Brownian survival among Poissonian traps with random shapes at critical intensity

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 ν(t) and whose shapes are drawn randomly and independently according to a probability distribution &UPi;, 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 ∈ (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 S-t 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 &UPi;. In one of the regimes there is a collapse transition at a critical value c* ∈ (0, ∞), 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 &UPi; and its slope at c = c* is continuous.