Abstract
We show that sample paths of Brownian motion (and other stable processes) intersect the same sets as certain random Cantor sets constructed by a branching process. With this approach, the classical result that two independent Brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process, and estimates due to Lawler (1982) for the long-range intersection probability of several random walk paths, reduce to Kolmogrov's 1938 law for the lifetime of a critical branching process. Extensions to random walks with long jumps and applications to Hausdorff dimension are also derived.
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