Almost sure asymptotics for the continuous parabolic Anderson model
type: Item
instance of: scholarly article
title: Almost sure asymptotics for the continuous parabolic Anderson model
zbMATH Open document ID: 0972.60056
publication date: 2001-11-07T00:00:00Z
review text: The authors study the almost sure growth of \(u(t,0)\), where \(u(t,x)\) solves the parabolic Anderson equation \[ \partial_t u(t,x)=\kappa \Delta u(t,x)+\xi(x) u(t,x),\quad u(0,x)=1,\quad s \in {\mathbb R}^d, t>0, \] where \(\kappa>0\) and \(\xi\) is either a stationary Gaussian or a shot--noise Poisson field. The main result states that under slight regularity assumptions \((1/t)\log u(t,0)=h_t-(\chi + o(1))\sqrt{h_t}\) almost surely as \(t \to \infty\), where \(\chi\) is an explicit constant and \(h_t\) solves \(L(h_t)=d \log t\), where \(L\) is the Legendre transform of the cumulant generating function of \(\xi(0)\).
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 82C44
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1563075
zbMATH Keywords: parabolic Anderson problem
zbMATH Keywords: random potential
zbMATH Keywords: intermittency
author: Jürgen Gärtner
author: W. König
author: S. Molchanov
reviewed by: Michael Scheutzow
MaRDI profile type: Publication
DOI: 10.1007/S004400000096
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1563075