A two cities theorem for the parabolic Anderson model
Wolfgang König, Hubert Lacoin, Peter Mörters, and Nadia Sidorova
Source: Ann. Probab. Volume 37, Number 1
(2009), 347-392.
Abstract
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 , 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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Keywords: Parabolic Anderson problem; Anderson Hamiltonian; random potential; intermittency; localization; pinning effect; heavy tail; polynomial tail; Pareto distribution; Feynman–Kac formula
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1234881693
Digital Object Identifier: doi:10.1214/08-AOP405
Mathematical Reviews number (MathSciNet): MR2489168
Zentralblatt MATH identifier: 1183.60024
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