{
  "id": "2008.11768",
  "version": "v1",
  "published": "2020-08-26T19:07:04.000Z",
  "updated": "2020-08-26T19:07:04.000Z",
  "title": "Density of imaginary multiplicative chaos via Malliavin calculus",
  "authors": [
    "Juhan Aru",
    "Antoine Jego",
    "Janne Junnila"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "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 $d \\geq 1$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-26T19:07:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G20",
      "60G57",
      "60G60",
      "60H07",
      "82B21"
    ],
    "keywords": [
      "malliavin calculus",
      "imaginary multiplicative chaos",
      "imaginary chaos",
      "imaginary gaussian multiplicative chaos",
      "complex wick exponential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}