What is the probability of intersecting the set of Brownian double points?

Abstract

We give potential theoretic estimates for the probability that a set $A$ contains a double point of planar Brownian motion run for unit time. Unlike the probability for $A$ to intersect the range of a Markov process, this cannot be estimated by a capacity of the set $A$. Instead, we introduce the notion of a capacity with respect to two gauge functions simultaneously. We also give a polar decomposition of $A$ into a set that never intersects the set of Brownian double points and a set for which intersection with the set of Brownian double points is the same as intersection with the Brownian path.

Published in at http://dx.doi.org/10.1214/009117907000000169 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)