On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian

TL;DR

This paper investigates the large-scale structure of tall peaks in stochastic heat equations with fractional Laplacian, revealing multi-fractal behavior and computing the macroscopic Hausdorff dimensions for both linear and parabolic Anderson models.

Contribution

It extends previous work to a broader class of stochastic heat equations, analyzing the large-scale structure and fractal dimensions of tall peaks.

Findings

Both models exhibit multi-fractal behavior at macroscopic scale.

The largest order of tall peaks is determined.

Macroscopic Hausdorff dimensions of peaks are computed.

Abstract

We consider stochastic heat equations with fractional Laplacian on $\mathbb{R}^d$. Here, the driving noise is generalized Gaussian which is white in time but spatially homogenous and the spatial covariance is given by the Riesz kernels. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the tall peaks and compute the macroscopic Hausdorff dimensions of the tall peaks for both (i) and (ii). These results imply that both (i) and (ii) exhibit multi-fractal behavior in a macroscopic scale even though (i) is not intermittent and (ii) is intermittent. This is an extension of a recent result by Khoshnevisan et al to a wider class of stochastic heat equations.