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

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

2004-04-05

arxiv.org/abs/math/0404105v3

Annals of Probability 2007, Vol. 35, No. 6, 2044-2062

Mathematics

Probability

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

Scientific paper

10.1214/009117907000000169

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.

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