{
  "id": "1706.02402",
  "version": "v1",
  "published": "2017-06-07T22:33:12.000Z",
  "updated": "2017-06-07T22:33:12.000Z",
  "title": "Weak Moment of a Class of Stochastic Heat Equation with Martingale-valued Harmonic Function",
  "authors": [
    "Ejighikeme Mcsylvester Omaba"
  ],
  "comment": "16 pages",
  "journal": "Asian Research Journal of Mathematics Asian Research Journal of Mathematics, 3(1): 1-16, Article no.ARJOM.31665 (2017)",
  "categories": [
    "math.PR"
  ],
  "abstract": "A study of a non-linear parabolic SPDEs of the form $\\partial_{t}u=\\mathcal{L}\\,u + \\sigma(u)f(B_t^x,t)\\dot{w}$ with $\\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function $\\sigma:\\mathbb{R}\\rightarrow\\mathbb{R}$ is Lipschitz continuous and $\\mathcal{L}$ the $L^2$-generator of a L\\'{e}vy process. Some precise condition for existence and uniqueness of the solution were given and we show that the solution grows weakly(in law/distribution) in time (for large $t$) at most a precise exponential rate for the $\\mathcal{L}$; and grows in time at most a precise exponential rate for the case of $\\mathcal{L}=-(-\\Delta)^{\\alpha/2},\\,\\,\\alpha\\in(1,2]$ generator of an alpha-stable process.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-07T22:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R60",
      "60H15",
      "82B44"
    ],
    "keywords": [
      "stochastic heat equation",
      "martingale-valued harmonic function",
      "weak moment",
      "precise exponential rate",
      "space-time harmonic function"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}