Macroscopic multi-fractality of Gaussian random fields and linear SPDEs with colored noise

TL;DR

This paper demonstrates that solutions to linear stochastic heat and wave equations with colored Gaussian noise exhibit macroscopic multi-fractality, characterized by tall peaks and complex fractal structures across space and time.

Contribution

It extends previous work on multi-fractality to more general linear SPDEs driven by colored Gaussian noise, providing new insights into their macroscopic fractal properties.

Findings

Solutions exhibit tall peaks at macroscopic scales.

The macroscopic Hausdorff dimension of peaks is computed.

The study extends multi-fractality results to broader classes of SPDEs.

Abstract

We consider the linear stochastic heat and wave equations with generalized Gaussian noise that is white in time and spatially correlated. Under the assumption that the homogeneous spatial correlation $f$ satisfies some mild conditions, we show that the solutions to the linear stochastic heat and wave equations exhibit tall peaks in macroscopic scales, which means they are macroscopically multi-fractal. We compute the macroscopic Hausdorff dimension of the peaks for Gaussian random fields with vanishing correlation and then apply this result to the solution of the linear stochastic heat and wave equations. We also study the spatio-temporal multi-fractality of the linear stochastic heat and wave equations. Our result is an extension of Khoshnevisan, Kim, and Xiao \cite{KKX,KKX2} and Kim \cite{K} to a more general class of the linear stochastic partial differential equations and Gaussian random fields.