LIMIT-THEOREMS FOR THE FRONTIER OF A ONE-DIMENSIONAL BRANCHING DIFFUSION

• 成果类型: Article
• 署名作者: LALLEY, S; SELLKE, T
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989693
• 发表日期: 1992
• 页码: 1310-1340
• 关键词: brownian-motion

摘要

Let R(t) be the position of the rightmost particle at time t in a time-homogeneous one-dimensional branching diffusion process. Let gamma(alpha, t) be the alpha-th quantile of R(t) under P0, where P(x) denotes the probability measure of the branching diffusion process starting with a single particle at position x. We show that gamma(alpha, t) is a limiting quantile of R(t) under P(x) in the sense that lim(t-->infinity) P(x){R(t) less-than-or-equal-to gamma(alpha, t)} exists for all alpha is-an-element-of (0, 1) and all x is-an-element-of R. If the underlying diffusion is recurrent, we show that, after an appropriate rescaling of space, the P(x) distribution of R(t) - t converges weakly to a nontrivial limiting distribution w(x).