Multifractal analysis of Gaussian multiplicative chaos and applications

Abstract

Let $M_{\mathit{\gamma }}$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D\subset {\mathbb{R}}^d$, $d\ge 1$. We find an explicit formula for its singularity spectrum by showing that $M_{\mathit{\gamma }}$ 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.