Density of imaginary multiplicative chaos via Malliavin calculus

Abstract

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 $d\ge 1$ dimensions. We prove a basic density result, showing that for any nonzero continuous test function $f$, the complex-valued random variable ${\mu }_{\beta }(f)$ has a smooth density w.r.t. the Lebesgue measure on $\mathbb{C}$. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.

arXiv commentMinor changes. To appear in Probability Theory and Related Fields

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Cited by in corpus (4)

• Superconvergence phenomenon in Wiener chaoses
• The density of imaginary multiplicative chaos is positive
• Noise-like analytic properties of imaginary chaos
• Regularity of laws via Dirichlet forms -- Application to quadratic forms in independent and identically distributed random variables