Geometric characterization of intermittency in the parabolic Anderson model

Details Geometric characterization of intermittency in the parabolic Anderson model Geometric characterization of intermittency in the parabolic Anderson model

2005-07-28

arxiv.org/abs/math/0507585v3

Annals of Probability 2007, Vol. 35, No. 2, 439-499

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000764 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117906000000764

We consider the parabolic Anderson problem $\partial_tu=\Delta u+\xi(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=\delta_0(x)$ and random i.i.d. potential $\xi$. Under the assumption that the distribution of $\xi(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to\infty$, the overwhelming contribution to the total mass $\sum_xu(t,x)$ 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 $\xi$ in a box of side length $2t\log^2t$ for which the (local) principal Dirichlet eigenvalue of the random operator $\Delta+\xi$ is close to the top of the spectrum in the box. We also prove that the shape of $\xi$ in these regions is nonrandom and that $u(t,\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.

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