Large deviations for Brownian motion in a random scenery

Abstract

We prove large deviations principles in large time, for the Brownian occupation time in random scenery ${{\frac{{1}}{{t}} \int_0^t \xi(B_s) \, ds}}$ . The random field is constant on the elements of a partition of ℝ d into unit cubes. These random constants, say ${{{{\left\lbrace{{ \xi(j), j \in \mathbb{{Z}}^d}}\right\rbrace}} }}$ consist of i.i.d. bounded variables, independent of the Brownian motion {B s ,s≥0}. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched'' and ``annealed'' settings.