A scaling limit theorem for the parabolic Anderson model with exponential potential

TL;DR

This paper investigates the long-term behavior of the parabolic Anderson model with exponential spatial potential, establishing limit theorems for mass growth and spatial localization, revealing trapping phenomena similar to heavy-tailed cases.

Contribution

It provides the first scaling limit theorem for the model with exponential potential, characterizing mass distribution and trapping in a single island.

Findings

Mass growth follows specific limit laws.

Mass localizes in a single island with high probability.

Spatial spread converges to a scaling limit.

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper we consider potentials which are constant in time and independent exponentially distributed in space. We study the growth rate of the total mass of the solution in terms of weak and almost sure limit theorems, and the spatial spread of the mass in terms of a scaling limit theorem. The latter result shows that in this case, just like in the case of heavy tailed potentials, the mass gets trapped in a single relevant island with high probability.