Journal of Theoretical Probability

, Volume 32, Issue 1, pp 447–459 | Cite as

Ergodic-Type Limit Theorem for Fundamental Solutions of Critical Schrödinger Operators

  • Masaki WadaEmail author


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.


Feynman–Kac functional Symmetric stable process Schrödinger form Criticality Dirichlet form Heat kernel 

Mathematics Subject Classification (2010)

60J45 60J40 35J10 



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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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Human Development and CultureFukushima UniversityFukushimaJapan

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