Open Access
April, 1992 Brownian Motion in Denjoy Domains
Christopher J. Bishop
Ann. Probab. 20(2): 631-651 (April, 1992). DOI: 10.1214/aop/1176989795


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 $|E| = 0$.


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Christopher J. Bishop. "Brownian Motion in Denjoy Domains." Ann. Probab. 20 (2) 631 - 651, April, 1992.


Published: April, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0753.60074
MathSciNet: MR1159563
Digital Object Identifier: 10.1214/aop/1176989795

Primary: 60J65
Secondary: 60J50

Keywords: Brownian motion , Cauchy process , Denjoy domain , Hausdorff dimension , Martin boundary

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 2 • April, 1992
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