. (Engl. Ser.) 16, 369-394 (2000)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Tohoku UniversitySendaiJapan

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