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

TL;DR

This paper analyzes the long-time behavior of solutions to the parabolic Anderson model with bounded random potential, revealing asymptotics of moments and almost-sure growth, and establishing Lifshitz tails for the associated Schrödinger operator.

Contribution

It identifies the asymptotics of moments and almost-sure behavior of the solution based on potential distribution, and establishes Lifshitz tail behavior at the spectrum's bottom.

Findings

Asymptotics of moments of u(t,0) are characterized by variational problems.

Almost-sure asymptotics of u(t,0) are determined for large t.

Lifshitz tail exponents range from d/2 to infinity, with power-law behavior and corrections.

Abstract

We consider the parabolic Anderson problem $\partial_t u=\kappa\Delta u+\xi u$ on $(0,\infty)\times \Z^d$ with random i.i.d. potential $\xi=(\xi(z))_{z\in\Z^d}$ and the initial condition $u(0,\cdot)\equiv1$. Our main assumption is that $\esssup\xi(0)=0$. Depending on the thickness of the distribution $\prob(\xi(0)\in\cdot)$ 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 Schr\"odinger 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.