The Annals of Probability

Localisation and ageing in the parabolic Anderson model with Weibull potential

Nadia Sidorova and Aleksander Twarowski

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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.

Article information

Source
Ann. Probab., Volume 42, Number 4 (2014), 1666-1698.

Dates
First available in Project Euclid: 3 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1404394075

Digital Object Identifier
doi:10.1214/13-AOP882

Mathematical Reviews number (MathSciNet)
MR3262489

Zentralblatt MATH identifier
1295.30078

Subjects
Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60F10: Large deviations

Keywords
Parabolic Anderson model Anderson Hamiltonian random potential intermittency localisation Weibull tail Weibull distribution Feynman–Kac formula

Citation

Sidorova, Nadia; Twarowski, Aleksander. Localisation and ageing in the parabolic Anderson model with Weibull potential. Ann. Probab. 42 (2014), no. 4, 1666--1698. doi:10.1214/13-AOP882. https://projecteuclid.org/euclid.aop/1404394075


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