×

Large deviation principles for generalized Feynman-Kac functionals and its applications. (English) Zbl 1348.31005

Summary: Large deviation principles of occupation distribution for generalized Feynman-Kac functionals are presented in the framework of symmetric Markov processes having doubly Feller or strong Feller property. As a consequence, we obtain the \(L^p\)-independence of spectral radius of our generalized Feynman-Kac functionals. We also prove Fukushima’s decomposition in the strict sense for functions locally in the domain of Dirichlet form having energy measure of Dynkin class without assuming no inside killing.

MSC:

31C25 Dirichlet forms
60J45 Probabilistic potential theory
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193-240. · Zbl 0293.60071 · doi:10.24033/bsmf.1778
[2] R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263-273. · Zbl 0107.12401 · doi:10.2307/1993291
[3] Z.-Q. Chen, Uniform integrability of exponential martingales and spectral bounds of non-local Feynman-Kac semigroups, Stochastic Analysis and Applications to Finance, Essays in Honor of Jia-an Yan. Eds by T. Zhang and X. Zhou, 2012. · doi:10.1142/9789814383585_0004
[4] Z.-Q. Chen, \(L^p\)-independence of spectral bounds of generalized non-local Feynman-Kac semigroups, J. Funct. Anal. 262 (2012), no. 9, 4120-4139. · Zbl 1263.47053 · doi:10.1016/j.jfa.2012.02.011
[5] Z.-Q. Chen, P. J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang, Absolute continuity of symmetric Markov processes, Ann. Probab. 32 (2004), no. 3, 2067-2098. · Zbl 1053.60084 · doi:10.1214/009117904000000432
[6] Z.-Q. Chen, P. J. Fitzsimmons, K. Kuwae and T.-S. Zhang, Stochastic calculus for symmetric Markov processes, Ann. Probab. 36 (2008), no. 3, 931-970. · Zbl 1142.31005 · doi:10.1214/07-AOP347
[7] Z.-Q. Chen, P. J. Fitzsimmons, K. Kuwae and T.-S. Zhang, On general perturbations of symmetric Markov processes, J. Math. Pures et Appliquées 92 (2009), no. 4, 363-374. · Zbl 1176.60070 · doi:10.1016/j.matpur.2009.05.012
[8] Z.-Q. Chen and K. Kuwae, On doubly Feller property, Osaka J. Math. 46, (2009), no. 4, 909-930. · Zbl 1221.60111
[9] Z.-Q. Chen and R. Song, Drift transforms and Green function estimates for discontinuous processes, J. Funct. Anal. 201 (2003), no. 1, 262-281. · Zbl 1031.60071 · doi:10.1016/S0022-1236(03)00087-9
[10] Z.-Q. Chen and T.-S. Zhang, Girsanov and Feynman-Kac type transformations for symmetric Markov processes, Ann. Inst. H. Poincaré Probab. Statist. 38 (2002), no. 4, 475-505. · Zbl 1004.60077 · doi:10.1016/S0246-0203(01)01086-X
[11] K. L. Chung, Doubly-Feller process with multiplicative functional, Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 63-78, Progr. Probab. Statist. 12, Birkhäuser Boston, Boston, MA, 1986. · doi:10.1007/978-1-4684-6748-2_4
[12] G. De Leva, D. Kim and K. Kuwae, \(L^p\)-independence of spectral bounds of Feynman-Kac semigroups by continuous additive functionals, J. Funct. Anal. 259 (2010), no. 3, 690-730. · Zbl 1190.47048 · doi:10.1016/j.jfa.2010.01.017
[13] W. Feller, The birth and death processes as diffusion processes, J. Math. Pures Appl. (9) 38 (1959), 301-345. · Zbl 0090.10902
[14] P. J. Fitzsimmons, Absolute continuity of symmetric diffusions, Ann. Probab. 25 (1997), no. 1, 230-258. · Zbl 0873.60054 · doi:10.1214/aop/1024404287
[15] M. Fukushima, On a decomposition of additive functionals in the strict sense for a symmetric Markov process, Dirichlet forms and stochastic processes (Beijing, 1993), 155-169, de Gruyter, Berlin, 1995. · Zbl 0842.60074
[16] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes. Second revised and extended edition. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. · Zbl 0838.31001
[17] M. Fukushima and M. Takeda, A transformation of a symmetric Markov process and the Donsker-Varadhan theory, Osaka J. Math. 21 (1984), no. 2, 311-326. · Zbl 0542.60077
[18] I. W. Herbst and A. D. Sloan, Perturbation of translation invariant positivity preserving semigroups on \(L^2(\mathbb{R}^n)\), Trans. Amer. Math. Soc. 236 (1978), 325-360. · Zbl 0388.47022 · doi:10.2307/1997790
[19] D. Kim, Asymptotic properties for continuous and jump type’s Feynman-Kac functionals, Osaka J. Math. 37 (2000), no. 1, 147-173. · Zbl 0957.60034
[20] D. Kim and K. Kuwae, Analytic characterizations of gaugeability for generalized Feynman-Kac functionals, (2016), to appear in Transactions of AMS. · Zbl 1352.31005
