The Annals of Probability

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

Jean-Francois Le Gall

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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.

Article information

Ann. Probab., Volume 14, Number 4 (1986), 1219-1244.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60G17: Sample path properties 60J55: Local time and additive functionals

Wiener sausage intersection local time Hausdorff measure multiple points


Gall, Jean-Francois Le. Sur La Saucisse De Wiener et les Points Multiples du Mouvement Brownien. Ann. Probab. 14 (1986), no. 4, 1219--1244. doi:10.1214/aop/1176992364.

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