Multifractal analysis of Gaussian multiplicative chaos and applications

TL;DR

This paper derives an explicit formula for the singularity spectrum of subcritical Gaussian multiplicative chaos measures, confirming the multifractal formalism, and applies it to analyze the multifractal properties of random walk and Liouville Brownian motion.

Contribution

It provides a precise formula for the singularity spectrum of Gaussian multiplicative chaos and verifies the multifractal formalism for these measures.

Findings

Singularity spectrum matches the Legendre-Fenchel transform of the $L^q$-spectrum.

Explicit formulas for the lower singularity spectrum of multifractal random walk.

Application to Liouville Brownian motion's multifractal analysis.

Abstract

Let $M_{\gamma}$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_{\gamma}$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.