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Shape theorem, Lyapounov exponents, and large deviations for Brownian motion in a Poissonian potential. (English) Zbl 0814.60022
Author’s abstract: We 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 Lyapunov 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.

60F10 Large deviations
60J65 Brownian motion
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