On the expected volume of intersection of independent Wiener sausages and the asymptotic behaviour of some related integrals

Let K be a compact, non-polar set in Euclidean space Rm(m⩾3) and let TK be the first hitting time of K by a Brownian motion. We obtain the leading asymptotic behaviour as t→∞ of ∫Rmdx(Px[TK0 is a constant.