{
  "id": "1908.01990",
  "version": "v1",
  "published": "2019-08-06T07:19:13.000Z",
  "updated": "2019-08-06T07:19:13.000Z",
  "title": "On the Stochastic Processes on $7$-Dimensional Spheres",
  "authors": [
    "Nurfarisha",
    "Adhitya Ronnie Effendie",
    "Muhammad Farchani Rosyid"
  ],
  "categories": [
    "math-ph",
    "math.DG",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We studied isometric stochastic flows of a Stratonovich stochastic differential equation on spheres, i.e. on the standard sphere and Gromoll-Meyer exotic sphere. The standard sphere $S^7_s$ can be constructed as the quotient manifold $\\mathrm{Sp}(2, \\mathbb{H})/S^3$ with the so-called ${\\bullet}$-action of $S^3$, whereas the Gromoll-Meyer exotic sphere $\\Sigma^7_{GM}$ as the quotient manifold $\\mathrm{Sp}(2, \\mathbb{H})/S^3$ with respect to the so-called ${\\star}$-action of $S^3$. The Stratonovich stochastic differential equation which describes a continuous-time stochastic process on the standard sphere is constructed and studied. The corresponding continuous-time stochastic process and its properties on the Gromoll-Meyer exotic sphere can be obtained by constructing a homeomorphism $h: S^7_s\\rightarrow \\Sigma^7_{GM}$. The corresponding Fokker-Planck equation and entropy rate in the Stratonovich approach is also investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-06T07:19:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G20",
      "60H10",
      "51H25",
      "57R22",
      "57R25",
      "57R50",
      "57R55",
      "57S15",
      "G.3.3"
    ],
    "keywords": [
      "gromoll-meyer exotic sphere",
      "dimensional spheres",
      "stratonovich stochastic differential equation",
      "standard sphere",
      "continuous-time stochastic process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}