A tightness property of a symmetric Markov process and the uniform large deviation principle

Takeda, Masayoshi

doi:10.1090/S0002-9939-2013-11696-5

A tightness property of a symmetric Markov process and the uniform large deviation principle
HTML articles powered by AMS MathViewer

by Masayoshi Takeda

Proc. Amer. Math. Soc. 141 (2013), 4371-4383

DOI: https://doi.org/10.1090/S0002-9939-2013-11696-5

Published electronically: August 21, 2013

PDF | Request permission

Abstract:

Previously, we considered a large deviation for occupation measures of a symmetric Markov processes under the condition that its resolvent possesses a kind of tightness property. In this paper, we prove that if the Markov process is conservative, then the tightness property implies the uniform hyper-exponential recurrence, which leads us to the uniform large deviation principle.

References

Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639–4679. MR 1926893, DOI 10.1090/S0002-9947-02-03059-3
Zhen-Qing Chen, Uniform integrability of exponential martingales and spectral bounds of non-local Feynman-Kac semigroups, Stochastic analysis and applications to finance, Interdiscip. Math. Sci., vol. 13, World Sci. Publ., Hackensack, NJ, 2012, pp. 55–75. MR 2986841, DOI 10.1142/9789814383585_{0}004
Zhen-Qing Chen, $L^p$-independence of spectral bounds of generalized non-local Feynman-Kac semigroups, J. Funct. Anal. 262 (2012), no. 9, 4120–4139. MR 2899989, DOI 10.1016/j.jfa.2012.02.011
K. L. Chung, Doubly-Feller process with multiplicative functional, Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985) Progr. Probab. Statist., vol. 12, Birkhäuser Boston, Boston, MA, 1986, pp. 63–78. MR 896735
E. B. Davies, $L^1$ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417–436. MR 806008, DOI 10.1112/blms/17.5.417
M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time. I. II, Comm. Pure Appl. Math. 28 (1975), 1–47; ibid. 28 (1975), 279–301. MR 386024, DOI 10.1002/cpa.3160280102
Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606
Kiyoshi Itô, Essentials of stochastic processes, Translations of Mathematical Monographs, vol. 231, American Mathematical Society, Providence, RI, 2006. Translated from the 1957 Japanese original by Yuji Ito. MR 2239081, DOI 10.1090/mmono/231
N. Kajino, Equivalence of recurrence and Liouville property for symmetric Dirichlet forms, preprint (2010).
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
Sadao Sato, An inequality for the spectral radius of Markov processes, Kodai Math. J. 8 (1985), no. 1, 5–13. MR 776702, DOI 10.2996/kmj/1138036992
Peter Stollmann and Jürgen Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138. MR 1378151, DOI 10.1007/BF00396775
D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154, DOI 10.1007/978-1-4613-8514-1
M. Takeda, Branching Brownian motions on Riemannian manifolds: expectation of the number of branches hitting closed sets, Potential Anal. 27 (2007), no. 1, 61–72. MR 2314189, DOI 10.1007/s11118-007-9039-3
Masayoshi Takeda, Large deviations for additive functionals of symmetric stable processes, J. Theoret. Probab. 21 (2008), no. 2, 336–355. MR 2391248, DOI 10.1007/s10959-007-0111-0
Masayoshi Takeda, A large deviation principle for symmetric Markov processes with Feynman-Kac functional, J. Theoret. Probab. 24 (2011), no. 4, 1097–1129. MR 2851247, DOI 10.1007/s10959-010-0324-5
M. Takeda and Y. Tawara, A large deviation principle for symmetric Markov processes normalized by Feynman$-$Kac functionals, to appear in Osaka J. Math.
Yoshihiro Tawara, $L^p$-independence of spectral bounds of Schrödinger-type operators with non-local potentials, J. Math. Soc. Japan 62 (2010), no. 3, 767–788. MR 2648062
Liming Wu, Some notes on large deviations of Markov processes, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 3, 369–394. MR 1787093, DOI 10.1007/PL00011549