## Kyoto Journal of Mathematics

### Diffusions with Bessel-like drifts

#### Abstract

Linear diffusions which are not far from Bessel diffusions are considered. Specifically, the regular variation of Feller’s canonical measure is interpreted in terms of the drift coefficient. As an application, the asymptotic behavior of the transition probability at large times is discussed.

#### Article information

Source
Kyoto J. Math., Volume 55, Number 4 (2015), 773-797.

Dates
Received: 21 November 2013
Revised: 4 September 2014
Accepted: 16 September 2014
First available in Project Euclid: 25 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1448460078

Digital Object Identifier
doi:10.1215/21562261-3157739

Mathematical Reviews number (MathSciNet)
MR3479309

Zentralblatt MATH identifier
1331.60157

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 34B24: Sturm-Liouville theory [See also 34Lxx]

#### Citation

Kasahara, Yuji; Kotani, Shin’ichi. Diffusions with Bessel-like drifts. Kyoto J. Math. 55 (2015), no. 4, 773--797. doi:10.1215/21562261-3157739. https://projecteuclid.org/euclid.kjm/1448460078

#### References

• [1] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl. 27, Cambridge Univ. Press, Cambridge, 1987.
• [2] A. N. Borodin and P. Salminen, Handbook of Brownian Motion—Facts and Formulae, 2nd ed., Birkhäuser, Basel, 2002.
• [3] J. M. Harrison and L. A. Shepp, On skew Brownian motion, Ann. Probab. 9 (1981), 309–313.
• [4] K. Itô, Essentials of Stochastic Processes, Transl. Math. Monogr. 231, Amer. Math. Soc., Providence, 2006.
• [5] K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Grundlehren Math. Wiss. 125, Springer, New York, 1974.
• [6] J. Karamata, Sur un mode de croissance régulière. Théorèmes fondamentaux, Bull. Soc. Math. France 61 (1953), 55–62.
• [7] Y. Kasahara, Spectral theory of generalized second order differential operators and its applications to Markov processes, Japan. J. Math. (N.S.) 1 (1975/76), 67–84.
• [8] Y. Kasahara, Spectral function of Krein’s and Kotani’s string in the class $\Gamma$, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), 173–177.
• [9] Y. Kasahara and S. Watanabe, Asymptotic behavior of spectral measures of Krein’s and Kotani’s strings, Kyoto J. Math. 50 (2010), 623–644.
• [10] S. Kotani, Krein’s strings with singular left boundary, Rep. Math. Phys. 59 (2007), 305–316.
• [11] S. Kotani, Krein’s strings whose spectral functions are of polynomial growth, Kyoto J. Math. 53 (2013), 787–814.
• [12] M. Tomisaki and M. Yamazato, Limit theorems for hitting times of 1-dimensional generalized diffusions, Nagoya Math. J. 152 (1998), 1–37.
• [13] S. Watanabe, “Generalized arc-sine laws for one-dimensional diffusion processes and random walks” in Stochastic Analysis (Ithaca, NY, 1993), Proc. Sympos. Pure Math. 57, Amer. Math. Soc., Providence, 1995, 157–172.
• [14] Y. Yano, On the occupation time on the half line of pinned diffusion processes, Publ. Res. Inst. Math. Sci. 42 (2006), 787–802.