## Journal of the Mathematical Society of Japan

### Asymptotic expansion of resolvent kernels and behavior of spectral functions for symmetric stable processes

#### Abstract

We give a precise behavior of spectral functions for symmetric stable processes applying the asymptotic expansion of resolvent kernels.

#### Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 673-692.

Dates
First available in Project Euclid: 20 April 2017

https://projecteuclid.org/euclid.jmsj/1492653642

Digital Object Identifier
doi:10.2969/jmsj/06920673

#### Citation

WADA, Masaki. Asymptotic expansion of resolvent kernels and behavior of spectral functions for symmetric stable processes. J. Math. Soc. Japan 69 (2017), no. 2, 673--692. doi:10.2969/jmsj/06920673. https://projecteuclid.org/euclid.jmsj/1492653642

#### References

• R. M. Blumenthal and R. K. Getoor, Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263–273.
• M. Cranston, L. Koralov, S. Molchanov and B. Vainberg, Continuous model for homopolymers, Journal of Funct. Anal., 256 (2009), 2656–2696.
• M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter, Studies in Mathematics, 19, 2011.
• T. Kato, Perturbation theory for linear operators, Reprint of the 1980 Edition, Springer, 1995.
• M. Klaus and B. Simon, Coupling constant thresholds in nonrelativistic quantum mechanics, I, Short-range two-body case, Annals of Physics, 130 (1980), 251–281.
• V. N. Kolokoltsov, Markov processes, semigroups and generators, De Gruyter, Studies in Mathematics, 38, 2011.
• E. H. Lieb and M. Loss, Analysis, Second edition, Graduate Studies in Math., 14, American Mathematical Society, 2001.
• Y. Shiozawa, Exponential growth of the numbers of particles for branching symmetric $\alpha$-stable processes, J. Math. Soc. Japan, 60 (2008), 75–116.
• M. Takeda, Large deviations for additive functionals of symmetric stable processes, J. Theor. Probab., 21 (2008), 336–355.
• 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.
• M. Wada, Perturbation of Dirichlet forms and stability of fundamental solutions, Tohoku Math. Journal, 66 (2014), 523–537.