Abstract
Let \(\{X_t\}_{t \ge 0}\) be the symmetric \(\alpha \)-stable process with generator \(H = (-\Delta )^{\alpha /2}\) for \(0 < \alpha \le 2\). For a positive Radon measure \(\mu \), we define the Schrödinger operator \(H^\mu = H - \mu \) and consider the fundamental solution of the equation \(\partial u/\partial t = - H^{\mu } u\). If \(\mu \) is critical, the behavior of the fundamental solution is different from that of the transition density function of \(\{X_t\}_{t \ge 0}\). In this paper, we give a certain ergodic-type limit theorem for fundamental solutions of critical Schrödinger operators.
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References
Albeverio, S., Blanchard, P., Ma, Z.-M.: Feynman–Kac semigroups in terms of signed smooth measures, in random partial differential equations (Ober–wolfach, 1989), Birkhäuser. Inter. Ser. Num. Math. 102, 1–31 (1991)
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Process. Appl. 108, 27–62 (2003)
Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes, Studies in Mathematics, vol. 19. De Gruyter, Berlin (2011)
Grigor’yan, A.: Heat kernels on weighted manifolds and applications. Contemp. Math. 338, 93–191 (2006)
Pinchover, Y.: Large time behavior of the heat kernel and the behavior of the Green function near criticality for non-symmetric elliptic operators. J. Funct. Anal. 104, 54–70 (1992)
Song, R.: Two-sided estimates on the density of the Feynman–Kac semigroups of stable-like processes. Electron. J. Probab. 11, 146–161 (2006)
Takeda, M.: Gaugeability for Feynman–Kac functionals with applications to symmetric \(\alpha \)-stable processes. Proc. Am. Math. Soc. 134, 2729–2738 (2006)
Takeda, M.: Gaussian bounds of heat kernels for Schrödinger operators on Riemannian manifolds. Bull. Lond. Math. Soc. 39, 85–94 (2007)
Takeda, M., Tsuchida, K.: Differentiability of spectral functions for symmetric \(\alpha \)-stable processes. Trans. Am. Math. Soc. 359, 4031–4054 (2007)
Takeda, M., Wada, M.: Large time asymptotic of Feynman–Kac functionals for symmetric stable processes. Math. Nachr. 289, 2069–2082 (2016)
Wada, M.: Perturbation of Dirichlet forms and stability of fundamental solutions. Tohoku Math. J. 66, 523–537 (2014)
Wada, M.: Feynman–Kac penalization problem for critical measures of symmetric \(\alpha \)-stable processes. Electron. Commun. Probab. 21(79), 14 (2016)
Wada, M.: Asymptotic expansion of resolvent kernel and behavior of spectral functions for symmetric stable processes. J. Math. Soc. Jpn. 69, 673–692 (2017)
Acknowledgements
This research was partly supported by Grant-in-Aid for Scientific Research (No. 17K14198), Japan Society for the Promotion of Science. The author is very grateful to Professors Masayoshi Takeda, Yuu Hariya, Kaneharu Tsuchida and Yuichi Shiozawa for their helpful comments. The author would like to thank the referees for their many helpful suggestions for this paper.
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Wada, M. Ergodic-Type Limit Theorem for Fundamental Solutions of Critical Schrödinger Operators. J Theor Probab 32, 447–459 (2019). https://doi.org/10.1007/s10959-017-0793-x
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DOI: https://doi.org/10.1007/s10959-017-0793-x