{
  "id": "1007.1957",
  "version": "v2",
  "published": "2010-07-12T18:42:30.000Z",
  "updated": "2011-08-18T03:40:38.000Z",
  "title": "Modulation spaces, Wiener amalgam spaces, and Brownian motions",
  "authors": [
    "Árpád Bényi",
    "Tadahiro Oh"
  ],
  "comment": "35 pages. The introduction is expanded. Appendices are added (A: derivation of Fourier-Wiener series, B: passing estimates from T to bounded intervals on R.) To appear in Adv. Math",
  "categories": [
    "math.FA",
    "math.AP",
    "math.PR"
  ],
  "abstract": "We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure \\mu on T, we show that (M^{p, q}_s (T), \\mu) and (W^{p, q}_s (T), \\mu) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space \\ft{b}^s_{p, \\infty} (T). Specifically, we prove that the Brownian motion belongs to \\ft{b}^s_{p, \\infty} (T) for (s-1) p = -1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces B_{p, q}^s, and indicate the endpoint large deviation estimates.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-18T03:40:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B35",
      "60G51",
      "42A61"
    ],
    "keywords": [
      "wiener amalgam spaces",
      "modulation spaces",
      "usual local besov spaces",
      "periodic brownian motion belongs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1007.1957B"
    }
  }
}