Hitting times of Bessel processes, volume of the Wiener sausages and zeros of Macdonald functions. (English) Zbl 1359.33005

Hamana, Yuji; Matsumoto, Hiroyuki

J. Math. Soc. Japan 68, No. 4, 1615-1653 (2016).

The ratio of the modified Bessel functions of the second kind \(K_{\nu}\) , so called Macdonald functions appears not only in classical analysis but also in probability theory. The authors present a simpler form for the ratio by means of the zeros of \(K_{\nu}\) and invert Laplace transform. They obtain an explicit expression for the Levy measure of the distribution of \(r_{a,b}^{(\nu)}\) which is the first hitting time to \(b\) of a Bessel process with index \(\nu\) starting at \(a\), (\(a,b\in \mathbb R\)). The expected volume of the Wiener sausage \(\{W(t)\}_{t\geq 0}\) for the even dimensional Brownian motion and the large time asymptotics for it are given. Finally they represent the zeros of Macdonald functions \(K_{\nu}\) as the roots of a polynomial of order \(N(\nu)\) for \(| \nu|\geq 3/2\).

Reviewer: Chrysoula G. Kokologiannaki (Patras)

MSC:

33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
41A60	Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
60E07	Infinitely divisible distributions; stable distributions
60G99	Stochastic processes

Keywords:

Bessel process; Levy measure; Macdonald function; Wiener sausage