Shape theorem, lyapounov exponents, and large deviations for brownian motion in a poissonian potential

Abstract

AbstractWe derive a large deviation principle governing the position of a d‐dimensional Brownian motion moving in a Poissonian potential. The derivation of this large deviation principle, and the form of the rate function rely on a result similar to the “shape theorem” of first passage percolation. This result produces certain constants which play in this multidimensional situation a similar role as the Lyapounov exponents in the one‐dimensional case. The large deviation principle enables us to investigate the transition of regime, which occurs between the small ∣h∣ and the large ∣h∣ case, for Brownian motion with a constant drift h moving in the same potential. © 1994 John Wiley & Sons, Inc.