{
  "id": "1404.2330",
  "version": "v2",
  "published": "2014-04-08T23:18:22.000Z",
  "updated": "2014-09-19T21:33:27.000Z",
  "title": "The Smoluchowski-Kramers limit of stochastic differential equations with arbitrary state-dependent friction",
  "authors": [
    "Scott Hottovy",
    "Giovanni Volpe",
    "Jan Wehr"
  ],
  "comment": "This paper has been withdrawn by the author due to a incomplete proof of Lemma 2",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study a class of systems of stochastic differential equations describing diffusive phenomena. The Smoluchowski-Kramers approximation is used to describe their dynamics in the small mass limit. Our systems have arbitrary state-dependent friction and noise coefficients. We identify the limiting equation {and,} in particular, the additional drift term which appears in the limit is expressed in terms of the solution to a Lyapunov matrix equation. The proof uses a theory of convergence of stochastic integrals developed by Kurtz and Protter. The result is sufficiently general to include systems driven by both white and Ornstein-Uhlenbeck colored noises. We discuss applications of the main theorem to several physical phenomena, including the experimental study of Brownian motion in a diffusion gradient.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-08T23:18:22.000Z",
      "comment": "19 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-09-19T21:33:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations",
      "arbitrary state-dependent friction",
      "smoluchowski-kramers limit",
      "differential equations describing diffusive phenomena",
      "lyapunov matrix equation"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00220-014-2233-4",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Jun",
      "pages": 1259,
      "volume": 336,
      "number": 3
    },
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.336.1259H"
    }
  }
}