  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 the Poisson point process. This model was first considered by
Pastur (1977) 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. As an application, we also derive the second
order asymptotics of the integrated density of states of the corresponding
random Schr\"odinger operator.
