INTERSECTION-EQUIVALENCE OF BROWNIAN PATHS AND CERTAIN BRANCHING-PROCESSES

We show that sample paths of Brownian motion (and other stable process es) intersect the same sets as certain random Canter sets constructed by a branching process. With this approach, the classical result that two independent Brownian paths in four dimensions do not intersect red uces to the dying out of a critical branching process, and estimates d ue to Lawler (1982) for the long-range intersection probability of sev eral random walk paths, reduce to Kolmogorov's 1938 law for the lifeti me of a critical branching process. Extensions to random walks with lo ng jumps and applications to Hausdorff dimension are also derived.