{
  "id": "2006.06808",
  "version": "v1",
  "published": "2020-06-11T20:48:37.000Z",
  "updated": "2020-06-11T20:48:37.000Z",
  "title": "Limit behavior of the invariant measure for Langevin dynamics",
  "authors": [
    "Gerardo Barrera"
  ],
  "comment": "13 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.DS",
    "math.MP"
  ],
  "abstract": "In this article, we consider the Langevin dynamics on $\\mathbb{R}^d$ with an overdamped vector field and driven by Brownian motion of small amplitude $\\sqrt{\\epsilon}$, $\\epsilon>0$. Under suitable conditions on the vector field, it is well-known that it possesses a unique invariant probability measure $\\mu^{\\epsilon}$. As $\\epsilon$ tends to zero, we prove that the probability measure $\\epsilon^{d/2} \\mu^{\\epsilon}(\\sqrt{\\epsilon}\\mathrm{d}x)$ converges in the $2$-Wasserstein distance to a Gaussian measure with zero-mean vector and non-degenerate covariance matrix which solves a Lyapunov matrix equation. We emphasize that generically no explicit formula for $\\mu^{\\epsilon}$ can be found.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T20:48:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "37M25",
      "60F05"
    ],
    "keywords": [
      "langevin dynamics",
      "invariant measure",
      "limit behavior",
      "unique invariant probability measure",
      "non-degenerate covariance matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}