Open Access
2023 Multifractal analysis of Gaussian multiplicative chaos and applications
Federico Bertacco
Author Affiliations +
Electron. J. Probab. 28: 1-36 (2023). DOI: 10.1214/22-EJP893


Let Mγ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain DRd, d1. We find an explicit formula for its singularity spectrum by showing that Mγ 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 Lq-spectrum. Then applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.

Funding Statement

The author is very grateful to the Royal Society for financial support through Prof. M. Hairer’s research professorship grant RP\R1\191065.


The author would like to thank Prof. M. Hairer for his constant support and guidance. We thank an anonymous referee for many helpful comments on an earlier version of this article.


Download Citation

Federico Bertacco. "Multifractal analysis of Gaussian multiplicative chaos and applications." Electron. J. Probab. 28 1 - 36, 2023.


Received: 13 June 2022; Accepted: 14 December 2022; Published: 2023
First available in Project Euclid: 4 January 2023

Digital Object Identifier: 10.1214/22-EJP893

Primary: 28A78 , 28A80 , 60G57 , 60G60

Keywords: Gaussian multiplicative chaos , Liouville Brownian motion , Multifractal analysis , multifractal formalism

Vol.28 • 2023
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