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Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field

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1.	Title	Title of document	Complete localisation and exponential shape of the parabolic Anderson model with Weibull potential field
2.	Creator	Author's name, affiliation, country	Artiom Fiodorov; University College London; United Kingdom
2.	Creator	Author's name, affiliation, country	Stephen Muirhead; University College London; United Kingdom
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Parabolic Anderson model; Anderson Hamiltonian; random Schrodinger operator; localisation; intermittency; Weibull tail; spectral gap
3.	Subject	Subject classification	60H25 (Primary); 82C44; 60F10; 35P05 (Secondary)
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	University College London; Leverhulme Trust
7.	Date	(YYYY-MM-DD)	2014-07-05
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/3203
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-3203
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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