{
  "id": "math/0602655",
  "version": "v1",
  "published": "2006-02-28T07:57:10.000Z",
  "updated": "2006-02-28T07:57:10.000Z",
  "title": "Large deviation for diffusions and Hamilton--Jacobi equation in Hilbert spaces",
  "authors": [
    "Jin Feng"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117905000000567 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2006, Vol. 34, No. 1, 321-385",
  "doi": "10.1214/009117905000000567",
  "categories": [
    "math.PR"
  ],
  "abstract": "Large deviation for Markov processes can be studied by Hamilton--Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications. This paper illustrates such a method applied to a class of Hilbert-space-valued small diffusion processes. The examples include stochastically perturbed Allen--Cahn, Cahn--Hilliard PDEs and a one-dimensional quasilinear PDE with a viscosity term. We prove the comparison principle using a variant of the Tataru method. We also discuss different notions of viscosity solution in infinite dimensions in such context.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-02-28T07:57:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F10",
      "60J25",
      "49L25",
      "60G99"
    ],
    "keywords": [
      "hilbert spaces",
      "markov processes",
      "exponential compact containment estimate",
      "comparison principle",
      "one-dimensional quasilinear pde"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......2655F"
    }
  }
}