{
  "id": "2305.00734",
  "version": "v1",
  "published": "2023-05-01T09:26:10.000Z",
  "updated": "2023-05-01T09:26:10.000Z",
  "title": "Convergence of processes time-changed by Gaussian multiplicative chaos",
  "authors": [
    "Takumu Ooi"
  ],
  "comment": "41 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "As represented by Liouville measure, Gaussian multiplicative chaos is a random measure constructed from Gaussian fields. We prove the convergence of processes time-changed by Gaussian multiplicative chaos under certain assumptions when Gaussian multiplicative chaos on bounded measurable sets is square integrable (the $L^2$-regime). As examples of the main result, we prove that, in half of the $L^2$-regime, the scaling limit of the ``Liouville simple random walk'' on $\\mathbb{Z}^2$ is Liouville Brownian motion and, in the whole $L^2$-regime, ``Liouville $\\alpha$-stable processes'' on $\\mathbb{R}$ converge weakly to the ``Liouville Cauchy process'' as $\\alpha \\to 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-01T09:26:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "31C25",
      "60J55",
      "60G57",
      "60G60"
    ],
    "keywords": [
      "gaussian multiplicative chaos",
      "convergence",
      "liouville simple random walk",
      "liouville cauchy process",
      "liouville brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}