{
  "id": "math/0404489",
  "version": "v1",
  "published": "2004-04-27T14:40:37.000Z",
  "updated": "2004-04-27T14:40:37.000Z",
  "title": "Integration by parts on the law of the reflecting Brownian motion",
  "authors": [
    "Lorenzo Zambotti"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove an integration by parts formula on the law of the reflecting Brownian motion $X:=|B|$ in the positive half line, where $B$ is a standard Brownian motion. In other terms, we consider a perturbation of $X$ of the form $X^\\epsilon = X+\\epsilon h$ with $h$ smooth deterministic function and $\\epsilon>0$ and we differentiate the law of $X^\\epsilon$ at $\\epsilon=0$. This infinitesimal perturbation changes drastically the set of zeros of $X$ for any $\\epsilon>0$. As a consequence, the formula we obtain contains an infinite dimensional generalized functional in the sense of Schwartz, defined in terms of Hida's renormalization of the squared derivative of $B$ and in terms of the local time of $X$ at 0. We also compute the divergence on the Wiener space of a class of vector fields not taking values in the Cameron-Martin space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-27T14:40:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H07",
      "60J65",
      "60H40"
    ],
    "keywords": [
      "reflecting brownian motion",
      "integration",
      "standard brownian motion",
      "smooth deterministic function",
      "infinite dimensional generalized functional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4489Z"
    }
  }
}