ASYMPTOTIC BEHAVIOR OF THE OCCUPANCY DENSITY FOR OBLIQUELY REFLECTED BROWNIAN MOTION IN A HALF-PLANE AND MARTIN BOUNDARY

• 成果类型: Article
• 署名作者: Ernst, Philip A.; Franceschi, Sandro
• 署名单位: Rice University; Universite Paris Saclay
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1681
• 发表日期: 2021
• 页码: 2991-3016
• 关键词: markov-processes; random-walks

摘要

Let pi be the occupancy density of an obliquely reflected Brownian motion in the half plane and let (rho, alpha) be the polar coordinates of a point in the upper half plane. This work determines the exact asymptotic behavior of pi(rho, alpha) as rho -> infinity with alpha epsilon (0, pi). We find explicit functions a, b, c such that pi(rho, alpha) similar to(rho -> infinity) a(alpha) rho(b(alpha)) e(-c(alpha)rho). This closes an open problem first stated by Professor J. Michael Harrison in August 2013. We also compute the exact asymptotics for the tail distribution of the boundary occupancy measure and we obtain an explicit integral expression for pi. We conclude by finding the Martin boundary of the process and giving all of the corresponding harmonic functions satisfying an oblique Neumann boundary problem.