{
  "id": "1305.0975",
  "version": "v2",
  "published": "2013-05-05T00:47:42.000Z",
  "updated": "2013-06-07T13:31:57.000Z",
  "title": "Singular Behavior of the Solution to the Stochastic Heat Equation on a Polygonal Domain",
  "authors": [
    "Felix Lindner"
  ],
  "comment": "49 pages, 1 figure; a few typos have been corrected",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We study the stochastic heat equation with trace class noise and zero Dirichlet boundary condition on a bounded polygonal domain O in R^2. It is shown that the solution u can be decomposed into a regular part u_R and a singular part u_S which incorporates the corner singularity functions for the Poisson problem. Due to the temporal irregularity of the noise, both u_R and u_S have negative L_2-Sobolev regularity of order s<-1/2 in time. The regular part u_R admits spatial Sobolev regularity of order r=2, while the spatial Sobolev regularity of u_S is restricted by r<1+\\pi/\\gamma, where \\gamma is the largest interior angle at the boundary of O. We obtain estimates for the Sobolev norm of u_R and the Sobolev norms of the coefficients of the singularity functions. The proof is based on a Laplace transform argument w.r.t. the time variable. The result is of interest in the context of numerical methods for stochastic PDEs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-06-07T13:31:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35B65",
      "35R60",
      "46E35"
    ],
    "keywords": [
      "stochastic heat equation",
      "polygonal domain",
      "singular behavior",
      "admits spatial sobolev regularity",
      "sobolev norm"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.0975L"
    }
  }
}