{
  "id": "1412.1912",
  "version": "v1",
  "published": "2014-12-05T07:50:42.000Z",
  "updated": "2014-12-05T07:50:42.000Z",
  "title": "Stationary solutions of stochastic partial differential equations in the space of tempered distributions",
  "authors": [
    "Suprio Bhar"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "In Rajeev (2013), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one to one correspondence between solutions of a class of finite dimensional SDEs and solutions of a class of SPDEs in $\\mathcal{S}'$, the space of tempered distributions, driven by the same Brownian motion. There the coefficients $\\bar{\\sigma}, \\bar{b}$ of the finite dimensional SDEs were related to the coefficients of the SPDEs in $\\mathcal{S}'$ in a special way, viz. through convolution with the initial value $y$ of the SPDEs. In this paper, we consider the situation where the solutions of the finite dimensional SDEs are stationary and ask whether the corresponding solutions of the equations in $\\mathcal{S}'$ are also stationary. We provide an affirmative answer, when the initial random variable takes value in a certain set $\\mathcal{C}$, which ensures that the coefficients of the finite dimensional SDEs are related to the coefficients of the SPDEs in the above `special' manner.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-05T07:50:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G10",
      "60H10",
      "60H15"
    ],
    "keywords": [
      "stochastic partial differential equations",
      "finite dimensional sdes",
      "tempered distributions",
      "stationary solutions",
      "coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.1912B"
    }
  }
}