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General analytic characterization of gaugeability for Feynman-Kac functionals

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Abstract

We give a general analytic characterization of the gaugeability for generalized Feynman-Kac functionals involving continous additive functionals locally of zero energy generalizing the earlier work on the analytic characterization of the gaugeability of Feynman-Kac functional for absorbing Brownian motion on a bounded domain given by Aizenman-Simon (Commun Pure Appl Math 35(2), 209–273, 1982). The result also extends the previous known results on the analytic characterization of the gaugeability for generalized Feynman-Kac functionals in the framework of symmetric Markov processes satisfying irreducibility condition and absolute continuity condition.

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References

  1. Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure. Appl. Math. 35(2), 209–273 (1982)

    Article  MATH  Google Scholar 

  2. Chen, Z.-Q.: Gaugeability and conditional gaugeability. Trans. Am. Math. Soc. 354(11), 4639–4679 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Z.-Q.: Analytic characterization of conditional gaugeability for non-local Feynman-Kac transforms. J. Funct. Anal. 202(1), 226–246 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, Z.-Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.-S.: Stochastic calculus for symmetric Markov processes. Ann. Probab. 36(3), 931–970 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Z.-Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.-S.: Perturbation of symmetric Markov processes. Probab. Theory Relat. Fields 140(1–2), 239–275 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Chen, Z.-Q., Fitzsimmons, P.J., Kuwae, K., Zhang, T.-S.: On general perturbations of symmetric Markov processes. J. Math. Pures et Appliquées 92(4), 363–374 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  8. Chen, Z.-Q., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Am. Math. Soc. 363(9), 5021–5055 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Z.-Q., Kim, P., Kumagai, T.: Global heat Kernel estimates for symmetric jump processes. Trans. Am. Math. Soc. 367(10), 7515 (2015)

    Article  MATH  Google Scholar 

  10. Chen, Z.-Q., Kim, P., Song, R.: Global heat Kernel estimates for relativistic stable processes in exterior open sets. J. Funct. Anal. 263(2), 448–475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, Z.-Q., Kim, P., Song, R.: Global heat Kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36(2), 235–261 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen, Z.-Q., Kumagai, T.: Heat Kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140(1–2), 277–317 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Chen, Z.-Q., Song, R.: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1), 262–281 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chung, K.L.: Doubly-Feller process with multiplicative functional, Seminar on stochastic processes, 1985 (Gainesville, Fla., 1985), 63–78, Progr. Probab. Statist., vol. 12, Birkhäuser Boston, Boston (1986)

  15. Chung, K.L.: Greenian bounds for Markov processes. Potential Anal. 1(1), 83–92 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 312. Springer, Berlin (1995)

    Google Scholar 

  17. De Leva, G., Kim, D., Kuwae, K.: \(L^p\)-independence of spectral bounds of Feynman-Kac semigroups by continuous additive functionals. J. Funct. Anal. 259(3), 690–730 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Feller, W.: The birth and death processes as diffusion processes. J. Math. Pures Appl. 38(9), 301–345 (1959)

    MathSciNet  MATH  Google Scholar 

  19. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, Second revised and extended edition. de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter & Co., Berlin (2011)

  20. Fuglede, B.: The quasi topology associated with a countably subadditive set function. Ann. Inst. Fourier (Grenoble) 21(fasc. 1), 123–169 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  21. Herbst, I.W., Sloan, A.D.: Perturbation of translation invariant positivity preserving semigroups on \(L^2({\mathbb{R}}^n)\). Trans. Am. Math. Soc. 236, 325–360 (1978)

    MathSciNet  MATH  Google Scholar 

  22. He, S.W., Wang, J.G., Yan, J.A.: Semimartingale Theory and Stochastic Calculus. Science Press, Beijing (1992)

    MATH  Google Scholar 

  23. Itô, K., Jr. McKean, H.P.: Diffusion processes and their sample paths, Second printing, corrected. Die Grundlehren der mathematischen Wissenschaften, Band, vol. 125. Springer, Berlin (1974)

  24. Kim, D., Kurniawaty, M., Kuwae, K.: A refinement of analytic characterizations of gaugeability for generalized Feynman-Kac functionals. Ill. J. Math. 59(3), 717–771 (2015)

    MathSciNet  MATH  Google Scholar 

  25. Kim, D., Kuwae, K.: Analytic characterizations of gaugeability for generalized Feynman-Kac functionals. Trans. Am. Math. Soc. http://www.ams.org/journals/tran/0000-000-00/S0002-9947-2016-06702-1/ (2016)

  26. Kim, D., Kuwae, K., Tawara, Y.: Large deviation principle for generalized Feynman-Kac functionals and its applications. Tohoku Math. J. 68(2), 161–197 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kurniawaty, M., Kuwae, K., Tsuchida, K.: On doubly Feller property of resolvent, preprint (2015) (to appear in Kyoto J. Math)

  28. Kuwae, K., Takahashi, M.: Kato class measures of symmetric Markov processes under heat Kernel estimates. J. Funct. Anal. 250(1), 86–113 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer Universitext, Berlin (1992)

    Book  MATH  Google Scholar 

  30. Ryznar, M.Ł.: Estimates of Green function for relativistic \(\alpha \)-stable process. Potential Anal. 17(1), 1–23 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5(2), 109–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Takeda, M.: Conditional gaugeability and subcriticality of generalized Schrödinger operators. J. Funct. Anal. 191(2), 343–376 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Takeda, M., Uemura, T.: Subcriticality and gaugeability for symmetric \(\alpha \)-stable processes. Forum Math. 16(4), 505–517 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhao, Z.: Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116(2), 309–334 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhao, Z.: A probabilistic principle and generalized Schrödinger perturbation. J. Funct. Anal. 101(1), 162–176 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Professor Masayoshi Takeda for his valuable comments to the draft of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Engineering, Graduate School of Science and Technology, Kumamoto University, Kumamoto, 860-8555, Japan

    Daehong Kim

  2. Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka, 814-0180, Japan

    Kazuhiro Kuwae

Authors
  1. Daehong Kim
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  2. Kazuhiro Kuwae
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Correspondence to Kazuhiro Kuwae.

Additional information

Communicated by Y. Giga.

Dedicated to the seventieth birthday of Professor Yoichi Oshima.

The first named author was partially supported by a Grant-in-Aid for Scientific Research (C) No. 23540147 from Japan Society for the Promotion of Science. The second named author was partially supported by a Grant-in-Aid for Scientific Research (B) No. 22340036 from Japan Society for the Promotion of Science.

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Kim, D., Kuwae, K. General analytic characterization of gaugeability for Feynman-Kac functionals. Math. Ann. 370, 1–37 (2018). https://doi.org/10.1007/s00208-017-1516-4

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  • DOI: https://doi.org/10.1007/s00208-017-1516-4

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