{
  "id": "1006.1255",
  "version": "v1",
  "published": "2010-06-07T13:43:04.000Z",
  "updated": "2010-06-07T13:43:04.000Z",
  "title": "From constructive field theory to fractional stochastic calculus. (I) The Lévy area of fractional Brownian motion with Hurst index $α\\in (1/8,1/4)$",
  "authors": [
    "J. Magnen",
    "J. Unterberger"
  ],
  "comment": "86 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $B=(B_1(t),\\ldots,B_d(t))$ be a $d$-dimensional fractional Brownian motion with Hurst index $\\alpha<1/4$. Defining properly iterated integrals of $B$ is a difficult task because of the low H\\\"older regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to $B$, or to solving differential equations driven by $B$. We show in this paper how to obtain second-order iterated integrals as the limit when the ultra-violet cut-off goes to infinity of iterated integrals of weakly interacting fields defined using the tools of constructive field theory, in particular, cluster expansion and renormalization. The construction extends to a large class of Gaussian fields with the same short-distance behaviour, called multi-scale Gaussian fields. Previous constructions \\cite{Unt-Holder,Unt-fBm} were of algebraic nature and did not provide such a limiting procedure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-06-07T13:43:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "60G15",
      "60G18",
      "60H05",
      "81T08",
      "81T18"
    ],
    "keywords": [
      "constructive field theory",
      "fractional stochastic calculus",
      "hurst index",
      "lévy area",
      "iterated integrals"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 86,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1006.1255M"
    }
  }
}