Brownian survival among Poissonian traps with random shapes at critical intensity

• 成果类型: Article
• 署名作者: van den Berg, M; Bolthausen, E; den Hollander, F
• 署名单位: University of Bristol; University of Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0393-4
• 发表日期: 2005
• 页码: 163-202
• 关键词: wiener sausage; asymptotics; volume; motion

摘要

In this paper we consider a standard Brownian motion in R-d, starting at 0 and observed until time t. The Brownian motion takes place in the presence of a Poisson random field of traps, whose centers have intensity nu(t) and whose shapes are drawn randomly and independently according to a probability distribution Pi, on the set of closed subsets of R-d, subject to appropriate conditions. The Brownian motion is killed as soon as it hits one of the traps. With the help of a large deviation technique developed in an earlier paper, we find the tail of the probability S-t that the Brownian motion survives up to time t when [GRAPHICS] where c is an element of (0,infinity) is a parameter. This choice of intensity corresponds to a critical scaling. We give a detailed analysis of the rate constant in the tail of S-t as a function of c, including its limiting behaviour as c --> infinity or c down arrow 0. For d >= 3, we find that there are two regimes, depending on the choice of Pi. In one of the regimes there is a collapse transition at a critical value c* is an element of (0, infinity), where the optimal survival strategy changes from being diffusive to being subdiffusive. At c*, the slope of the rate constant is discontinuous. For d = 2, there is again a collapse transition, but the rate constant is independent of Pi and its slope at c = c* is continuous.