Geometric characterization of intermittency in the parabolic Anderson model

Abstract

We consider the parabolic Anderson problem $\partial_tu=Δu+ξ(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=δ_0(x)$ 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 $t\to\infty$, 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 $ξ$ in a box of side length $2t\log^2t$ 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 $u(t,\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.