Abstract
We study here ad-dimensional Brownian motion in a random potentialV(·, ω) obtained as the sum of translations of a given fixed non negative shape function at the points of a Poisson cloud of constant intensityv. We are interested in the larget behavior for typical cloud configurations, of the Brownian path in timet under the influence of the natural Feynman-Kac weight associated toV(·, ω). In particular, we show that the location at timet of the process tends to be concentrated near points of suitably “low local eigenvalue” of −1/2Δ+V(·,ω), which lie almost at distancet from the origin. Near these points one can find in the cloud a “big hole” or “clearing” of size ≈ const(logt)1/d with volume like a ball of radiusR 0(d, v)(logt)1/d.
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Sznitman, AS. Brownian motion in a Poissonian potential. Probab. Th. Rel. Fields 97, 447–477 (1993). https://doi.org/10.1007/BF01192959
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DOI: https://doi.org/10.1007/BF01192959
Mathematics Subject Classification (1991)
- 60K40
- 82D30