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

TL;DR

This paper derives simplified formulas for Macdonald functions, enabling new probabilistic results on Bessel process hitting times and Wiener sausage volumes, and provides algebraic equations for Macdonald function zeros.

Contribution

It introduces simpler formulas for Macdonald functions, facilitating explicit calculations of Bessel process hitting times, Wiener sausage volumes, and zeros of Macdonald functions.

Findings

Formula for the Lévy measure of Bessel process hitting times

Explicit expected volume of Wiener sausages in even dimensions

Algebraic equations for zeros of Macdonald functions

Abstract

We derive formulae for some ratios of the Macdonald functions, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the L{\'e}vy measure of the distribution of the first hitting time of a Bessel process and the other is an explicit form for the expected volume of the Wiener sausage for an even dimensional Brownian motion. Moreover, the result enables us to write down the algebraic equations whose roots are the zeros of Macdonald functions.