Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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

Author: Masayoshi Takeda
Journal: Proc. Amer. Math. Soc. 141 (2013), 4371-4383
MSC (2010): Primary 60F10; Secondary 60J45, 31C25
Published electronically: August 21, 2013
MathSciNet review: 3105879
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639-4679 (electronic). MR 1926893 (2003i:60127),
  • [2] 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
  • [3] Z.-Q. Chen, $ L^p$-independence of spectral bounds of generalized non-local Feynman-Kac semigroups, J.Funct. Anal. 262 (2012), 4120-4139. MR 2899989
  • [4] 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 (88k:60128)
  • [5] E. B. Davies, $ L^1$ properties of second order elliptic operators, Bull. London Math. Soc. 17 (1985), no. 5, 417-436. MR 806008 (87g:58126),
  • [6] 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 0386024 (52 #6883)
  • [7] 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 (2011k:60249)
  • [8] 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 (2007i:60001)
  • [9] N. Kajino, Equivalence of recurrence and Liouville property for symmetric Dirichlet forms, preprint (2010).
  • [10] Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452 (96a:47025)
  • [11] Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225 (2001i:00001)
  • [12] Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375 (94d:60119)
  • [13] Sadao Sato, An inequality for the spectral radius of Markov processes, Kodai Math. J. 8 (1985), no. 1, 5-13. MR 776702 (86h:60144),
  • [14] Peter Stollmann and Jürgen Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109-138. MR 1378151 (97e:47065),
  • [15] D. W. Stroock, An introduction to the theory of large deviations, Universitext, Springer-Verlag, New York, 1984. MR 755154 (86h:60067a)
  • [16] 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 (2008g:60263),
  • [17] Masayoshi Takeda, Large deviations for additive functionals of symmetric stable processes, J. Theoret. Probab. 21 (2008), no. 2, 336-355. MR 2391248 (2009k:60175),
  • [18] 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,
  • [19] M. Takeda and Y. Tawara, A large deviation principle for symmetric Markov processes normalized by Feynman$ -$Kac functionals, to appear in Osaka J. Math.
  • [20] 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 (2012a:60208)
  • [21] Liming Wu, Some notes on large deviations of Markov processes, Acta Math. Sin. (Engl. Ser.) 16 (2000), no. 3, 369-394. MR 1787093 (2001i:60048),

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60F10, 60J45, 31C25

Retrieve articles in all journals with MSC (2010): 60F10, 60J45, 31C25

Additional Information

Masayoshi Takeda
Affiliation: Mathematical Institute, Tohoku University, Aoba, Sendai, 980-8578, Japan

Keywords: Large deviation, symmetric Markov process, Dirichlet form
Received by editor(s): November 1, 2011
Received by editor(s) in revised form: February 13, 2012
Published electronically: August 21, 2013
Additional Notes: The author was supported in part by Grant-in-Aid for Scientific Research No. 22340024 (B), Japan Society for the Promotion of Science
Communicated by: Edward C. Waymire
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society