Winding and intersection of Brownian motions

TL;DR

This paper investigates the geometric properties of points around which two independent Brownian motions wind multiple times, revealing asymptotic behaviors of the set's area and its relation to the intersection measure of the trajectories.

Contribution

It establishes the asymptotic equivalence of the area of winding points to the intersection measure, providing new insights into the geometric structure of Brownian motion intersections.

Findings

The area of winding points is asymptotically equivalent to a constant times the intersection measure.

The scaled Lebesgue measure of winding points converges weakly to the intersection measure.

Results hold in both $L^p$ and almost sure senses.

Abstract

We study the set of points $\mathcal{D}_{n,m}$ around which two independent Brownian motions wind at least $n$ (resp. $m$) times. We prove that its area is asymptotically equivalent, in $L^p$ and almost surely, to $\frac{\ell(\mathbb{R}^2)}{4\pi^2 n m}$, where $\ell$ is the intersection measure of the two trajectories. We also prove that the properly scaled Lebesgue measure carried by $\mathcal{D}_{n,m}$ converges almost surely weakly toward $\ell$.