{
  "id": "1601.06998",
  "version": "v1",
  "published": "2016-01-26T12:47:32.000Z",
  "updated": "2016-01-26T12:47:32.000Z",
  "title": "Integral equation for the transition density of the multidimensional Markov random flight",
  "authors": [
    "Alexander D. Kolesnik"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the Markov random flight $\\bold X(t)$ in the Euclidean space $\\Bbb R^m, \\; m\\ge 2,$ starting from the origin $\\bold 0\\in\\Bbb R^m$ that, at Poisson-paced times, changes its direction at random according to arbitrary distribution on the unit $(m-1)$-dimensional sphere $S^m(\\bold 0,1)$ having absolutely continuous density. For any time instant $t>0$, the convolution-type recurrent relations for the joint and conditional densities of process $\\bold X(t)$ and of the number of changes of direction, are obtained. Using these relations, we derive an integral equation for the transition density of $\\bold X(t)$ whose solution is given in the form of a uniformly converging series composed of the multiple double convolutions of the singular component of the density with itself. Two important particular cases of the uniform distribution on $S^m(\\bold 0,1)$ and of the Gaussian distributions on the unit circumference $S^2(\\bold 0,1)$ are separately considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-26T12:47:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60K99",
      "60J60",
      "60J65",
      "82C41",
      "82C70"
    ],
    "keywords": [
      "multidimensional markov random flight",
      "integral equation",
      "transition density",
      "convolution-type recurrent relations",
      "gaussian distributions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160106998K"
    }
  }
}