Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour

TL;DR

This paper establishes stable limit theorems and conditions for strong laws of large numbers for the parabolic Anderson model in large boxes, analyzing the transition between quenched and annealed behaviors over various scales.

Contribution

It provides a comprehensive set of stable limit laws for the model across all scaling regimes and identifies conditions for strong laws of large numbers.

Findings

Derived stable limit theorems for the model

Identified conditions for strong law of large numbers

Analyzed behavior across different growth rates of boxes

Abstract

We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution depending on the growth rate of the boxes. Furthermore, we give sufficient conditions for a strong law of large numbers.