Winding and intersection of Brownian motions

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 $\frac{\ell ({\mathbb{R}}^2)}{4{\pi }^{2}nm}$, 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 $\ell$.

35 pages