{
  "id": "1601.00639",
  "version": "v1",
  "published": "2016-01-04T20:57:43.000Z",
  "updated": "2016-01-04T20:57:43.000Z",
  "title": "On the equivalence of probability spaces",
  "authors": [
    "Daniel Alpay",
    "Palle Jorgensen",
    "David Levanony"
  ],
  "comment": "To appear in Journal of Theoretical Probability",
  "categories": [
    "math.PR"
  ],
  "abstract": "For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\\mathscr F$ of a general measure space $(M,\\mathscr F, \\sigma)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $\\sigma(A)$, for $A\\in\\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivalence formula, realizing an equivalence between the two approaches to Gaussian processes, one where the choice of sample space is the traditional path-space, and the other where it is Schwartz' space of tempered distributions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-04T20:57:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability spaces",
      "gaussian processes",
      "general class",
      "quadratic variation representation",
      "general measure space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160100639A"
    }
  }
}