The first exit time of a Brownian motion from an unbounded convex domain

• 成果类型: Article
• 署名作者: Li, WBV
• 署名单位: University of Delaware
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1078-1096
• 关键词: small ball probabilities; gaussian markov-processes; moving boundaries; 1st-passage density; small deviations; Metric Entropy; sup-norm; cones; INEQUALITY; bounds

摘要

Consider the first exit time tau(D) of a (d + 1)-dimensional Brownian motion from an unbounded open domain D = {(x, y) is an element of Rd+1 : y > f(x), x is an element of R-d} starting at (x(0), f(x(0)) + 1) is an element of Rd+1 for some x(0) is an element of R-d, where the function f(x) on R-d is convex and f(x) --> infinity as the Euclidean norm \x\ --> infinity. Very general estimates for the asymptotics of logP(tau(D) > t) are given by using Gaussian techniques. In particular, for f(x) exp{\x\(P)}, p > 0, [GRAPHICS] where v = (d - 2)/2 and j(nu) is the smallest positive zero of the Bessel function J(nu).