BROWNIAN-MOTION IN DENJOY DOMAINS

• 成果类型: Article
• 署名作者: BISHOP, CJ
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989795
• 发表日期: 1992
• 页码: 631-651
• 关键词: harmonic-functions; BOUNDARY; rn

摘要

A planar domain whose boundary E hes 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 (Absolute value of E = 0), then there is a set F subset-of E of full harmonic measure such that every Brownian path which exits at x is-an-element-of F hits both (- infinity, x) and (x, infinity) 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 infinity and we give an example to show dim(E) < 1 cannot be replaced by Absolute value of E = 0.