Almost sure asymptotic behaviour of the r-neighbourhood surface area of Brownian paths

Abstract

We show that whenever the q-dimensional Minkowski content of a subset A ⊂ ℝd exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ℝd, d ⩾ 3.