Density of imaginary multiplicative chaos via Malliavin calculus

Juhan Aru, Antoine Jego, Janne Junnila

Published 2020-08-26Version 1

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential $\mu_\beta := :e^{i\beta \Gamma(x)}:$ for a log-correlated Gaussian field $\Gamma$ in $d \geq 1$ dimensions. We show that for any nonzero and bounded test function $f$, the complex-valued random variable $\mu_\beta(f)$ has a smooth density w.r.t. the Lebesgue measure on $\mathbb{C}$. Our main tool is Malliavin calculus, which seems to be well-adapted to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop some estimates on imaginary chaos that could be of independent interest.