{
  "id": "1305.3325",
  "version": "v1",
  "published": "2013-05-14T23:51:23.000Z",
  "updated": "2013-05-14T23:51:23.000Z",
  "title": "On the spatial dynamics of the solution to the stochastic heat equation",
  "authors": [
    "Sigurd Assing",
    "James Bichard"
  ],
  "comment": "36 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "We consider the solution of $\\partial_t u=\\partial_x^2 u+\\partial_x\\partial_t B,\\,(x,t)\\in R\\times(0,\\infty)$, subject to $u(x,0)=0,\\,x\\in R$, where $B$ is a Brownian sheet. We show that $u$ also satisfies $\\partial_x^2 u +[\\,(-\\partial_t^2)^{1/2}+\\sqrt{2}\\partial_x(-\\partial_t^2)^{1/4}\\,]\\,u^a= \\partial_x\\partial_t{\\tilde B}$ in $R\\times(0,\\infty)$ where $u^a$ stands for the extension of $u(x,t)$ to $(x,t)\\in R^2$ which is antisymmetric in $t$ and $\\tilde{B}$ is another Brownian sheet. The new SPDE allows us to prove the strong Markov property of the pair $(u,\\partial_x u)$ when seen as a process indexed by $x\\ge x_0$, $x_0$ fixed, taking values in a state space of functions in $t$. The method of proof is based on enlargement of filtration and we discuss how our method could be applied to other quasi-linear SPDEs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-05-14T23:51:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15"
    ],
    "keywords": [
      "stochastic heat equation",
      "spatial dynamics",
      "brownian sheet",
      "strong markov property",
      "quasi-linear spdes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1305.3325A"
    }
  }
}