{
  "id": "2109.09387",
  "version": "v1",
  "published": "2021-09-20T09:20:58.000Z",
  "updated": "2021-09-20T09:20:58.000Z",
  "title": "Amplitude equations for SPDEs driven by fractional additive noise with small Hurst parameter",
  "authors": [
    "Dirk Blömker",
    "Alexandra Neamtu"
  ],
  "comment": "Preprint, 32 pages",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\\in(0,1)$. Close to a change of stability measured with a small parameter $\\varepsilon$, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. We prove that up to an error term bounded by a power of $\\varepsilon$ depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so called amplitude equation, and in second order by a fast infinite dimensional Ornstein-Uhlenbeck process. To this aim we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-20T09:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60G22",
      "37H20"
    ],
    "keywords": [
      "small hurst parameter",
      "fractional additive noise",
      "amplitude equation",
      "infinite dimensional ornstein-uhlenbeck process",
      "spdes driven"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}