{
  "id": "2212.04178",
  "version": "v1",
  "published": "2022-12-08T10:33:26.000Z",
  "updated": "2022-12-08T10:33:26.000Z",
  "title": "Lower deviation for the supremum of the support of super-Brownian motion",
  "authors": [
    "Yan-Xia Ren",
    "Renming Song",
    "Rui Zhang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the asymptotic behavior of the supremum $M_t$ of the support of a supercritical super-Brownian motion. In our recent paper (Stoch. Proc. Appl. 137 (2021), 1-34), we showed that, under some conditions, $M_t-m(t)$ converges in distribution to a randomly shifted Gumbel random variable, where $m(t)=c_0t-c_1\\log t$. In the same paper, we also studied the upper large deviation of $M_t$, i.e., the asymptotic behavior of $\\mathbb{P}(M_t>\\delta c_0t) $ for $\\delta\\ge 1$. In this paper, we study the lower large deviation of $M_t$, i.e., the asymptotic behavior of $\\mathbb{P}(M_t\\le \\delta c_0t|\\mathcal{S}) $ for $\\delta<1$, where $\\mathcal{S}$ is the survival event.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-08T10:33:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "super-brownian motion",
      "lower deviation",
      "asymptotic behavior",
      "shifted gumbel random variable",
      "upper large deviation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}