{
  "id": "2208.09498",
  "version": "v1",
  "published": "2022-08-19T18:22:56.000Z",
  "updated": "2022-08-19T18:22:56.000Z",
  "title": "Solutions of kinetic-type equations with perturbed collisions",
  "authors": [
    "Dariusz Buraczewski",
    "Piotr Dyszewski",
    "Alexander Marynych"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a class of kinetic-type differential equations $\\partial \\phi_t/\\partial t+\\phi_t=\\widehat{\\mathcal{Q}}\\phi_t$, where $\\widehat{\\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\\geq 0$, $\\phi_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\\widehat{\\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertained to $\\widehat{\\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\\to\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-19T18:22:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J85",
      "82C40",
      "60F05"
    ],
    "keywords": [
      "kinetic-type equations",
      "perturbed collisions",
      "continuous time branching random walk",
      "differential equation possesses",
      "kinetic-type differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}