Existence and Uniqueness of Quasi-stationary Distributions for Symmetric Markov Processes with Tightness Property

Abstract

Let X be an irreducible symmetric Markov process with the strong Feller property. We assume, in addition, that X is explosive and has a tightness property. We then prove the existence and uniqueness of quasi-stationary distributions of X.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Aikawa, H.: Intrinsic ultracontractivity via capacitary width. Rev. Mat. Iberoam. 31, 1041–1106 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Cattiaux, P., Collet, P., Lamberrt, A., Martinez, S., Meleard, S., San Martin, J.: Quasi-stationary distributions and diffusion models in population dynamics. Ann. Probab. 37, 1926–1969 (2009)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chen, Z.-Q., Fitzsimmons, P.J., Takeda, M., Ying, J., Zhang, T.-S.: Absolute continuity of symmetric Markov processes. Ann. Probab. 32, 2067–2098 (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change and Boundary Theory, London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)

    Google Scholar 

  5. 5.

    Chen, Z.-Q., Kim, D., Kuwae, K.: \(L^p\)-independence of spectral radius for generalized Feynman–Kac semigroups. Math. Ann. (to appear)

  6. 6.

    Collet, P., Martínez, S., San Martín, J.: Quasi-stationary Distributions, Markov Chains, Diffusions and Dynamical Systems. Springer, Berlin (2013)

    Google Scholar 

  7. 7.

    Fukushima, M.: A note on irreducibility and ergodicity of symmetric Markov processes. Springer Lecture Notes in Physics vol. 173, pp. 200–207 (1982)

  8. 8.

    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2nd edn. de Gruyter, Berlin (2010)

    Google Scholar 

  9. 9.

    Itô, K.: Essentials of Stochastic Processes. American Mathematical Society, Providence (2006)

    Google Scholar 

  10. 10.

    Kajino, N.: Equivalence of recurrence and Liouville property for symmetric Dirichlet forms. Math. Phys. Comput. Simul. 40, 89–98 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Kaleta, K., Kulczycki, T.: Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians. Potential Anal. 33, 313–339 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math. 46, 325–334 (1998)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Knobloch, R., Partzsch, L.: Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal. 33, 107–136 (2010)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kwaśnicki, M.: Intrinsic ultracontractivity for stable semigroups on unbounded open sets. Potential Anal. 31, 57–77 (2009)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Miura, Y.: Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32, 591–601 (2014)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Pinsky, R.G.: On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab. 13, 363378 (1985)

    MathSciNet  Google Scholar 

  17. 17.

    Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 1, 2nd edn. Cambridge University Press, Foundations (2000)

    Google Scholar 

  18. 18.

    Takeda, M.: A tightness property of a symmetric Markov process and the uniform large deviation principle. Proc. Am. Math. Soc. 141, 4371–4383 (2013)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Takeda, M.: A variational formula for Dirichlet forms and existence of ground states. J. Funct. Anal. 266, 660–675 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Takeda, M.: Criticality and subcriticality of generalized Schrödinger forms. Ill. J. Math. 58, 251–277 (2014)

    Article  Google Scholar 

  21. 21.

    Takeda, M.: Compactness of symmetric Markov semi-groups and boundedness of eigenfunctions. Trans. Amer. Math. Soc. (to appear)

  22. 22.

    Takeda, M., Tawara, Y., Tsuchida, K.: Compactness of Markov and Schrödinger semi-groups: a probabilistic approach. Osaka J. Math. 54, 517–532 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Tomisaki, M.: Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators. J. Funct. Anal. 251, 289–324 (2007)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Masayoshi Takeda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The author was supported in part by Grant-in-Aid for Scientific Research (No. 26247008(A)) and Grant-in-Aid for Challenging Exploratory Research (No. 25610018), Japan Society for the Promotion of Science.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Takeda, M. Existence and Uniqueness of Quasi-stationary Distributions for Symmetric Markov Processes with Tightness Property. J Theor Probab 32, 2006–2019 (2019). https://doi.org/10.1007/s10959-019-00878-0

Download citation

Keywords

  • Quasi-stationary distribution
  • Symmetric Markov process
  • Dirichlet form
  • Yaglom limit
  • Tightness

Mathematics Subject Classification (2010)

  • 60B10
  • 60J25
  • 37A30
  • 31C25