{
  "id": "1503.00871",
  "version": "v1",
  "published": "2015-03-03T09:56:25.000Z",
  "updated": "2015-03-03T09:56:25.000Z",
  "title": "Linear forms of the telegraph random processes driven by partial differential equations",
  "authors": [
    "Alexander D. Kolesnik"
  ],
  "comment": "23 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider $n$ independent Goldstein-Kac telegraph processes $X_1(t), \\dots ,X_n(t), \\; n\\ge 2, \\; t\\ge 0,$ on the real line $\\Bbb R$. Each the process $X_k(t), \\; k=1,\\dots,n,$ describes a stochastic motion at constant finite speed $c_k>0$ of a particle that, at the initial time instant $t=0$, starts from some initial point $x_k^0=X_k(0)\\in\\Bbb R$ and whose evolution is controlled by a homogeneous Poisson process $N_k(t)$ of rate $\\lambda_k>0$. The governing Poisson processes $N_k(t), \\; k=1,\\dots,n,$ are supposed to be independent as well. Consider the linear form of the processes $X_1(t), \\dots ,X_n(t), \\; n\\ge 2,$ defined by $$L(t) = \\sum_{k=1}^n a_k X_k(t) ,$$ where $a_k, \\; k=1,\\dots,n,$ are arbitrary real non-zero constant coefficients. We obtain a hyperbolic system of first-order partial differential equations for the joint probability densities of the process $L(t)$ and of the directions of motions at arbitrary time $t>0$. From this system we derive a partial differential equation of order $2^n$ for the transition density of $L(t)$ in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. The weak convergence of $L(t)$ to a homogeneous Wiener process, under Kac's scaling conditions, is proved. Initial-value problems for the transition densities of the sum and difference $S^{\\pm}(t)=X_1(t) \\pm X_2(t)$ of two independent telegraph processes with arbitrary parameters, are also posed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-03T09:56:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "60J60",
      "35G10",
      "35G05",
      "35L55",
      "60E10"
    ],
    "keywords": [
      "partial differential equation",
      "telegraph random processes driven",
      "linear form",
      "arbitrary real non-zero constant coefficients"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}