Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 68, Number 4 (2016), 1615-1653.
Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions
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.
J. Math. Soc. Japan, Volume 68, Number 4 (2016), 1615-1653.
First available in Project Euclid: 24 October 2016
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
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