Geometric characterization of intermittency in the parabolic Anderson model

Abstract

We consider the parabolic Anderson problem ${\partial }_{t}u=\Delta u+\xi (x)u$ 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 ${\sum }_{x}u(t,x)$ 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 $2t{\log }^{2}t$ 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.

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

References in corpus (1)

• The universality classes in the parabolic Anderson model

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