Correlation structure of intermittency in the parabolic Anderson model
type: Item
instance of: scholarly article
title: Correlation structure of intermittency in the parabolic Anderson model
zbMATH Open document ID: 0951.60069
publication date: 2000-12-18T00:00:00Z
full work available at URL: http://hdl.handle.net/1887/63147
review text: This paper is concerned with the parabolic Anderson model, which corresponds to the Cauchy problem with the Anderson Hamiltonian \(\kappa\Delta+ \xi\), \(\Delta\) being the discrete Laplacian, \(\kappa\in(0,\infty)\) a diffusion constant and \(\xi=[\xi(x)\), \(x\in\mathbb{R}^d]\) an i.i.d. random field taking values in \(\mathbb{R}\). Depending on \(\kappa\) and the marginal law for \(\xi\), the model can represent various physical and chemical properties. In a paper of \textit{J. Gärtner} and \textit{S. A. Molchanov} [Commun. Math. Phys. 132, No. 3, 613-655 (1990; Zbl 0711.60055)], it is shown that if \(\xi\) is not constant, the system is intermittent. That is, qualitatively, the solution field develops sparsely distributed high peaks as \(t\) increases. Here it is shown that there is a qualitative change in the structure of the intermittent peaks when the law of \(\xi(0)\) is in the vicinity of the double exponential law: \(P(\xi(\theta) >s)= \exp (-e^{-s/ \theta})\). Indeed, this law is the only one producing a nondegenerate correlation system for the solution.
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 82C44
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 1309029
zbMATH Keywords: random media
zbMATH Keywords: intermittency
zbMATH Keywords: large deviations
zbMATH Keywords: nonlinear difference equation
author: Jürgen Gärtner
author: F. den Hollander
MaRDI profile type: Publication
OpenAlex ID: W2065805988
DOI: 10.1007/S004400050220
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1309029