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

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.