Brownian survival among Poissonian traps with random shapes at critical intensity

Abstract

In this paper we consider a standard Brownian motion in Kd, 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 vt and whose shapes are drawn randomly and independently according to a probability distribution n, on the set of closed subsets of Rd, 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, that the Brownian motion survives up to time / when where c e (0. ∞) 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 St as a function of c, including its limiting behaviour as c → ∞ or c ↓ 0. For d ≥ 3, we find that there are two regimes, depending on the choice of n. In one of the regimes there is a collapse transition at a critical value c* e (0, oo), 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 n and its slope at c = c* is continuous.