{
  "id": "1910.09271",
  "version": "v1",
  "published": "2019-10-21T11:46:14.000Z",
  "updated": "2019-10-21T11:46:14.000Z",
  "title": "Fractional moments of the Stochastic Heat Equation",
  "authors": [
    "Sayan Das",
    "Li-Cheng Tsai"
  ],
  "comment": "19 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the solution $\\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\\mathcal{Z}(0,x) = \\delta(x)$. For any real $p>0$, we obtained detailed estimates of the $p$-th moment of $e^{t/12}\\mathcal{Z}(2t,0)$, as $t\\to\\infty$, and from these estimates establish the one-point upper-tail large deviation principle of the Kardar-Parisi-Zhang equation. The deviations have speed $t$ and rate function $\\Phi_+(y)=\\frac{4}{3}y^{3/2}$. Our result confirms the existing physics predictions [Le Doussal, Majumdar, Schehr 16] and also [Kamenev, Meerson, Sasorov 16].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-21T11:46:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional moments",
      "one-point upper-tail large deviation principle",
      "one-dimensional stochastic heat equation",
      "multiplicative spacetime white noise",
      "delta initial data"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}