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Brownian motion in a Poissonian potential
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  • Published: December 1993

Brownian motion in a Poissonian potential

  • Alain-Sol Sznitman1 

Probability Theory and Related Fields volume 97, pages 447–477 (1993)Cite this article

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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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Authors and Affiliations

  1. Department Mathematik, ETH-Zentrum, CH-8092, Zürich, Switzerland

    Alain-Sol Sznitman

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  1. Alain-Sol Sznitman
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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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  • Received: 17 January 1992

  • Revised: 25 May 1993

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01192959

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Mathematics Subject Classification (1991)

  • 60K40
  • 82D30
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