Complete localisation in the parabolic Anderson model with Pareto-distributed potential

TL;DR

This paper proves that in the parabolic Anderson model with Pareto-distributed potential, the solution becomes completely localized at a single point over time, with the localization point following a predictable asymptotic behavior.

Contribution

It establishes complete localization in the parabolic Anderson model with Pareto potential and characterizes the asymptotic behavior of the localization point.

Findings

Solution localizes at a single point with high probability

The localization point follows a weak limit theorem

Asymptotic behavior of the localization point is identified

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation $\partial_t u(t,z)=\Delta u(t,z)+\xi(z) u(t,z)$ on $(0,\infty)\times {\mathbb Z}^d$ with random potential $(\xi(z) \colon z\in {\mathbb Z}^d)$. We consider independent and identically distributed potential variables, such that Prob$(\xi(z)>x)$ decays polynomially as $x\uparrow\infty$. If $u$ is initially localised in the origin, i.e. if $u(0,x)=\one_0(x)$, we show that, at any large time $t$, the solution is completely localised in a single point with high probability. More precisely, we find a random process $(Z_t \colon t\ge 0)$ with values in $\Z^d$ such that $\lim_{t \uparrow\infty} u(t,Z_t)/\sum_{z\in\Z^d} u(t,z) =1,$ in probability. We also identify the asymptotic behaviour of $Z_t$ in terms of a weak limit theorem.