{
  "id": "2106.03510",
  "version": "v1",
  "published": "2021-06-07T11:00:10.000Z",
  "updated": "2021-06-07T11:00:10.000Z",
  "title": "Cooling down stochastic differential equations: almost sure convergence",
  "authors": [
    "S. Dereich",
    "S. Kassing"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider almost sure convergence of the SDE $dX_t=\\alpha_t d t + \\beta_t d W_t$ under the existence of a $C^2$-Lyapunov function $F:\\mathbb R^d \\to \\mathbb R$. More explicitly, we show that on the event that the process stays local we have almost sure convergence in the Lyapunov function $(F(X_t))$ as well as $\\nabla F(X_t)\\to 0$, if $|\\beta_t|=\\mathcal O( t^{-\\beta})$ for a $\\beta>1/2$. If, additionally, one assumes that $F$ is a Lojasiewicz function, we get almost sure convergence of the process itself, given that $|\\beta_t|=\\mathcal O(t^{-\\beta})$ for a $\\beta>1$. The assumptions are shown to be optimal in the sense that there is a divergent counterexample where $|\\beta_t|$ is of order $t^{-1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-07T11:00:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J60",
      "60H10",
      "65C35"
    ],
    "keywords": [
      "sure convergence",
      "stochastic differential equations",
      "lyapunov function",
      "process stays local",
      "divergent counterexample"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}