[21] D. Kim, M. Takeda and J. Ying, Some variational formulas on additive functionals of symmetric Markov chains, Proc. Amer. Math. Soc. 130 (2002), no. 7, 2115-2123. · Zbl 0990.60021 · doi:10.1090/S0002-9939-01-06308-0
[22] K. Kuwae and M. Takahashi, Kato class functions of Markov processes under ultracontractivity, Potential theory in Matsue, 193-202, Adv. Stud. Pure Math. 44, Math. Soc. Japan, Tokyo, 2006. · Zbl 1116.31005
[23] K. Kuwae and M. Takahashi, Kato class measures of symmetric Markov processes under heat kernel estimates, J. Funct. Anal. 250 (2007), no. 1, 86-113. · Zbl 1132.47033 · doi:10.1016/j.jfa.2006.10.010
[24] K. Kuwae and S. Nakao, Time changes in Dirichlet space theory, Osaka J. Math. 28 (1991), no. 4, 847-865. · Zbl 0764.31005
[25] K. Kuwae, Stochastic calculus over symmetric Markov processes without time reversal, Ann. Probab. 38 (2010), no. 4 1532-1569. · Zbl 1206.31009 · doi:10.1214/09-AOP516
[26] K. Kuwae, Errata Stochastic calculus over symmetric Markov processes without time reversal, Ann. Probab. 40 (2012), no. 6, 2705-2706. · Zbl 1262.31011 · doi:10.1214/11-AOP700
[27] Z.-M. Ma, W. Sun and L.-F. Wang, Fukushima type decomposition for semi-Dirichlet forms, Tohoku Math. J. 68 (2016), no. 1, 1-27. · Zbl 1343.31007 · doi:10.2748/tmj/1458248859
[28] Y. Ogura and M. Tomisaki, One dimensional diffusion processes, Abstracts of Summer School of Probability Theory, Kyushu University. (2004) (in Japanese). http://www.math.kyoto-u.ac.jp/probability/sympo/PSS04.html S. C. Port, The first hitting distribution of a sphere for symmetric stable processes, Trans. Amer. Math. Soc. 135 (1969), 115-125. · doi:10.1090/S0002-9947-1969-0233426-7
[29] M. Ł. Ryznar, Estimates of Green function for relativistic \(\alpha\)-stable process, Potential Anal. 17 (2002), no. 1, 1-23. · Zbl 1004.60047 · doi:10.1023/A:1015231913916
[30] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109-138. · Zbl 0861.31004 · doi:10.1007/BF00396775
[31] M. Takeda, On a large deviation for symmetric Markov processes with finite life time, Stochastics Stochastic Reports 59 (1996), no. 1-2, 143-167. · Zbl 0868.60018 · doi:10.1080/17442509608834086
[32] M. Takeda, Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal. 9 (1998), no. 3, 261-291. · Zbl 0920.60009 · doi:10.1023/A:1008656907265
[33] M. Takeda, \(L^p\)-independence of the spectral radius of symmetric Markov semigroups, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999), 613-623, CMS Conf. Proc. 29, Amer. Math. Soc., Providence, RI, 2000.
[34] M. Takeda, Conditional gaugeability and subcriticality of generalized Schrödinger operators, J. Funct. Anal. 191 (2002), no. 2, 343-376. · Zbl 1002.60063 · doi:10.1006/jfan.2001.3864
[35] M. Takeda, \(L^p\)-independence of spectral bounds of Schrödinger type semigroups, J. Funct. Anal. 252 (2007), no. 2, 550-565. · Zbl 1130.47023 · doi:10.1016/j.jfa.2007.08.003
[36] M. Takeda, A large deviation principle for symmetric Markov processes with Feynman-Kac functional, J. Theoret. Probab. 24 (2011), no. 4, 1097-1129. · Zbl 1237.60065 · doi:10.1007/s10959-010-0324-5
[37] M. Takeda, \(L^p\)-independence of growth bounds of Feynman-Kac semigroups, Surveys in Stochastic Processes, eds. J. Blath, P. Imkeller, S. Roelly, Proceedings of the 33rd SPA Conference in Berlin, 2009. 201-226, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011. · Zbl 1235.60103 · doi:10.4171/072-1/10
[38] M. Takeda and Y. Tawara, \(L^p\)-independence of spectral bounds of non-local Feynman-Kac semigroups, Forum Math. 21 (2009), no. 6, 1067-1080. · Zbl 1187.60058 · doi:10.1515/FORUM.2009.053
[39] M. Takeda and Y. Tawara, A large deviation principle for symmetric Markov processes normalized by Feynman-Kac functionals, Osaka J. Math. 50 (2013), no. 2, 287-307. · Zbl 1272.60014
[40] M. Takeda and T.-S. Zhang, Asymptotic properties of additive functionals of Brownian motion, Ann. Probab. 25 (1997), no. 2, 940-952. · Zbl 0887.60077 · doi:10.1214/aop/1024404425
[41] Y. 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. · Zbl 1205.60138 · doi:10.2969/jmsj/06230767
[42] Y. Tawara, \(L^p\)-independence of growth bounds of generalized Feynman-Kac semigroups, Doctor’s Degree Thesis, Mathematical Institute, Tohoku University, 2009.
[43] J. Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math. 34 (1997), no. 4, 933-952. · Zbl 0903.60062
[44] T.-S. Zhang, Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties, Potential Anal. 14 (2001), no. 4, 387-408. · Zbl 0984.31006 · doi:10.1023/A:1011200525751
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.