Geometric properties of 2-dimensional Brownian paths

Abstract

Let A be the set of all points of the plane ℂ, visited by 2-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 ℂ\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 ℂ\A within suitable wedges. © 1989 Springer-Verlag.