Geometric Properties of 2-dimensional Brownian Paths

Abstract

Let A be the set of all points of the plane C, visited by two-dimensional Brownian motion before time 1. With probability 1, all points of A are "twist points" except a set of harmonic measure zero. "Twist points" may be continuously approached in [the set that contains all those elements of complex numbers that are not in] A only along a special spiral. Although negligible in the sense of harmonic measure, various classes of "cone points" are dense in A, with probability 1. "Cone points" may be approached in [the set that contains all those elements of complex numbers that are not in] A within suitable wedges.