Planar Brownian motion winds evenly along its trajectory

TL;DR

This paper proves that as the winding number threshold increases, the scaled winding measure of planar Brownian motion converges to its occupation measure, revealing a deep connection between winding behavior and spatial occupation.

Contribution

It establishes the almost sure weak convergence of the winding measure scaled by the winding number to the occupation measure of planar Brownian motion.

Findings

Winding measure converges to occupation measure as N increases

The convergence is almost sure and weak

Provides insight into the relationship between winding and occupation in Brownian motion

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.