Annealed survival asymptotics for Brownian motion in a scaled Poissonian potential

[摘要] 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 Feymnan-Kac weight associated to beta pi (-2)(t) V. We prove that for d greater than or equal to 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. (C) 2001 Elsevier Science B.V. All rights reserved.

[发布日期] 2001-12-01

[关键词] Brownian motion in random potentials;random Schrodinger operators;phase transition;Wiener sausage