{
  "id": "math/0409274",
  "version": "v1",
  "published": "2004-09-16T09:23:09.000Z",
  "updated": "2004-09-16T09:23:09.000Z",
  "title": "Long time behavior of the solutions to non-linear Kraichnan equations",
  "authors": [
    "Alice Guionnet",
    "Christian Mazza"
  ],
  "comment": "32 pages",
  "categories": [
    "math.PR",
    "cond-mat.dis-nn",
    "math.OA"
  ],
  "abstract": "We consider the solution of a nonlinear Kraichnan equation $$\\partial_s H(s,t)=\\int_t^s H(s,u)H(u,t) k(s,u) du,\\quad s\\ge t$$ with a covariance kernel $k$ and boundary condition $H(t,t)=1$. We study the long time behaviour of $H$ as the time parameters $t,s$ go to infinity, according to the asymptotic behaviour of $k$. This question appears in various subjects since it is related with the analysis of the asymptotic behaviour of the trace of non-commutative processes satisfying a linear differential equation, but also naturally shows up in the study of the so-called response function and aging properties of the dynamics of some disordered spin systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-09-16T09:23:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B44",
      "46L54",
      "45G10"
    ],
    "keywords": [
      "long time behavior",
      "non-linear kraichnan equations",
      "asymptotic behaviour",
      "linear differential equation",
      "long time behaviour"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......9274G"
    }
  }
}