Let be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain , . We find an explicit formula for its singularity spectrum by showing that 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 -spectrum. Then applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.
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.
"Multifractal analysis of Gaussian multiplicative chaos and applications." Electron. J. Probab. 28 1 - 36, 2023. https://doi.org/10.1214/22-EJP893