Brownian Motion in Denjoy Domains

Abstract

A planar domain whose boundary $E$ lies in the real line is called a Denjoy domain. In this article we consider some geometric properties of Brownian motion in such a domain. The first result is that if $E$ has zero length $(|E| = 0)$, then there is a set $F \subset E$ of full harmonic measure such that every Brownian path which exits at $x \in F$ hits both $(-\infty, x)$ and $(x,\infty)$ with probability 1, verifying a conjecture of Burdzy. Next we show that if $\dim(E) < 1$, then almost every Brownian path forms infinitely many loops separating its exit point from $\infty$ and we give an example to show $\dim(E) < 1$ cannot be replaced by $|E| = 0$.