{
  "id": "1802.04190",
  "version": "v1",
  "published": "2018-02-08T20:44:39.000Z",
  "updated": "2018-02-08T20:44:39.000Z",
  "title": "On the approximation of the probability density function of the randomized heat equation",
  "authors": [
    "J. Calatayud",
    "J. -C. Cortes",
    "M. Jornet"
  ],
  "comment": "Pages: 35; Figures: 31 Tables: 5",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coeffcient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this randomized partial differential equation problem is a stochastic process, which is given by a random series obtained via the classical method of separation of variables. Any stochastic process is determined by its finite-dimensional joint distributions. In this paper, the goal is to obtain approximations to the probability density function of the solution (the first finite-dimensional distributions) under mild conditions. Since the solution is expressed as a random series, we perform approximations of its probability density function. We use two approaches: broadly speaking, first, dealing with the random Fourier coefficients of the random series, and second, taking advantage of the Karhunen-Loeve expansion of the initial condition stochastic process. Finally, several numerical examples illustrating the potentiality of our findings with regard to both approaches are presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-08T20:44:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H35",
      "60H10",
      "37H10"
    ],
    "keywords": [
      "probability density function",
      "randomized heat equation",
      "random series",
      "approximation",
      "initial condition stochastic process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}