{
  "id": "2403.03638",
  "version": "v2",
  "published": "2024-03-06T11:51:57.000Z",
  "updated": "2024-03-08T08:10:06.000Z",
  "title": "Stochastic partial differential equations for superprocesses in random environments",
  "authors": [
    "Jieliang Hong",
    "Jie Xiong"
  ],
  "comment": "It turns out that our results overlap with a previous paper",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X=(X_t, t\\geq 0)$ be a superprocess in a random environment described by a Gaussian noise $W^g=\\{W^g(t,x), t\\geq 0, x\\in \\mathbb{R}^d\\}$ white in time and colored in space with correlation kernel $g(x,y)$. We show that when $d=1$, $X_t$ admits a jointly continuous density function $X_t(x)$ that is a unique in law solution to a stochastic partial differential equation \\begin{align*} \\frac{\\partial }{\\partial t}X_t(x)=\\frac{\\Delta}{2} X_t(x)+\\sqrt{X_t(x)} \\dot{V}(t,x)+X_t(x)\\dot{W}^g(t, x) , \\quad X_t(x)\\geq 0, \\end{align*} where $V=\\{V(t,x), t\\geq 0, x\\in \\mathbb{R}\\}$ is a space-time white noise and is orthogonal with $W^g$. When $d\\geq 2$, we prove that $X_t$ is singular and hence density does not exist.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-08T08:10:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60G57",
      "60J80"
    ],
    "keywords": [
      "stochastic partial differential equation",
      "random environment",
      "superprocess",
      "space-time white noise",
      "law solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}