Time correlations for the parabolic Anderson model

TL;DR

This paper derives precise asymptotics for time correlation functions and moments in the parabolic Anderson model, revealing the structure and longevity of intermittency peaks in the system.

Contribution

It provides exact asymptotics for time correlations and moments, and characterizes the spatial distribution and aging properties of intermittency peaks.

Findings

Exact asymptotics for time correlation functions

Description of potential peaks and their spatial distribution

Analysis of the longevity of intermittency peaks

Abstract

We derive exact asymptotics of time correlation functions for the parabolic Anderson model with homogeneous initial condition and time-independent tails that decay more slowly than those of a double exponential distribution and have a finite cumulant generating function. We use these results to give precise asymptotics for statistical moments of positive order. Furthermore, we show what the potential peaks that contribute to the intermittency picture look like and how they are distributed in space. We also investigate for how long intermittency peaks remain relevant in terms of ageing properties of the model.