{
  "id": "1312.6069",
  "version": "v2",
  "published": "2013-12-20T18:42:27.000Z",
  "updated": "2014-04-23T13:05:41.000Z",
  "title": "A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter",
  "authors": [
    "Alexandre Richard"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space $(0,1/2] \\times L^2(T,m)$, $(T,m)$ a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as L\\'evy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and H\\\"older regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on L\\'evy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local H\\\"older regularity on general indexing collections.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-04-23T13:05:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fractional brownian field",
      "varying hurst parameter",
      "levy fractional brownian motion",
      "multiparameter fractional brownian motions",
      "small ball probabilities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.6069R"
    }
  }
}