Journal of the Mathematical Society of Japan

Lp-independence of spectral bounds of Schrödinger-type operators with non-local potentials

Yoshihiro TAWARA

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Abstract

We establish a necessary and sufficient condition for spectral bounds of a non-local Feynman-Kac semigroup being Lp-independent. This result is an extension of that in [24] to more general symmetric Markov processes; in [24], we only treated a symmetric stable process on Rd. For example, we consider a symmetric stable process on the hyperbolic space, the jump process generated by the fractional power of the Laplace-Beltrami operator, and prove that by adding a non-local potential, the associated Feynman-Kac semigroup satisfies the Lp-independence.

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 3 (2010), 767-788.

Dates
First available in Project Euclid: 30 July 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1280496819

Digital Object Identifier
doi:10.2969/jmsj/06230767

Mathematical Reviews number (MathSciNet)
MR2648062

Zentralblatt MATH identifier
1205.60138

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J75: Jump processes

Keywords
Feynman-Kac formula Lp-independence Dirichlet form Markov process

Citation

TAWARA, Yoshihiro. L p -independence of spectral bounds of Schrödinger-type operators with non-local potentials. J. Math. Soc. Japan 62 (2010), no. 3, 767--788. doi:10.2969/jmsj/06230767. https://projecteuclid.org/euclid.jmsj/1280496819


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References

  • S. Albeverio, P. Blanchard and Z.-M. Ma, Feynman-Kac semigroups in terms of signed smooth measures, In: Random Partial Differential Equations, Internat. Ser. Numer. Math., 102, Birkhäuser, Boston, 1991, pp.,1–31.
  • S. Albeverio and Z.-M. Ma, Perturbation of Dirichlet forms – lower semiboundedness, closability, and form cores, J. Funct. Anal., 99 (1991), 332–356.
  • S. Albeverio and Z.-M. Ma, Additive functionals, nowhere Radon and Kato class smooth measures associated with Dirichlet forms, Osaka J. Math., 29 (1992), 247–265.
  • R. Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France, 102 (1974), 193–240.
  • R. Bass and D. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., 354 (2002), 2933–2953.
  • Z.-Q. Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc., 354 (2002), 4639–4679.
  • Z.-Q. Chen and R. Song, Conditional gauge theorem for non-local Feynman-Kac transforms, Probab. Theory Relat. Fields, 125 (2003), 45–72.
  • K. L. Chung, Doubly Feller process with multiplicative functional, Seminar on Stochastic Processes, 1985, Birkhäuser, Boston, 1986, pp.,63–78.
  • E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter, Berlin, 1994.
  • A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135–249.
  • A. Grigor'yan and M. Noguchi, The heat kernel on hyperbolic space, Bull. London Math. Soc., 30 (1998), 643–650.
  • S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992.
  • D. Kim, Asymptotic properties for continuous and jump type's Feynman-Kac functionals, Osaka J. Math., 37 (2000), 147–173.
  • I. McGillivray, A recurrence condition for some subordinated strongly local Dirichlet forms, Forum Math., 9 (1997), 229–246.
  • H. Ôkura, Recurrence and transience criteria for subordinated symmetric Markov processes, Forum Math., 14 (2002), 121–146.
  • M. Sharpe, General Theory of Markov Processes, Pure and Applied Mathematics, 133, Academic Press, 1988.
  • B. Simon, Brownian motion, ${L}^{p}$ properties of Schrödinger semigroups and the localization of binding, J. Funct. Anal., 35 (1982), 215–229.
  • P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal., 5 (1996), 109–138.
  • K.-Th. Sturm, On the ${L}^{p}$-spectrum of uniformly elliptic operators on Riemannian manifolds, J. Funct. Anal., 118 (1993), 442–453.
  • K.-Th. Sturm, Schrödinger semigroups on manifolds, J. Funct. Anal., 118 (1993), 309–350.
  • M. Takeda, Asymptotic properties of generalized Feynman-Kac functionals, Potential Anal., 9 (1998), 261–291.
  • M. Takeda, ${L}^{p}$-independence of spectral bounds of Schrödinger type semigroups, J. Funct. Anal., 252 (2007), 550–565.
  • M. Takeda and Y. Tawara, ${L}^{p}$-independence of spectral bounds of non-local Feynman-Kac semigroups, Forum Math., 21 (2009), 1067–1080.
  • J. Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math., 34 (1997), 933–952.
  • K. Yosida, Functinal Analysis, 6th ed., Springer-Verlag, Berlin, 1980.