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 of a Euclidean space 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 dimension greater or equal to 3.

Keywords

• Minkowski content
• Kneser function
• Brownian motion
• Wiener sausage