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July 2014 Localisation and ageing in the parabolic Anderson model with Weibull potential
Nadia Sidorova, Aleksander Twarowski
Ann. Probab. 42(4): 1666-1698 (July 2014). DOI: 10.1214/13-AOP882


The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case when $\{\xi(z):z\in\mathbb{Z}^{d}\}$ is a collection of independent identically distributed random variables with Weibull distribution with parameter $0<\gamma<2$, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing.


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Nadia Sidorova. Aleksander Twarowski. "Localisation and ageing in the parabolic Anderson model with Weibull potential." Ann. Probab. 42 (4) 1666 - 1698, July 2014.


Published: July 2014
First available in Project Euclid: 3 July 2014

zbMATH: 1295.30078
MathSciNet: MR3262489
Digital Object Identifier: 10.1214/13-AOP882

Primary: 60H25
Secondary: 60F10 , 82C44

Keywords: Anderson Hamiltonian , Feynman–Kac formula , Intermittency , localisation , Parabolic Anderson model , Random potential , Weibull distribution , Weibull tail

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • July 2014
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