{
  "id": "2307.02160",
  "version": "v1",
  "published": "2023-07-05T10:01:32.000Z",
  "updated": "2023-07-05T10:01:32.000Z",
  "title": "Invariance principle for Lifts of Geodesic Random Walks",
  "authors": [
    "Jonathan Junné",
    "Frank Redig",
    "Rik Versendaal"
  ],
  "categories": [
    "math.PR",
    "math.DG"
  ],
  "abstract": "We consider a certain class of Riemannian submersions $\\pi : N \\to M$ and study lifted geodesic random walks from the base manifold $M$ to the total manifold $N$. Under appropriate conditions on the distribution of the speed of the geodesic random walks, we prove an invariance principle; i.e., convergence to horizontal Brownian motion for the lifted walks. This gives us a natural probabilistic proof of the geometric identity relating the horizontal Laplacian $\\Delta_\\H$ on $N$ and the Laplace-Beltrami operator $\\Delta_M$ on $M$. In particular, when $N$ is the orthonormal frame bundle $O(M)$, this identity is central in the Malliavin-Eells-Elworthy construction of Riemannian Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-05T10:01:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60K35",
      "58J65"
    ],
    "keywords": [
      "invariance principle",
      "study lifted geodesic random walks",
      "orthonormal frame bundle",
      "natural probabilistic proof",
      "riemannian brownian motion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}