{
  "id": "2209.04771",
  "version": "v1",
  "published": "2022-09-11T02:59:50.000Z",
  "updated": "2022-09-11T02:59:50.000Z",
  "title": "Invariant measures for the nonlinear stochastic heat equation with no drift term",
  "authors": [
    "Le Chen",
    "Nicholas Eisenberg"
  ],
  "comment": "29 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\\partial u /\\partial t - \\frac{1}{2}\\Delta u = b(u)\\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the noise $\\dot{W}$ is a centered and spatially homogeneous Gaussian noise that is white in time. Using the moment formulas obtained in [9, 10], we identify a set of conditions on the initial data, the correlation measure and the weight function $\\rho$, which will together guarantee the existence of an invariant measure in the weighted space $L^2_\\rho(\\mathbb{R}^d)$. In particular, our result includes the parabolic Anderson model (i.e., the case when $b(u) = \\lambda u$) starting from the Dirac delta measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-11T02:59:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H07",
      "60F05"
    ],
    "keywords": [
      "nonlinear stochastic heat equation",
      "invariant measure",
      "drift term",
      "long term behavior",
      "parabolic anderson model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}