The expected volume and surface area of the Wiener sausage in odd dimensions

Abstract

We consider the Wiener sausage up to time $t$ associated with a closed ball. A formula for the expected volume of the Wiener sausage is obtained in odd dimensions. In these cases, we also find that the formula leads to the asymptotic expansion for large $t$ and each coefficient is represented by zeros of a modified Bessel function of the second kind. Moreover we obtain a formula for the expected surface area of the Wiener sausage.