{
  "id": "math/0504551",
  "version": "v3",
  "published": "2005-04-27T16:18:40.000Z",
  "updated": "2008-11-22T15:01:17.000Z",
  "title": "Stochastic 2-microlocal analysis",
  "authors": [
    "Erick Herbin",
    "Jacques Lévy-Véhel"
  ],
  "comment": "35 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "A lot is known about the H\\\"older regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-) differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-11-22T15:01:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62G05",
      "60G15",
      "60G17",
      "60G18"
    ],
    "keywords": [
      "gaussian processes",
      "stochastic processes",
      "sure lower bounds",
      "incremental covariance yields",
      "multifractional brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......4551H"
    }
  }
}