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.
"Annealed Brownian motion in a heavy tailed Poissonian potential." Ann. Probab. 41 (5) 3462 - 3493, September 2013. https://doi.org/10.1214/12-AOP754