{
  "id": "2007.10911",
  "version": "v1",
  "published": "2020-07-21T16:04:18.000Z",
  "updated": "2020-07-21T16:04:18.000Z",
  "title": "On regularization by a small noise of multidimensional ODEs with non-Lipschitz coefficients",
  "authors": [
    "Alexei Kulik",
    "Andrey Pilipenko"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we solve a selection problem for multidimensional SDE $d X^\\varepsilon(t)=a(X^\\varepsilon(t)) d t+\\varepsilon \\sigma(X^\\varepsilon(t))\\, d W(t)$, where the drift and diffusion are locally Lipschitz continuous outside of a fixed hyperplane $H$. It is assumed that $X^\\varepsilon(0)=x^0\\in H$, the drift $a(x)$ has a Hoelder asymptotics as $x$ approaches $H$, and the limit ODE $d X(t)=a(X(t))\\, d t$ does not have a unique solution. We show that if the drift pushes the solution away of $H$, then the limit process with certain probabilities selects some extreme solutions to the limit ODE. If the drift attracts the solution to $H$, then the limit process satisfies an ODE with some averaged coefficients. To prove the last result we formulate an averaging principle, which is quite general and new.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-21T16:04:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "60H10",
      "34F05"
    ],
    "keywords": [
      "multidimensional odes",
      "non-lipschitz coefficients",
      "small noise",
      "limit ode",
      "regularization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}