Fluctuation Results for the Wiener Sausage

Abstract

Some fluctuation results are proved for the volume of the Wiener sausage associated with a $d$-dimensional Brownian motion and a compact set of positive capacity. In high dimensions, the limiting distribution is normal, whereas, if $d = 2$, it is that of a renormalized local time of self-intersections of planar Brownian motion. For $d = 2$ or 3, these limit theorems are closely linked with the renormalization results for self-intersections of Brownian paths.