Brownian survival among Poissonian traps with random shapes at critical intensity

Abstract

In this paper we consider a standard Brownian motion in ℝd, 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 νt and whose shapes are drawn randomly and independently according to a probability distribution Π, 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 St that the Brownian motion survives up to time t when