The Annals of Probability

A two cities theorem for the parabolic Anderson model

Wolfgang König, Hubert Lacoin, Peter Mörters, and Nadia Sidorova

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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 $u(0,{z})={\mathbh1}_{0}({z})$, 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.

Article information

Source
Ann. Probab. Volume 37, Number 1 (2009), 347-392.

Dates
First available in Project Euclid: 17 February 2009

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

Subjects
Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 82C44: Dynamics of disordered systems (random Ising systems, etc.) 60F10: Large deviations

Keywords
Parabolic Anderson problem Anderson Hamiltonian random potential intermittency localization pinning effect heavy tail polynomial tail Pareto distribution Feynman–Kac formula

Citation

König, Wolfgang; Lacoin, Hubert; Mörters, Peter; Sidorova, Nadia. A two cities theorem for the parabolic Anderson model. The Annals of Probability 37 (2009), no. 1, 347--392. doi:10.1214/08-AOP405. http://projecteuclid.org/euclid.aop/1234881693.


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