The Annals of Probability

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

Tobias Povel

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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.

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Ann. Probab., Volume 25, Number 4 (1997), 1735-1773.

First available in Project Euclid: 7 June 2002

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Primary: 60F10: Large deviations 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Large deviations Poisson potential Brownian motion with drift


Povel, Tobias. Critical large deviations of one-dimensional annealed Brownian motion in a Poissonian potential. Ann. Probab. 25 (1997), no. 4, 1735--1773. doi:10.1214/aop/1023481109.

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