{
  "id": "0908.1839",
  "version": "v1",
  "published": "2009-08-13T07:53:14.000Z",
  "updated": "2009-08-13T07:53:14.000Z",
  "title": "Lack of strong completeness for stochastic flows",
  "authors": [
    "Xue-Mei Li",
    "Michael Scheutzow"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "It is well-known that a stochastic differential equation (SDE) on a Euclidean space driven by a Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. When the coefficients are only locally Lipschitz, then a maximal continuous flow still exists but explosion in finite time may occur. If -- in addition -- the coefficients grow at most linearly, then this flow has the property that for each fixed initial condition $x$, the solution exists for all times almost surely. If the exceptional set of measure zero can be chosen independently $x$, then the maximal flow is called {\\em strongly complete}. The question, whether an SDE with locally Lipschitz continuous coefficients satisfying a linear growth condition is strongly complete was open for many years. In this paper, we construct a 2-dimensional SDE with coefficients which are even bounded (and smooth) and which is {\\em not} strongly complete thus answering the question in the negative.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-08-13T07:53:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10"
    ],
    "keywords": [
      "stochastic flow",
      "strong completeness",
      "lipschitz continuous coefficients satisfying",
      "strongly complete",
      "locally lipschitz"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0908.1839L"
    }
  }
}