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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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

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

First available in Project Euclid: 17 February 2009

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Zentralblatt MATH identifier

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

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


König, Wolfgang; Lacoin, Hubert; Mörters, Peter; Sidorova, Nadia. A two cities theorem for the parabolic Anderson model. Ann. Probab. 37 (2009), no. 1, 347--392. doi:10.1214/08-AOP405.

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