Sur La Saucisse De Wiener et les Points Multiples du Mouvement Brownien

Abstract

Let $B$ be a Brownian motion with values in Euclidean space $R^d$, where $d = 2 \text{or} 3$. The Wiener sausage with radius $\varepsilon$ associated with $B$ is defined as the set of points whose distance from the path is less than $\varepsilon$. Let $B'$ be another Brownian motion with values in $R^d$, independent of $B$. The Lebesgue measure of the intersection of the Wiener sausages associated with $B$ and $B'$, suitably normalized, is shown to converge, when $\varepsilon$ goes to 0, towards the intersection local time of $B$ and $B'$, as defined by German, Horowitz and Rosen. This approximation of the intersection local time is used to prove a conjecture of Taylor, relating to the Hausdorff measure of the set of multiple points of planar Brownian motion.