Second Order Asymptotics for Brownian Motion in a Heavy Tailed Poissonian Potential

#### R. Fukushima

2011, v.17, Issue 3, 447-482

ABSTRACT

We consider the Feynman - Kac functional associated with a Brownian motion in a random potential. The potential is defined by attaching a heavy tailed positive potential around a Poisson point process. This model was first considered by Pastur [L.A. Pastur, The behavior of certain Wiener integrals as $t\rightarrow \infty$ and the density of states of Schrodinger equations with random potential. Teor. Mat. Fiz., 1977, v. 32, N1, 88-95] and the first order term of the moment asymptotics was determined. In this paper, both moment and almost sure asymptotics are determined up to the second order. We also derive the second order asymptotics of the integrated density of states of the corresponding random Schrodinger operator.

Keywords: Brownian motion,parabolic Anderson model,random media,Poissonian potential