{
  "id": "1307.3155",
  "version": "v2",
  "published": "2013-07-11T16:00:36.000Z",
  "updated": "2015-07-01T16:56:04.000Z",
  "title": "If $B$ and $f(B)$ are Brownian motions, then $f$ is affine",
  "authors": [
    "Michael R. Tehranchi"
  ],
  "comment": "4 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "It is shown that if the processes $B$ and $f(B)$ are both Brownian motions then $f$ must be an affine function. As a by-product of the proof, it is shown that the only functions which are solutions to both the Laplace equation and the eikonal equation are affine.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-11T16:00:36.000Z",
      "comment": "3 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-07-01T16:56:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H30",
      "35C05"
    ],
    "keywords": [
      "brownian motions",
      "affine function",
      "eikonal equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3155T"
    }
  }
}