## Tohoku Mathematical Journal

### Large deviation principles for generalized Feynman-Kac functionals and its applications

#### Abstract

Large deviation principles of occupation distribution for generalized Feyn-man-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.

#### Article information

Source
Tohoku Math. J. (2), Volume 68, Number 2 (2016), 161-197.

Dates
First available in Project Euclid: 17 June 2016

https://projecteuclid.org/euclid.tmj/1466172769

Digital Object Identifier
doi:10.2748/tmj/1466172769

Mathematical Reviews number (MathSciNet)
MR3514698

Zentralblatt MATH identifier
1348.31005

#### Citation

Kim, Daehong; Kuwae, Kazuhiro; Tawara, Yoshihiro. Large deviation principles for generalized Feynman-Kac functionals and its applications. Tohoku Math. J. (2) 68 (2016), no. 2, 161--197. doi:10.2748/tmj/1466172769. https://projecteuclid.org/euclid.tmj/1466172769

#### References

• R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240.
• R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc. 95 (1960), 263–273.
• 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.
• 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.
• 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.
• 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.
• 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.
• Z.-Q. Chen and K. Kuwae, On doubly Feller property, Osaka J. Math. 46, (2009), no. 4, 909–930.
• Z.-Q. Chen and R. Song, Drift transforms and Green function estimates for discontinuous processes, J. Funct. Anal. 201 (2003), no. 1, 262–281.
• 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.
• 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.
• 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.
• W. Feller, The birth and death processes as diffusion processes, J. Math. Pures Appl. (9) 38 (1959), 301–345.
• P. J. Fitzsimmons, Absolute continuity of symmetric diffusions, Ann. Probab. 25 (1997), no. 1, 230–258.
• 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.
• 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.
• 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.
• 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.
• D. Kim, Asymptotic properties for continuous and jump type's Feynman-Kac functionals, Osaka J. Math. 37 (2000), no. 1, 147–173.
• D. Kim and K. Kuwae, Analytic characterizations of gaugeability for generalized Feynman-Kac functionals, (2016), to appear in Transactions of AMS.
• 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.
• 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.
• 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.
• K. Kuwae and S. Nakao, Time changes in Dirichlet space theory, Osaka J. Math. 28 (1991), no. 4, 847–865.
• K. Kuwae, Stochastic calculus over symmetric Markov processes without time reversal, Ann. Probab. 38 (2010), no. 4 1532–1569.
• K. Kuwae, Errata Stochastic calculus over symmetric Markov processes without time reversal, Ann. Probab. 40 (2012), no. 6, 2705–2706.
• 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.
• 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.
• M. Ł. Ryznar, Estimates of Green function for relativistic $\alpha$-stable process, Potential Anal. 17 (2002), no. 1, 1–23.
• P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138.
• M. Takeda, On a large deviation for symmetric Markov processes with finite life time, Stochastics Stochastic Reports 59 (1996), no. 1–2, 143–167.
• M. Takeda, Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal. 9 (1998), no. 3, 261–291.
• 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.
• M. Takeda, Conditional gaugeability and subcriticality of generalized Schrödinger operators, J. Funct. Anal. 191 (2002), no. 2, 343–376.
• M. Takeda, $L^p$-independence of spectral bounds of Schrödinger type semigroups, J. Funct. Anal. 252 (2007), no. 2, 550–565.
• M. Takeda, A large deviation principle for symmetric Markov processes with Feynman-Kac functional, J. Theoret. Probab. 24 (2011), no. 4, 1097–1129.
• 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.
• 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.
• 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.
• M. Takeda and T.-S. Zhang, Asymptotic properties of additive functionals of Brownian motion, Ann. Probab. 25 (1997), no. 2, 940–952.
• 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.
• Y. Tawara, $L^p$-independence of growth bounds of generalized Feynman-Kac semigroups, Doctor's Degree Thesis, Mathematical Institute, Tohoku University, 2009.
• J. Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math. 34 (1997), no. 4, 933–952.
• T.-S. Zhang, Generalized Feynman-Kac semigroups, associated quadratic forms and asymptotic properties, Potential Anal. 14 (2001), no. 4, 387–408.