Intersection-Equivalence of Brownian Paths and Certain Branching Processes

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 Kolmogorov's 1938 law for the lifetime of a critical branching process. Extensions to random walks with long jumps and applications to Hausdorr dimension are also derived.