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, which are simpler and easier to treat than known formulae. The result gives two applications in probability theory. One is the formula for the Lévy 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.

On an identity for zeros of Bessel functions

“…. } has exactly n zeros with multiplicity one in C − := {z ∈ C : Re z < 0} and that the non-real zeros are complex conjugate in pairs (see [5]). We also know that for ν = n + 1 2 , n ∈ N the value K ν (z) can be represented as…”

Asymptotic Expansion of the Expected Volume of the Wiener Sausage in Even Dimensions

“…[5]). It is shown in [6,9] that the right-hand side of (2.1) can be inverted. Before giving the result on L(t), we recall several facts on modified Bessel functions.…”

“…In comparison with the odd-dimensional case, the explicit form in the even-dimensional case has an additional part which concerns the integral of some function consisting of modified Bessel functions. The formula is given in [9]. The asymptotic behavior of the mean volume of the Wiener sausage is also discussed in [9].…”

“…The formula is given in [9]. The asymptotic behavior of the mean volume of the Wiener sausage is also discussed in [9]. It is remarkable that it has the term of log t/t d−3 , which does not appear in the odd-dimensional case.…”

“…It is remarkable that it has the term of log t/t d−3 , which does not appear in the odd-dimensional case. The purpose of this paper is to give the asymptotic expansion of the expected volume of the Wiener sausage, which means the improvement of the result in [9], and the main results are described in the next section.…”