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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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

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

First available in Project Euclid: 24 October 2016

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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

Bessel process Lévy measure Macdonald function Wiener sausage


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.

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