Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential

We consider d-dimensional Brownian motion evolving in a scaled Poissonian potential [beta][phi]-2(t)V, where [beta]>0 is a constant, [phi] is the scaling function which typically tends to infinity, and V is obtained by translating a fixed non-negative compactly supported shape function to all the particles of a d-dimensional Poissonian point process. We are interested in the large t behavior of the annealed partition sum of Brownian motion up to time t under the influence of the natural Feynman-Kac weight associated to [beta][phi]-2(t)V. We prove that for d[greater-or-equal, slanted]2 there is a critical scale [phi] and a critical constant [beta]c(d)>0 such that the annealed partition sum undergoes a phase transition if [beta] crosses [beta]c(d). In d=1 this picture does not hold true, which can formally be interpreted that on the critical scale [phi] we have [beta]c(1)=0.