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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.

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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. We consider the case when the potential is a collection of independent identically distributed random variables with Weibull distribution 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
Additional information:The paper has changed substantially and now proves complete localisation for Weibull potentials with parameter between 0 and 2 (rather than 0 and 1). In particular, this covers the exponential distribution and beyond. This result is much stronger and requires significantly more delicate analysis, as one has to deal with the spectral gap going to zero (rather than to infinity). 27 pages
UCL classification:UCL > School of BEAMS > Faculty of Maths and Physical Sciences > Mathematics

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