{
  "id": "1812.09708",
  "version": "v1",
  "published": "2018-12-23T13:39:46.000Z",
  "updated": "2018-12-23T13:39:46.000Z",
  "title": "A family of stable diffusions",
  "authors": [
    "François Ledrappier",
    "Lin Shu"
  ],
  "categories": [
    "math.DS",
    "math.PR"
  ],
  "abstract": "Consider a $C^{\\infty}$ closed connected Riemannian manifold $(M, g)$ with negative curvature. The unit tangent bundle $SM$ is foliated by the (weak) stable foliation $\\mathcal{W}^s$ of the geodesic flow. Let $\\Delta^s$ be the leafwise Laplacian for $\\mathcal{W}^s$ and let $\\overline{X}$ be the geodesic spray, i.e., the vector field that generates the geodesic flow. For each $\\lambda$, the operator $\\mathcal{L}_{\\lambda}:=\\Delta^s+\\lambda \\overline{X}$ generates a diffusion for $\\mathcal{W}^s$. We show that, as $\\lambda\\to -\\infty$, the unique stationary probability measure for the leafwise diffusion of $\\mathcal{L}_{\\lambda}$ converges to the normalized Lebesgue measure on $SM$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-23T13:39:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37D40",
      "58J65"
    ],
    "keywords": [
      "stable diffusions",
      "unique stationary probability measure",
      "geodesic flow",
      "unit tangent bundle",
      "normalized lebesgue measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}