## The Annals of Probability

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

Tobias Povel

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 7 June 2002

https://projecteuclid.org/euclid.aop/1023481109

Digital Object Identifier
doi:10.1214/aop/1023481109

Mathematical Reviews number (MathSciNet)
MR1487434

Zentralblatt MATH identifier
0911.60014

#### Citation

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. https://projecteuclid.org/euclid.aop/1023481109

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