A landscape of peaks: The intermittency islands of the stochastic heat equation with L\'evy noise

TL;DR

This paper explores the complex spatial structure of solutions to the stochastic heat equation driven by non-Gaussian Lévy noise, revealing multiple layers of intermittency islands and their fractal properties, contrasting with Gaussian noise behavior.

Contribution

It uncovers the layered structure of intermittency islands in the solution's spatial profile under non-Gaussian noise and analyzes their fractal dimensions and self-similarity properties.

Findings

Large peaks are taller under heavy-tailed noise with multiplicative effects.

For finite second moments, peak heights are similar for additive and multiplicative noise.

A second layer of peaks exists only under multiplicative noise, observable on a lattice.

Abstract

We show that the spatial profile of the solution to the stochastic heat equation features multiple layers of intermittency islands if the driving noise is non-Gaussian. On the one hand, as expected, if the noise is sufficiently heavy-tailed, the largest peaks of the solution will be taller under multiplicative than under additive noise. On the other hand, surprisingly, as soon as the noise has a finite moment of order $\frac2d$, where $d$ is the spatial dimension, the largest peaks will be of the same order for both additive and multiplicative noise, which is in sharp contrast to the behavior of the solution under Gaussian noise. However, in this case, a closer inspection reveals a second layer of peaks, beneath the largest peaks, that is exclusive to multiplicative noise and that can be observed by sampling the solution on the lattice. Finally, we compute the macroscopic Hausdorff and Minkowski dimensions of the intermittency islands of the solution. Under both additive and multiplicative noise, if it is not too heavy-tailed, the largest peaks will be self-similar in terms of their large-scale multifractal behavior. But under multiplicative noise, this type of self-similarity is not present in the peaks observed on the lattice.