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Localisation and ageing in the parabolic Anderson model with Weibull potential

Sidorova, N; Twarowski, A; (2012) Localisation and ageing in the parabolic Anderson model with Weibull potential. Annals of Probability , 42 (4) 1666 - 1698. 10.1214/13-AOP882.

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Abstract

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.

Type:Article
Title:Localisation and ageing in the parabolic Anderson model with Weibull potential
DOI:10.1214/13-AOP882
Publisher version:http://dx.doi.org/10.1214/13-AOP882
Additional information:Published in at http://dx.doi.org/10.1214/13-AOP882 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
UCL classification:UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics

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