{
  "id": "2009.09658",
  "version": "v1",
  "published": "2020-09-21T07:48:29.000Z",
  "updated": "2020-09-21T07:48:29.000Z",
  "title": "Limit theorems for time-dependent averages of nonlinear stochastic heat equations",
  "authors": [
    "Kunwoo Kim",
    "Jaeyun Yi"
  ],
  "comment": "24 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study limit theorems for time-dependent average of the form $X_t:=\\frac{1}{2L(t)}\\int_{-L(t)}^{L(t)} u(t, x) \\, dx$, as $t\\to \\infty$, where $L(t)=e^{\\lambda t}$ and $u(t, x)$ is the solution to a stochastic heat equation on $\\mathbb{R}_+\\times \\mathbb{R}$ driven by space-time white noise with $u_0(x)=1$ for all $x\\in \\mathbb{R}$. We show that for $X_t$ (i) the weak law of large numbers holds when $\\lambda>\\lambda_1$, (ii) the strong law of large numbers holds when $\\lambda>\\lambda_2$, (iii) the central limit theorem holds when $\\lambda>\\lambda_3$, (iv) the quantitative central limit theorem holds when $\\lambda>\\lambda_4$, where $\\lambda_i$'s are positive constants depending on the moment Lyapunov exponents of $u(t, x)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-21T07:48:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60F15",
      "60F05"
    ],
    "keywords": [
      "nonlinear stochastic heat equations",
      "time-dependent average",
      "large numbers holds",
      "quantitative central limit theorem holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}