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2014 Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field
Artiom Fiodorov, Stephen Muirhead
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-3203

Abstract

We consider the parabolic Anderson model with Weibull potential field, for all values of the Weibull parameter. We prove that the solution is eventually localised at a single site with overwhelming probability (complete localisation) and, moreover, that the solution has exponential shape around the localisation site. We determine the localisation site explicitly, and derive limit formulae for its distance, the profile of the nearby potential field and its ageing behaviour. We also prove that the localisation site is determined locally, that is, by maximising a certain time-dependent functional that depends only on: (i) the value of the potential field in a neighbourhood of fixed radius around a site; and (ii) the distance of that site to the origin. Our results extend the class of potential field distributions for which the parabolic Anderson model is known to completely localise; previously, this had only been established in the case where the potential field distribution has sub-Gaussian tail decay, corresponding to a Weibull parameter less than two.

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Artiom Fiodorov. Stephen Muirhead. "Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-3203

Information

Accepted: 5 July 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1316.60109
MathSciNet: MR3238778
Digital Object Identifier: 10.1214/EJP.v19-3203

Subjects:
Primary: 60H25
Secondary: 35P05 , 60F10 , 82C44

Keywords: Anderson Hamiltonian , Intermittency , localisation , Parabolic Anderson model , random Schrodinger operator , spectral gap , Weibull tail

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