Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential

Abstract

We derive a large deviation principle for the position at large times $t$ of a one-dimensional annealed Brownian motion in a Poissonian potential in the critical spatial scale $t^{1/3}$. Here “annealed” means that averages are taken with respect to both the path and environment measures. In contrast to the $d$-dimensional case for $d \geq 2$ in the critical scale $t^{d/(d+2)}$ as treated by Sznitman, the rate function which measures the large deviations exhibits three different regimes. These regimes depend on the position of the path at time $t$. Our large deviation principle has a natural application to the study of a one-dimensional annealed Brownian motion with a constant drift in a Poissonian potential.