Osaka Journal of Mathematics

Tauberian theorem for harmonic mean of Stieltjes transforms and its applications to linear diffusions

Yuji Kasahara and Shin'ichi Kotani

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When two Radon measures on the half line are given, the harmonic mean of their Stieltjes transforms is again the Stieltjes transform of a Radon measure. We study the relationship between the asymptotic behavior of the resulting measure and those of the original ones. The problem comes from the spectral theory of second--order differential operators and the results are applied to linear diffusions neither boundaries of which is regular.

Article information

Osaka J. Math., Volume 53, Number 1 (2016), 221-251.

First available in Project Euclid: 19 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 40E05: Tauberian theorems, general 34B24: Sturm-Liouville theory [See also 34Lxx]


Kasahara, Yuji; Kotani, Shin'ichi. Tauberian theorem for harmonic mean of Stieltjes transforms and its applications to linear diffusions. Osaka J. Math. 53 (2016), no. 1, 221--251.

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  • N.H. Bingham, C.M. Goldie and J.L. Teugels: Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge Univ. Press, Cambridge, 1987.
  • A.N. Borodin and P. Salminen: Handbook of Brownian Motion–-Facts and Formulae, second edition, Probability and its Applications, Birkhäuser, Basel, 2002.
  • K. Itô: Essentials of Stochastic Processes, Translations of Mathematical Monographs 231, Amer. Math. Soc., Providence, RI, 2006.
  • K. Itô and H.P. McKean, Jr.: Diffusion Processes and Their Sample Paths, Springer, Berlin, 1974.
  • Y. Kasahara: Asymptotic behavior of the transition density of an ergodic linear diffusion, Publ. Res. Inst. Math. Sci. 48 (2012), 565–578.
  • Y. Kasahara and S. Kotani: Diffusions with Bessel-like drifts, preprint.
  • Y. Kasahara and S. Watanabe: Asymptotic behavior of spectral measures of Krein's and Kotani's strings, Kyoto J. Math. 50 (2010), 623–644.
  • S. Kotani and S. Watanabe: Kreĭ n's spectral theory of strings and generalized diffusion processes; in Functional Analysis in Markov Processes (Katata/Kyoto, 1981), Lecture Notes in Math. 923, Springer, Berlin, 1982, 235–259.
  • Y. Yano: On the occupation time on the half line of pinned diffusion processes, Publ. Res. Inst. Math. Sci. 42 (2006), 787–802.