On the regularity of complex multiplicative chaos

Abstract

Denote by ${\mu }_{\beta }="\exp (\beta X)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset {\mathbb{R}}^d$. The case $\beta \in \mathbb{R}$ has been studied quite intensively, and then ${\mu }_{\beta }$ is a random measure on $U$. It is known that ${\mu }_{\beta }$ can also be defined for complex values $\beta$ lying in certain subdomain of $\mathbb{C}$, and then the realizations of ${\mu }_{\beta }$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $\beta$ was considered) by studying the Besov-regularity of ${\mu }_{\beta }$ and the finiteness of moments for general complex values of $\beta$.