Localisation and ageing in the parabolic Anderson model with Weibull potential

TL;DR

This paper studies the long-term localization and ageing phenomena in the parabolic Anderson model with Weibull-distributed potential, showing complete localization at a single site and linear ageing intervals.

Contribution

It provides a rigorous analysis of localization and ageing in the parabolic Anderson model with Weibull potential, identifying the asymptotic behavior of the localization site and the linear growth of relocalization intervals.

Findings

Solution localizes at a single site with high probability as time goes to infinity.

Intervals between relocalizations grow linearly over time.

The asymptotic behavior of the localization site is characterized.

Abstract

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential $\xi$. We consider the case when $\{\xi(z):z\in\mathbb{Z}^d\}$ is a collection of independent identically distributed random variables with Weibull distribution with parameter $0<\gamma<2$, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution relocalises from one site to another increase linearly over time, a phenomenon known as ageing.