A two cities theorem for the parabolic Anderson model

TL;DR

This paper studies the long-term localization of solutions to the parabolic Anderson model with i.i.d. potentials, showing that solutions concentrate in one or two points over time, with precise asymptotic behavior.

Contribution

It establishes a two cities theorem for the parabolic Anderson model with polynomially decaying potential distribution, revealing detailed localization behavior.

Findings

Solutions localize in two points almost surely as time goes to infinity.

In high probability, solutions localize in a single point.

Asymptotic distribution of concentration sites is characterized by a weak limit theorem.

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_tu(t,z)=\Delta u(t,z)+\xi(z)u(t,z)$ on $(0,\infty)\times {\mathbb{Z}}^d$ with random potential $(\xi(z):z\in{\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\xi(z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.