{
  "id": "1810.06754",
  "version": "v1",
  "published": "2018-10-15T23:47:28.000Z",
  "updated": "2018-10-15T23:47:28.000Z",
  "title": "On the Peaks of a Stochastic Heat Equation on a Sphere with a Large Radius",
  "authors": [
    "Weicong Su"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For every $R>0$, consider the stochastic heat equation $\\partial_{t} u_{R}(t\\,,x)=\\tfrac12 \\Delta_{S_{R}^{2}}u_{R}(t\\,,x)+\\sigma(u_{R}(t\\,,x)) \\xi_{R}(t\\,,x)$ on $S_{R}^{2}$, where $\\xi_{R}=\\dot{W_{R}}$ are centered Gaussian noises with the covariance structure given by $E [\\dot{W_{R}}(t,x)\\dot{W_{R}}(s,y)]=h_{R}(x,y)\\delta_{0}(t-s)$, where $h_{R}$ is symmetric and semi-positive definite and there exist some fixed constants $-2< C_{h_{up}}< 2$ and $\\frac{1}{2}C_{h_{up}}-1 <C_{h_{lo}}\\le C_{h_{up}}$ such that for all $R>0$ and $x\\,,y \\in S_{R}^{2}$, $(\\log R)^{C_{h_{lo}}/2}=h_{lo}(R)\\leq h_{R}(x,y) \\leq h_{up}(R)=(\\log R)^{C_{h_{up}}/2}$, $\\Delta_{S_{R}^{2}}$ denotes the Laplace-Beltrami operator defined on $S_{R}^{2}$ and $\\sigma:R \\mapsto R$ is Lipschitz continuous, positive and uniformly bounded away from $0$ and $\\infty$. Under the assumption that $u_{R,0}(x)=u_{R}(0\\,,x)$ is a nonrandom continuous function on $x \\in S_{R}^{2}$ and the initial condition that there exists a finite positive $U$ such that $\\sup_{R>0}\\sup_{x \\in S_{R}^{2}}\\vert u_{R,0}(x)\\vert \\le U$, we prove that for every finite positive $t$, there exist finite positive constants $C_{low}(t)$ and $C_{up}(t)$ which only depend on $t$ such that as $R \\to \\infty$, $\\sup_{x \\in S_{R}^{2}}\\vert u_{R}(t\\,,x)\\vert$ is asymptotically bounded below by $C_{low}(t)(\\log R)^{1/4+C_{h_{lo}}/4-C_{h_{up}}/8}$ and asymptotically bounded above by $C_{up}(t)(\\log R)^{1/2+C_{h_{up}}/4}$ with high probability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-15T23:47:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic heat equation",
      "large radius",
      "centered gaussian noises",
      "laplace-beltrami operator",
      "nonrandom continuous function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}