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Fluctuation results for Brownian motion in a Poissonian potential. (English) Zbl 0909.60073

Summary: We consider Brownian motion in a truncated Poissonian potential conditioned to reach a remote location. If the Brownian motion starts in 0 and ends in the closed ball with center \(y\in \mathbb{R}^d\) and radius 1, then the transverse fluctuation is expected to be of order \(| y|^\xi\). We prove that \(\xi\leq 3/4\) and \(\xi\geq 1/(d+1)\), whereas for the lower bound we have to assume that the dimension \(d\geq 3\) or that we have a potential with lower bound \(\lambda>0\). As a second result we prove, in dimension \(d=2\), that \(\chi\geq 1/8\), where \(\chi\) is the critical exponent for the fluctuation for certain naturally defined random distance functions.

MSC:

60J65 Brownian motion
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