{
  "id": "1903.06066",
  "version": "v1",
  "published": "2019-03-14T15:13:31.000Z",
  "updated": "2019-03-14T15:13:31.000Z",
  "title": "Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities",
  "authors": [
    "Matteo Beccari",
    "Martin Hutzenthaler",
    "Arnulf Jentzen",
    "Ryan Kurniawan",
    "Felix Lindner",
    "Diyora Salimova"
  ],
  "comment": "65 pages",
  "categories": [
    "math.NA",
    "math.PR"
  ],
  "abstract": "The explicit Euler scheme and similar explicit approximation schemes (such as the Milstein scheme) are known to diverge strongly and numerically weakly in the case of one-dimensional stochastic ordinary differential equations with superlinearly growing nonlinearities. It remained an open question whether such a divergence phenomenon also holds in the case of stochastic partial differential equations with superlinearly growing nonlinearities such as stochastic Allen-Cahn equations. In this work we solve this problem by proving that full-discrete exponential Euler and full-discrete linear-implicit Euler approximations diverge strongly and numerically weakly in the case of stochastic Allen-Cahn equations. This article also contains a short literature overview on existing numerical approximation results for stochastic differential equations with superlinearly growing nonlinearities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-14T15:13:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "60H35",
      "65C30"
    ],
    "keywords": [
      "stochastic partial differential equations",
      "superlinearly growing nonlinearities",
      "weak divergence",
      "stochastic ordinary differential equations",
      "linear-implicit euler approximations diverge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}