{
  "id": "2204.09896",
  "version": "v1",
  "published": "2022-04-21T05:57:34.000Z",
  "updated": "2022-04-21T05:57:34.000Z",
  "title": "{A direct construction of the Wiener measure on $\\textbf{C}[0, \\infty)$",
  "authors": [
    "R. P. Pakshirajan",
    "M. Sreehari"
  ],
  "comment": "16 pages. arXiv admin note: text overlap with arXiv:2011.05584",
  "categories": [
    "math.PR"
  ],
  "abstract": "Our construction of the Wiener measure on $\\textbf{C}=\\textbf{C}[0, \\infty)$ consists in first defining a set function $\\varphi$\\ on the class of all compact sets based on certain $n$-dimensional normal distributions, $n = 1,\\ 2,\\ldots$\\ using the structural relation at (\\ref{E1.2}) below. This structural relation, discovered by the first author, is recorded in his book (2013) on page 130. We then define a measure $\\mu$ on the Borel $\\sigma$-field of subsets of $\\textbf{C}$ which is the Wiener measure. This is done via a similar construction of the Wiener measure on $\\textbf{C}_a=\\textbf{C}[0, a)$ where $a > 0$ is an arbitrary real number. The traditional way is to first construct the Brownian Motion process (BMP) and then, by proving it is a measurable mapping into $(\\textbf{C},\\ \\mathscr{C}_\\infty)$, call the measure induced by the BMP on $\\textbf{C}$\\ the Wiener measure. In the present paper, we define the Wiener measure directly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-21T05:57:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60G15"
    ],
    "keywords": [
      "wiener measure",
      "direct construction",
      "structural relation",
      "brownian motion process",
      "arbitrary real number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}