Open Access
January 2009 A two cities theorem for the parabolic Anderson model
Wolfgang König, Hubert Lacoin, Peter Mörters, Nadia Sidorova
Ann. Probab. 37(1): 347-392 (January 2009). DOI: 10.1214/08-AOP405


The parabolic Anderson problem is the Cauchy problem for the heat equation tu(t, z)=Δu(t, z)+ξ(z)u(t, z) on (0, ∞)×ℤd with random potential (ξ(z):z∈ℤd). We consider independent and identically distributed potentials, such that the distribution function of ξ(z) converges polynomially at infinity. If u is initially localized in the origin, that is, if $u(0,{z})=𝟙_{0}({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.


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Wolfgang König. Hubert Lacoin. Peter Mörters. Nadia Sidorova. "A two cities theorem for the parabolic Anderson model." Ann. Probab. 37 (1) 347 - 392, January 2009.


Published: January 2009
First available in Project Euclid: 17 February 2009

zbMATH: 1183.60024
MathSciNet: MR2489168
Digital Object Identifier: 10.1214/08-AOP405

Primary: 60H25
Secondary: 60F10 , 82C44

Keywords: Anderson Hamiltonian , Feynman–Kac formula , heavy tail , Intermittency , Localization , Parabolic Anderson problem , Pareto distribution , pinning effect , Polynomial tail , Random potential

Rights: Copyright © 2009 Institute of Mathematical Statistics

Vol.37 • No. 1 • January 2009
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