Journal of the Mathematical Society of Japan

Feynman-Kac penalization problem for additive functionals with jumping functions


Full-text: Open access


Takeda ([30]) solved the Feynman-Kac penalization problem for positive continuous additive functionals. We extend his result to additive functionals with jumps. We further give concrete examples of jumping functions.

Article information

J. Math. Soc. Japan, Volume 66, Number 1 (2014), 223-245.

First available in Project Euclid: 24 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 60J57: Multiplicative functionals 60H20: Stochastic integral equations

symmetric stable process Girsanov transform additive functional gaugeability Kato class Chacon-Ornstein type ergodic theorem semimartingale Doléans-Dade formula


MATSUURA, Masakuni. Feynman-Kac penalization problem for additive functionals with jumping functions. J. Math. Soc. Japan 66 (2014), no. 1, 223--245. doi:10.2969/jmsj/06610223.

Export citation


  • M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure. Appl. Math., 35 (1982), 209–273.
  • R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Pure Appl. Math., 29, Academic Press, New York, London, 1968.
  • R. F. Bass and D. A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933–2953.
  • M. Brancovan, Fonctionnelles additives spéciales des processus récurrents au sens de Harris, Z. Wahrsch. Verw. Gebiete., 47 (1979), 163–194.
  • Z.-Q. Chen, P. J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang, Absolute continuity of symmetric Markov processes, Ann. Probab., 32 (2004), 2067–2098.
  • Z.-Q. Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc., 354 (2002), 4639–4679.
  • Z.-Q. Chen, Analytic characterization of conditional gaugeability for non-local Feynman-Kac transforms, J. Funct. Anal., 202 (2003), 226–246.
  • Z.-Q. Chen, Uniform integrability of exponential martingales and spectral bounds of non-local Feynman-Kac semigroups, arXiv:1105.3020.
  • Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140 (2008), 277–317.
  • Z.-Q. Chen and R. Song, Drift transforms and Green function estimates for discontinuous processes, J. Funct. Anal., 201 (2003), 262–281.
  • Z.-Q. Chen and R. Song, General gauge and conditional gauge theorems, Ann. Probab., 30 (2002), 1313–1339.
  • K. L. Chung and Z. Zhao, From Brownian Motion to Schrödinger's Equation, Grundlehren Math. Wiss., 312, Springer-Verlag, Berlin, Heidelberg, 1995.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., 19, Walter de Gruyter, Berlin, 1994.
  • M. Fukushima, Dirichlet forms and Markov processes, Kinokuni-ya, Tokyo, 1975 (in Japanese).
  • S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
  • N. Jacob, Pseudo-Differential Operators and Markov Processes. vol.,I, Imperial College Press, London, 2001.
  • K. Kuwae and M. Takahashi, Kato class measures of symmetric Markov processes under heat kernel estimates, J. Funct. Anal., 250 (2007), 86–113.
  • K. Kuwae and M. Takahashi, Kato class functions of Markov processes under ultracontractivity, In: Potential Theory in Matsue, (eds. H. Aikawa, T. Kumagai, Y. Mizuta and N. Suzuki), Adv. Stud. Pure Math., 44, Math. Soc. Japan, Tokyo, 2006, pp.,193–202.
  • H. Kumano-go, Pseudo-differential Operators, Iwanami Shoten, Tokyo, 1974 (in Japanese).
  • L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. vol.,2, Itô Calculus, Wiley Ser. Probab. Math. Statist. Probab. Math. Statist., John Wiley & Sons Ltd., New York, 1987.
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. third edition, Grundlehren Math. Wiss., 293, Springer-Verlag, Berlin, 1999.
  • B. Roynette, P. Vallois and M. Yor, Some penalisations of the Wiener measure, Jpn. J. Math., 1 (2006), 263–290.
  • B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by normalized exponential weights. I, Studia Sci. Math. Hungar., 43 (2006), 171–246.
  • B. Roynette, P. Vallois and M. Yor, Limiting laws associated with Brownian motion perturbed by its maximum, minimum and local time. II, Studia Sci. Math. Hungar., 43 (2006), 295–360.
  • B. Roynette, P. Vallois and M. Yor, Penalisations of multidimensional Brownian motion. VI, ESAIM Probab. Stat., 13 (2009), 152–180.
  • B. Simon, Brownian motion, $L^p$ properties of Schrödinger operators and the localization of binding, J. Funct. Anal., 35 (1980), 215–229.
  • B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.), 7 (1982), 447–526.
  • E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Ann. of Math. Stud., 63, Princeton University Press, 1970.
  • P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal., 5 (1996), 109–138.
  • M. Takeda, Feynman-Kac penalisations of symmetric stable processes, Electron Commun. Probab., 15 (2010), 32–43.
  • M. Takeda, Large deviations for additive functionals of symmetric stable processes, J. Theoret. Probab., 21 (2008), 336–355.
  • Y. Tawara, $L^p$-independence of Growth Bounds of Generalized Feynman-Kac Semigroups, Doctor Thesis, Tohoku University, 2008.
  • M. Takeda and K. Tsuchida, Differentiability of spectral functions for symmetric $\alpha$-stable processes, Trans. Amer. Math. Soc., 359 (2007), 4031–4054.
  • M. Takeda and T. Uemura, Subcriticality and gaugeability for symmetric $\alpha$-stable processes, Forum Math., 16 (2004), 505–517.
  • K. Yano, Y. Yano and M. Yor, Penalising symmetric stable Lévy paths, J. Math. Soc. Japan, 61 (2009), 757–798.