{
  "id": "1804.05418",
  "version": "v1",
  "published": "2018-04-15T19:58:43.000Z",
  "updated": "2018-04-15T19:58:43.000Z",
  "title": "Self-similar solutions of kinetic-type equations: the critical case",
  "authors": [
    "Kamil Bogus",
    "Dariusz Buraczewski",
    "Alexander Marynych"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "For a time dependent family of probability measures $(\\rho_t)_{t\\ge 0}$ we consider a kinetic-type evolution equation $\\partial \\phi_t/\\partial t + \\phi_t = \\widehat{Q} \\phi_t$ where $\\widehat{Q}$ is a smoothing transform and $\\phi_t$ is the Fourier--Stieltjes transform of $\\rho_t$. Assuming that the initial measure $\\rho_0$ belongs to the domain of attraction of a stable law, we describe asymptotic properties of $\\rho_t$, as $t\\to\\infty$. We consider the critical regime when the standard normalization leads to a degenerate limit and find an appropriate scaling ensuring a non-degenerate self-similar limit. Our approach is based on a probabilistic representation of probability measures $(\\rho_t)_{t\\ge 0}$ that refines the corresponding construction proposed in Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928--1961, 2012].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-15T19:58:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "82C40"
    ],
    "keywords": [
      "kinetic-type equations",
      "self-similar solutions",
      "critical case",
      "probability measures",
      "kinetic-type evolution equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}