{
  "id": "1604.08362",
  "version": "v1",
  "published": "2016-04-28T10:06:11.000Z",
  "updated": "2016-04-28T10:06:11.000Z",
  "title": "Asymptotic relation for the transition density of the three-dimensional Markov random flight on small time intervals",
  "authors": [
    "Alexander D. Kolesnik"
  ],
  "comment": "19 pages, 5 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Markov random flight $\\bold X(t), \\; t>0,$ in the three-dimensional Euclidean space $\\Bbb R^3$ with constant finite speed $c>0$ and the uniform choice of the initial and each new direction at random time instants that form a homogeneous Poisson flow of rate $\\lambda>0$. Series representations for the conditional characteristic functions of $\\bold X(t)$ corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as $t\\to 0$, for the unconditional characteristic function of $\\bold X(t)$ is derived. By inverting it, we obtain an asymptotic relation for the transition density of the process. We show that the error in this formula has the order $o(t^3)$ and, therefore, it gives a good approximation on small time intervals whose lengths depend on $\\lambda$. Estimate of the accuracy of the approximation is analysed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-28T10:06:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K99",
      "60J60",
      "60J65",
      "82C41",
      "82C70"
    ],
    "keywords": [
      "three-dimensional markov random flight",
      "small time intervals",
      "asymptotic relation",
      "transition density",
      "unconditional characteristic function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}