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 $\mathrm{esssup} \xi(0)=0$. Depending on the thickness of the distribution $\mathrm{Prob} (\xi(0) \in \cdot)$ close to its essential supremum, we identify both the asymptotics of the moments of $u(t, 0)$ and the almost­sure asymptotics of $u(t, 0)$ as $t \to \infty$ in terms of variational problems. As a by­product, 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.