OBLIQUELY REFLECTED BROWNIAN MOTION IN NONSMOOTH PLANAR DOMAINS

• 成果类型: Article
• 署名作者: Burdzy, Krzysztof; Chen, Zhen-Qing; Marshall, Donald; Ramanan, Kavita
• 署名单位: University of Washington; University of Washington Seattle; Brown University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1130
• 发表日期: 2017
• 页码: 2971-3037
• 关键词: poisson point-processes; MARKOV-PROCESSES; DIFFUSIONS

摘要

We construct obliquely reflected Brownian motions in all bounded simply connected planar domains, including nonsmooth domains, with general reflection vector fields on the boundary. Conformal mappings and excursion theory are our main technical tools. A key intermediate step, which may be of independent interest, is an alternative characterization of reflected Brownian motions in smooth bounded planar domains with a given field of angles of oblique reflection on the boundary in terms of a pair of quantities, namely an integrable positive harmonic function, which represents the stationary distribution of the process, and a real number that represents, in a suitable sense, the asymptotic rate of rotation of the process around a reference point in the domain. Furthermore, we also show that any obliquely reflected Brownian motion in a simply connected Jordan domain can be obtained as a suitable limit of obliquely reflected Brownian motions in smooth domains.