{
  "id": "1811.05069",
  "version": "v1",
  "published": "2018-11-13T02:07:27.000Z",
  "updated": "2018-11-13T02:07:27.000Z",
  "title": "The first passage time density of Brownian motion and the heat equation with Dirichlet boundary condition in time dependent domains",
  "authors": [
    "JM Lee"
  ],
  "comment": "18 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "In [4], it is proved that we can have a continuous first-passage-time density function of one dimensional standard Brownian motion when the boundary is H\\\"older continuous with exponent greater than 1/2. For the purpose of extending [4] into multidimensional domains, we show that there exists a continuous first-passage-time density function of standard $d$-dimensional Brownian motion in moving boundaries in $\\mathbb{R}^{d}$, $d\\geq 2$, under a $C^{3}$-diffeomorphism. Similarly as in [4], by using a property of local time of standard $d$-dimensional Brownian motion and the heat equation with Dirichlet boundary condition, we find a sufficient condition for the existence of the continuous density function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-13T02:07:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "first passage time density",
      "dirichlet boundary condition",
      "time dependent domains",
      "heat equation",
      "continuous first-passage-time density function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}