The density of imaginary multiplicative chaos is positive

Consider a log-correlated Gaussian field $\Gamma$ and its associated imaginary multiplicative chaos $:e^{i \beta \Gamma}:$ where $\beta$ is a real parameter. In our companion paper, we showed that for any nonzero test function $f$, the law of $\int f :e^{i \beta \Gamma}:$ possesses a smooth density...

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Published inElectronic communications in probability Vol. 29; no. none
Main Authors Aru, Juhan, Jego, Antoine, Junnila, Janne
Format Journal Article
LanguageEnglish
Published Institute of Mathematical Statistics (IMS) 01.01.2024
Subjects
Mathematics
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ISSN1083-589X
1083-589X
DOI10.1214/24-ECP630

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Abstract Consider a log-correlated Gaussian field $\Gamma$ and its associated imaginary multiplicative chaos $:e^{i \beta \Gamma}:$ where $\beta$ is a real parameter. In our companion paper, we showed that for any nonzero test function $f$, the law of $\int f :e^{i \beta \Gamma}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.
AbstractList Consider a log-correlated Gaussian field $\Gamma$ and its associated imaginary multiplicative chaos $:e^{i \beta \Gamma}:$ where $\beta$ is a real parameter. In our companion paper, we showed that for any nonzero test function $f$, the law of $\int f :e^{i \beta \Gamma}:$ possesses a smooth density with respect to Lebesgue measure on $\mathbb{C}$. In this note, we show that this density is strictly positive everywhere on $\mathbb{C}$. Our simple and direct strategy could be useful for studying other functionals on Gaussian spaces.
Author Junnila, Janne
Aru, Juhan
Jego, Antoine
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  fullname: Junnila, Janne
  organization: University of Helsinki, Finland
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