{
  "id": "1912.05350",
  "version": "v1",
  "published": "2019-12-11T14:27:08.000Z",
  "updated": "2019-12-11T14:27:08.000Z",
  "title": "Stochastic comparisons for stochastic heat equation",
  "authors": [
    "Le Chen",
    "Kunwoo Kim"
  ],
  "comment": "38 pages, 0 figure",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on $\\mathbb{R}^d$ \\[ \\left(\\frac{\\partial }{\\partial t} -\\frac{1}{2}\\Delta \\right) u(t,x) = \\rho(u(t,x)) \\:\\dot{M}(t,x), \\] where $\\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $\\rho$ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang's condition, namely, $\\int_{\\mathbb{R}^d}(1+|\\xi|^2)^{-1}\\hat{f}(\\text{d} \\xi)<\\infty$, where $\\hat{f}$ is the spectral measure of the noise. We establish the comparison principles by comparing either the diffusion coefficient $\\rho$ or the correlation function of the noise $f$. As corollaries, we obtain Slepian's inequality for SPDEs and SDEs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-11T14:27:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60G60",
      "35R60"
    ],
    "keywords": [
      "nonlinear stochastic heat equation",
      "stochastic comparison principles",
      "moment comparison principle",
      "rough initial data",
      "spatially homogeneous gaussian noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}