# A macroscopic multifractal analysis of parabolic stochastic PDEs

## TL;DR

This paper rigorously demonstrates that solutions to certain stochastic PDEs with multiplicative noise exhibit complex, multi-scale multifractal structures in both space and time, confirming long-standing conjectures about their peak behavior.

## Contribution

It provides a rigorous proof that the spatio-temporal peaks of these stochastic PDEs form infinitely many multifractals across various scales, extending previous qualitative insights.

## Key findings

- Solutions form macroscopic multifractals with large peaks
- Existence of infinitely many multifractals on multiple scales
- Similar structures found in constant-coefficient cases

## Abstract

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+u\xi$, where $\xi$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.