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.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Aikawa, H.: Intrinsic ultracontractivity via capacitary width. Rev. Mat. Iberoam. 31, 1041–1106 (2015)
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)
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)
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)
Chen, Z.-Q., Kim, D., Kuwae, K.: \(L^p\)-independence of spectral radius for generalized Feynman–Kac semigroups. Math. Ann. (to appear)
Collet, P., Martínez, S., San Martín, J.: Quasi-stationary Distributions, Markov Chains, Diffusions and Dynamical Systems. Springer, Berlin (2013)
Fukushima, M.: A note on irreducibility and ergodicity of symmetric Markov processes. Springer Lecture Notes in Physics vol. 173, pp. 200–207 (1982)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, 2nd edn. de Gruyter, Berlin (2010)
Itô, K.: Essentials of Stochastic Processes. American Mathematical Society, Providence (2006)
Kajino, N.: Equivalence of recurrence and Liouville property for symmetric Dirichlet forms. Math. Phys. Comput. Simul. 40, 89–98 (2017)
Kaleta, K., Kulczycki, T.: Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians. Potential Anal. 33, 313–339 (2010)
Kulczycki, T.: Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math. 46, 325–334 (1998)
Knobloch, R., Partzsch, L.: Uniform conditional ergodicity and intrinsic ultracontractivity. Potential Anal. 33, 107–136 (2010)
Kwaśnicki, M.: Intrinsic ultracontractivity for stable semigroups on unbounded open sets. Potential Anal. 31, 57–77 (2009)
Miura, Y.: Ultracontractivity for Markov semigroups and quasi-stationary distributions. Stoch. Anal. Appl. 32, 591–601 (2014)
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)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes, and Martingales, vol. 1, 2nd edn. Cambridge University Press, Foundations (2000)
Takeda, M.: A tightness property of a symmetric Markov process and the uniform large deviation principle. Proc. Am. Math. Soc. 141, 4371–4383 (2013)
Takeda, M.: A variational formula for Dirichlet forms and existence of ground states. J. Funct. Anal. 266, 660–675 (2014)
Takeda, M.: Criticality and subcriticality of generalized Schrödinger forms. Ill. J. Math. 58, 251–277 (2014)
Takeda, M.: Compactness of symmetric Markov semi-groups and boundedness of eigenfunctions. Trans. Amer. Math. Soc. (to appear)
Takeda, M., Tawara, Y., Tsuchida, K.: Compactness of Markov and Schrödinger semi-groups: a probabilistic approach. Osaka J. Math. 54, 517–532 (2017)
Tomisaki, M.: Intrinsic ultracontractivity and small perturbation for one-dimensional generalized diffusion operators. J. Funct. Anal. 251, 289–324 (2007)
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.
About this article
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
- Quasi-stationary distribution
- Symmetric Markov process
- Dirichlet form
- Yaglom limit
Mathematics Subject Classification (2010)