Superdiffusive behavior of two-dimensional Brownian motion in a Poissonian potential

Abstract

We consider $d$-dimensional Brownian motion in a truncated Poissonian potential conditioned to reach a remote location. If Brownian motion starts at the origin and ends in an hyperplane at distance $L$ from the origin, the transverse fluctuation of the path is expected to be of order $L^{\xi}$ We are interested in a lower bound for $\xi$. We first show that $\xi \geq 1/2$ in dimensions $d \geq 2$ and then we prove superdiffusive behavior for $d = 2$, resulting in $\xi \geq 3/5$.