Annealed Brownian motion in a heavy tailed Poissonian potential

TL;DR

This paper studies how a Brownian motion in a heavy-tailed Poissonian potential tends to localize around the origin, analyzing its scaling limit and the potential's shape under a weighted measure involving a Feynman-Kac functional.

Contribution

It introduces a model with a weighted measure for Brownian motion in a heavy-tailed Poissonian potential, revealing localization and scaling behaviors.

Findings

Brownian motion localizes around the origin under the weighted measure

Determines the scaling limit of the Brownian path

Identifies the limit shape of the random potential

Abstract

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.