Brownian asymptotics in a Poissonian environment

Summary We consider a Brownian motion moving in a random potential obtained by translating a given fixed non negative shape function at the points of a Poisson cloud. We derive the almost sure principal long time behavior of the expectation of the natural Feynman Kac functional, which is insensitive to the detail of the shape function. We also study the situation of hard obstacles where Brownian motion is killed once it comes within distancea of a point of the cloud. The nature of the results then changes between the case whena is small or large in connection with the presence, or absence of an infinite component in the complement of the obstacles.