# Density of imaginary multiplicative chaos via Malliavin calculus

Juhan Aru, Antoine Jego, Janne Junnila

Juhan Aru, Antoine Jego, Janne Junnila

**26/8/20**Published in : arXiv:2008.11768

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.