{
  "id": "2012.09040",
  "version": "v1",
  "published": "2020-12-16T16:00:41.000Z",
  "updated": "2020-12-16T16:00:41.000Z",
  "title": "Absolute continuity of the Super-Brownian motion with infinite mean",
  "authors": [
    "Rustam Mamin",
    "Leonid Mytnik"
  ],
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In this work we prove that for any dimension $d\\geq 1$ and any $\\gamma \\in (0,1)$ super-Brownian motion corresponding to the log-Laplace equation \\begin{equation*} \\begin{split} \\frac{\\partial v(t,x)}{\\partial t } & = \\frac{1}{2}\\bigtriangleup v(t,x) + v^\\gamma (t,x) ,\\: (t,x) \\in \\mathbb{R}_+\\times \\mathbb{R}^d,\\\\ v(0,x)&= f(x) \\end{split} \\end{equation*} is absolutely continuous with respect to the Lebesgue measure at any fixed time $t>0$. Our proof is based on properties of solutions of the \\LL\\ equation. We also prove that when initial datum $v(0,\\cdot)$ is a finite, non-zero measure, then the \\LL\\ equation has a unique, continuous solution. Moreover this solution continuously depends on initial data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-16T16:00:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "super-brownian motion",
      "infinite mean",
      "absolute continuity",
      "initial datum",
      "lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}