Brownian motion in a Poissonian potential
type: Item
instance of: scholarly article
title: Brownian motion in a Poissonian potential
zbMATH Open document ID: 0794.60110
DOI: 10.1007/BF01192959
author: Alain-Sol Sznitman
publication date: 1994-08-15T00:00:00Z
review text: We study here a \(d\)-dimensional Brownian motion in a random potential \(V (\cdot, \omega)\) obtained as the sum of translations of a given fixed nonnegative shape function at the points of a Poisson cloud of constant intensity \(v\). We are interested in the large \(t\) behavior for typical cloud configurations, of the Brownian path in time \(t\) under the influence of the natural Feynman-Kac weight associated to \(V(\cdot,\omega)\). In particular, we show that the location at time \(t\) of the process tends to be concentrated near points of suitably ``low local eigenvalue'' of \(-{1 \over 2} \Delta+V (\cdot, \omega)\), which lie almost at distance \(t\) from the origin. Near these points one can find in the cloud a ``big hole'' or ``claring'' of size \(\sim \text{const} (\log t)^{1/d}\) with volume like a ball of radius \(R_ 0(d,v) (\log t)^{1/d}\).
Mathematics Subject Classification ID: 60K40
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 568998
zbMATH Keywords: Brownian motion
zbMATH Keywords: Poisson cloud of constant intensity
zbMATH Keywords: Feynman-Kac weight
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192959
OpenAlex ID: W2006505079
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 568998