  Consider a d-dimensional Brownian motion in a random potential defined by
attaching a nonnegative and polynomially decaying potential around Poisson
points. We introduce a repulsive interaction between the Brownian path and the
Poisson points by weighting the measure by the Feynman-Kac functional. We show
that under the weighted measure, the Brownian motion tends to localize around
the origin. We also determine the scaling limit of the path and also the limit
shape of the random potential.
