{
  "id": "1404.2990",
  "version": "v3",
  "published": "2014-04-11T02:54:25.000Z",
  "updated": "2014-10-11T03:05:52.000Z",
  "title": "Gradient Estimates and Applications for SDEs in Hilbert Space with Multiplicative Noise and Log-Hölder Drif",
  "authors": [
    "Feng-Yu Wang"
  ],
  "comment": "37 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the stochastic evolution equation in a separable Hilbert space $\\H$ with a nice multiplicative noise and a locally log-H\\\"older continuous drift $b: [0,\\infty)\\times \\H\\to \\H,$ i.e. for any $n\\ge 1$ there exists a constant $K_n>0$ such that $$|b_t(x)-b_t(y)|^2 \\le \\ff{K_n}{\\log (\\e + |x-y|^{-1})},\\ \\ \\ t\\in [0,n], x,y\\in \\H, |x|\\lor |y|\\le n.$$ We prove that for any initial data the equation has a unique (possibly explosive) mild solution. Under a reasonable condition ensuring the non-explosion of the solution, the strong Feller property of the associated Markov semigroup is proved. Gradient estimates and log-Harnack inequalities are derived for the associated semigroup under certain global conditions, which are new even in finite-dimensions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-05-08T09:41:21.000Z",
      "comment": "35 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-10-11T03:05:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gradient estimates",
      "log-hölder drif",
      "applications",
      "stochastic evolution equation",
      "strong feller property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.2990W"
    }
  }
}