Advertisement

Mathematische Annalen

, Volume 374, Issue 1–2, pp 601–652 | Cite as

\(L^p\)-independence of spectral radius for generalized Feynman–Kac semigroups

  • Zhen-Qing Chen
  • Daehong Kim
  • Kazuhiro KuwaeEmail author
Article

Abstract

Under mild conditions on measures used in the perturbation, we establish the \(L^p\)-independence of spectral radius for generalized Feynman–Kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric Markov processes. These results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for Markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized Feynman–Kac functionals developed by the second and third authors.

Mathematics Subject Classification

Primary 31C25 60J45 60J57 Secondary 35J10 60J35 60J25 

References

  1. 1.
    Aizenman, M., Simon, B.: Brownian motion and Harnack inequality for Schrödinger operators. Commun. Pure Appl. Math. 35(2), 209–273 (1982)zbMATHCrossRefGoogle Scholar
  2. 2.
    Chen, Z.-Q.: Gaugeability and conditional gaugeability. Trans. Am. Math. Soc. 354(11), 4639–4679 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Chen, Z.-Q.: Analytic characterization of conditional gaugeability for non-local Feynman–Kac transforms. J. Funct. Anal. 202(1), 226–246 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Chen, Z.-Q.: Uniform integrability of exponential martingales and spectral bounds of non-local Feynman–Kac semigroups. In: Zhang, T., Zhou, X. (eds.) Stochastic Analysis and Applications to Finance, Essays in Honor of Jia-an Yan, pp. 55–75. (2012)Google Scholar
  5. 5.
    Chen, Z.-Q.: \(L^p\)-independence of spectral bounds of generalized non-local Feynman–Kac semigroups. J. Funct. Anal. 226(6), 4120–4139 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    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)zbMATHGoogle Scholar
  9. 9.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Am. Math. Soc. 363(9), 5021–5055 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Corrigendum to “Global heat kernel estimates for symmetric jump processes” [MR2806700]. Trans. Am. Math. Soc. 367(10), 7515 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    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)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chen, Z.-Q., Kuwae, K.: On doubly Feller property. Osaka J. Math. 46(4), 909–930 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chen, Z.-Q., Song, R.: Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1), 262–281 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Chen, Z.-Q., Zhang, T.-S.: Girsanov and Feynman–Kac type transformations for symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Stat. 38(4), 475–505 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Chung, K.L., Zhao, Z.X.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995)zbMATHCrossRefGoogle Scholar
  18. 18.
    Dembo, A., Zeitouni, O.: Large deviations techniques and applications. Corrected Reprint of the Second (1998) Edition. Stochastic Modelling and Applied Probability, vol. 38. Springer, Berlin (2010)Google Scholar
  19. 19.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations. Springer, Berlin (2000)zbMATHGoogle Scholar
  21. 21.
    Fuglede, B.: The quasi topology associated with a countably subadditive set function. Ann. Inst. Fourier (Grenoble) 21(1), 123–169 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    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)zbMATHGoogle Scholar
  23. 23.
    Hempel, R., Voigt, J.: The spectrum of a Schrödinger operator in \(L^p({ R}^{\nu })\) is \(p\)-independent. Commun. Math. Phys. 104(2), 243–250 (1986)zbMATHCrossRefGoogle Scholar
  24. 24.
    Hempel, R., Voigt, J.: On the \(L^p\)-spectrum of Schrödinger operators. J. Math. Anal. Appl. 121(1), 138–159 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    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)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Jiang, R., Li, H., Zhang, H.: Heat kernel bounds on metric measure spaces and some applications. Potential Anal. 44(3), 601–627 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kim, D., Kuwae, K.: Analytic characterizations of gaugeability for generalized Feynman–Kac functionals. Trans. Am. Math. Soc. 369(7), 4545–4596 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Kim, D., Kuwae, K.: General analytic characterization of gaugeability for Feynman–Kac functionals. Math. Ann. 370(1–2), 1–37 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Kim, D., Kuwae, K.: Stability of estimates for fundamental solutions under Feynman–Kac perturbations for symmetric Markov processes, (2016). PreprintGoogle Scholar
  31. 31.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Kuwae, K.: Functional calculus for Dirichlet forms. Osaka J. Math. 35(3), 683–715 (1998)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Kuwae, K.: Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38(4), 1532–1569 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kuwae, K.: Errata: Stochastic calculus over symmetric Markov processes without time reversal. Ann. Probab. 38(4), 1532–1569 (2010). Ann. Probab. 40(6), 2705–2706 (2012)Google Scholar
  35. 35.
    Kuwae, K.: Stochastic calculus over symmetric Markov processes with time reversal. Nagoya Math. J. 220(4), 91–148 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Kuwae, K.: Irreducible decomposition for Markov processes, preprint (2017)Google Scholar
  37. 37.
    Kuwae, K., Takahashi, M.: Kato class functions of Markov processes under ultracontractivity. Potential theory in Matsue, 193–202, Adv. Stud. Pure Math. 44, Math. Soc. Japan, Tokyo (2006)Google Scholar
  38. 38.
    Kuwae, K., Takahashi, M.: Kato class measures of symmetric Markov processes under heat kernel estimates. J. Funct. Anal. 250(1), 86–113 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin (1992)zbMATHCrossRefGoogle Scholar
  40. 40.
    Simon, B.: Brownian motion, \(L^p\)-properties of Schrödinger operators and the localization of binding. J. Funct. Anal. 35(2), 215–229 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Simon, B.: Schrödinger semigroups. Bull. Am. Math. Soc. 7(3), 447–526 (1982)zbMATHCrossRefGoogle Scholar
  42. 42.
    Stollmann, P., Voigt, J.: Perturbation of Dirichlet forms by measures. Potential Anal. 5(2), 109–138 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Sturm, K-Th: Schödinger semigroups on manifolds. J. Funct. Anal. 118(2), 309–350 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Sturm, K-Th: On the \(L^p\)-spectrum of uniformly elliptic operators on Riemannian manifolds. J. Funct. Anal. 118(2), 442–453 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Takeda, M.: \(L^p\)-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)Google Scholar
  46. 46.
    Takeda, M.: Conditional gaugeability and subcriticality of generalized Schrödinger operators. J. Funct. Anal. 191(2), 343–376 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Takeda, M.: \(L^p\)-independence of spectral bounds of Schrödinger type semigroups. J. Funct. Anal. 252(2), 550–565 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Takeda, M.: \(L^p\)-independence of growth bounds of Feynman–Kac semigroups. Surveys in stochastic processes, 201–226, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich (2011)Google Scholar
  49. 49.
    Takeda, M.: A large deviation principle for symmetric Markov processes with Feynman–Kac functional. J. Theor. Probab. 24(4), 1097–1129 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Takeda, M., Tawara, Y.: \(L^p\)-independence of spectral bounds of non-local Feynman–Kac semigroups. Forum Math. 21(6), 1067–1080 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Tawara, Y.: \(L^p\)-independence of spectral bounds of Schrödinger type operators with non-local potentials. J. Math. Soc. Jpn. 62(3), 767–788 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Terkelsen, F.: Some minimax theorems. Math. Scand. 31, 405–413 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Zhao, Z.: Subcriticality and gaugeability of the Schrödinger operator. Trans. Am. Math. Soc. 334(1), 75–96 (1992)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of Mathematics and Engineering, Graduate School of Science and TechnologyKumamoto UniversityKumamotoJapan
  3. 3.Department of Applied Mathematics, Faculty of ScienceFukuoka UniversityFukuokaJapan

Personalised recommendations