Journal of the Mathematical Society of Japan

Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions

Yuji HAMANA and Hiroyuki MATSUMOTO

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Abstract

We derive formulae for some ratios of the Macdonald functions by using their zeros, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory and one in classical analysis. We show a formula for the Lévy measure of the distribution of the first hitting time of a Bessel process and an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. In addition, we show that the complex zeros of the Macdonald functions are the roots of some algebraic equations with real coefficients.

Article information

Source
J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1615-1653.

Dates
First available in Project Euclid: 24 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1477327227

Digital Object Identifier
doi:10.2969/jmsj/06841615

Mathematical Reviews number (MathSciNet)
MR3564445

Zentralblatt MATH identifier
1359.33005

Subjects
Primary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Secondary: 41A60: Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 60E07: Infinitely divisible distributions; stable distributions 60G99: None of the above, but in this section

Keywords
Bessel process Lévy measure Macdonald function Wiener sausage

Citation

HAMANA, Yuji; MATSUMOTO, Hiroyuki. Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions. J. Math. Soc. Japan 68 (2016), no. 4, 1615--1653. doi:10.2969/jmsj/06841615. https://projecteuclid.org/euclid.jmsj/1477327227


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References

  • R. K. Getoor, Some asymptotic formulas involving capacity, Z. Wahrsch. Verw. Gebiete, 4 (1965), 248–252.
  • R. K. Getoor and M. J. Sharpe, Excursions of Brownian motion and Bessel processes, Z. Wahrsch. Verw. Gebiete, 47 (1979), 83–106.
  • I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed., Academic Press, Amsterdam, 2007.
  • L. C. Grove, Algebra, Academic Press, New York, 1983.
  • Y. Hamana, On the expected volume of the Wiener sausage, J. Math. Soc. Japan, 62 (2010), 1113–1136.
  • Y. Hamana, The expected volume and surface area of the Wiener sausage in odd dimensions, Osaka J. Math., 49 (2012), 853–868.
  • Y. Hamana and H. Matsumoto, The probability densities of the first hitting times of Bessel processes, J. Math-for-Ind., 4B (2012), 91–95.
  • Y. Hamana and H. Matsumoto, The probability distributions of the first hitting times of Bessel processes, Trans. Amer. Math. Soc., 365 (2013), 5237–5257.
  • Y. Hamana, H. Matsumoto and T. Shirai, On the zeros of the Macdonald functions, preprint, available at arXiv:1302.5154 [math.CA].
  • M. G. H. Ismail, Integral representations and complete monotonicity of various quotients of Bessel functions, Canad. J. Math., 29 (1977), 1198–1207.
  • M. G. H. Ismail and D. H. Kelker, Special functions, Stieltjes transforms and infinite divisibility, SIAM J. Math. Anal., 10 (1989), 884–901.
  • K. Itô and H. P. McKean, Diffusion Processes and Their Sample Paths, Springer-Verlag, Berlin-New York, 1974.
  • J. T. Kent, Some probabilistic properties of Bessel functions, Ann. Probab., 6 (1978), 760–770.
  • M. K. Kerimov and S. L. Skorokhodov, Calculation of complex zeros of a modified Bessel function of the second kind and its derivatives, Zh. Vychisl. Mat. i Mat. Fiz., 24 (1984), 1150–1163.
  • N. N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972.
  • J. -F. Le Gall, Sur une conjecture de M. Kac, Probab. Theory Related Fields, 78 (1988), 389–402.
  • J. -F. Le Gall, Wiener sausage and self-intersection local times, J. Funct. Anal., 88 (1990), 299–341.
  • S. C. Port, Asymptotic expansions for the expected volume of a stable sausage, Ann. Probab., 18 (1990), 492–523.
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Springer-Verlag, Berlin, 1999.
  • F. Spitzer, Electrostatic capacity, heat flow and Brownian motion, Z. Wahrsch. Verw. Gebiete, 3 (1964), 110–121.
  • G. N. Watson, A Treatise on the Theory of Bessel Functions, Reprinted of 2nd ed., Cambridge University Press, Cambridge, 1995.
  • M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes, Nagoya Math. J., 119 (1990), 143–172.