Long-time tails in the parabolic Anderson model with bounded potential

Abstract

We consider the parabolic Anderson problem ${\partial }_{t}u=\kappa \Delta u+\xi u$ on $(0,\infty )\times {\mathbb{Z}}^d$ with random i.i.d. potential $\xi =(\xi (z){)}_{z\in {\mathbb{Z}}^d}$ and the initial condition $u(0,\cdot )\equiv 1$. Our main assumption is that $\esssupξ(0)=0$. Depending on the thickness of the distribution $\prob(ξ(0)\in\cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t,0)$ and the almost-sure asymptotics of $u(t,0)$ as $t\to \infty$ in terms of variational problems. As a by-product, we establish Lifshitz tails for the random Schrödinger operator $-\kappa \Delta -\xi$ at the bottom of its spectrum. In our class of $\xi$ distributions, the Lifshitz exponent ranges from $d/2$ to $\infty$; the power law is typically accompanied by lower-order corrections.

40 pages, LaTeX 2e+times, version published in Ann. Probab