Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential

We consider d-dimensional Brownian motion evolving in a scaled Poissonian potential ß-2(t)V, where ß>0 is a constant, 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 ß-2(t)V. We prove that for d2 there is a critical scale and a critical constant ßc(d)>0 such that the annealed partition sum undergoes a phase transition if ß crosses ßc(d). In d=1 this picture does not hold true, which can formally be interpreted that on the critical scale we have ßc(1)=0.