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March 2007 Geometric characterization of intermittency in the parabolic Anderson model
Jürgen Gärtner, Wolfgang König, Stanislav Molchanov
Ann. Probab. 35(2): 439-499 (March 2007). DOI: 10.1214/009117906000000764


We consider the parabolic Anderson problem tuu+ξ(x)u on ℝ+×ℤd with localized initial condition u(0, x)=δ0(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑xu(t, x) comes from a slowly increasing number of “islands” which are located far from each other. These “islands” are local regions of those high exceedances of the field ξ in a box of side length 2t log2t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.


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Jürgen Gärtner. Wolfgang König. Stanislav Molchanov. "Geometric characterization of intermittency in the parabolic Anderson model." Ann. Probab. 35 (2) 439 - 499, March 2007.


Published: March 2007
First available in Project Euclid: 30 March 2007

zbMATH: 1126.60091
MathSciNet: MR2308585
Digital Object Identifier: 10.1214/009117906000000764

Primary: 60H25 , 82C44
Secondary: 35B40 , 60F10

Keywords: heat equation with random potential , Intermittency , Parabolic Anderson problem , quenched asymptotics , random environment

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 2 • March 2007
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