Summary
We study the behavior of ad dimensional Brownian motion in a soft repulsive Poissonian potential over long time intervals [0,t]. We introduce certaint and configuration dependent scales, which grow almost linearly witht. For typical configurations with probability tending to 1 ast goes to ∞, the size of displacements of the process is bounded above by these scales, (confinement effect). The proof involves calculations beyond “leading order”. To this end we use a coarse grained picture of the environment (method of enlargement of obstacles) and of the path (a backbone of excursions between clearings and forest parts in the environment). These coarse grained pictures are also used in the sequel [11] to the present article, when proving the pinning effect.
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