Planar Brownian motion winds evenly along its trajectory

Abstract

Let $D_N$ be the set of points around which a planar Brownian motion winds at least $N$ times. We prove that the random measure on the plane with density $2\pi N{1}_{D_N}$ with respect to the Lebesgue measure converges almost surely weakly, as $N$ tends to infinity, towards the occupation measure of the Brownian motion.