{
  "id": "math-ph/0508039",
  "version": "v1",
  "published": "2005-08-19T15:47:06.000Z",
  "updated": "2005-08-19T15:47:06.000Z",
  "title": "On Convergence to Equilibrium Distribution, II. The Wave Equation in Odd Dimensions, with Mixing",
  "authors": [
    "T. V. Dudnikova",
    "A. I. Komech",
    "N. E. Ratanov",
    "Yu. M. Suhov"
  ],
  "comment": "27 pages",
  "journal": "Journal of Statistical Physics 108 (2002), no.4, 1219-1253",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The paper considers the wave equation, with constant or variable coefficients in $\\R^n$, with odd $n\\geq 3$. We study the asymptotics of the distribution $\\mu_t$ of the random solution at time $t\\in\\R$ as $t\\to\\infty$. It is assumed that the initial measure $\\mu_0$ has zero mean, translation-invariant covariance matrices, and finite expected energy density. We also assume that $\\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing condition. The main result is the convergence of $\\mu_t$ to a Gaussian measure $\\mu_\\infty$ as $t\\to\\infty$, which gives a Central Limit Theorem (CLT) for the wave equation. The proof for the case of constant coefficients is based on an analysis of long-time asymptotics of the solution in the Fourier representation and Bernstein's `room-corridor' argument. The case of variable coefficients is treated by using a version of the scattering theory for infinite energy solutions, based on Vainberg's results on local energy decay.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-08-19T15:47:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35L05",
      "60F05"
    ],
    "keywords": [
      "wave equation",
      "odd dimensions",
      "equilibrium distribution",
      "convergence",
      "ibragimov-linnik-type space mixing condition"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.ph...8039D"
    }
  }
}