{
  "id": "1010.4868",
  "version": "v2",
  "published": "2010-10-23T10:15:53.000Z",
  "updated": "2011-03-23T14:12:20.000Z",
  "title": "Parabolic Anderson model with a finite number of moving catalysts",
  "authors": [
    "Fabienne Castell",
    "Onur Gün",
    "Grégory Maillard"
  ],
  "comment": "In honour of J\\\"urgen G\\\"artner on the occasion of his 60th birthday, 25 pages. Updated version following the referee's comments",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the parabolic Anderson model (PAM) which is given by the equation $\\partial u/\\partial t = \\kappa\\Delta u + \\xi u$ with $u\\colon\\, \\Z^d\\times [0,\\infty)\\to \\R$, where $\\kappa \\in [0,\\infty)$ is the diffusion constant, $\\Delta$ is the discrete Laplacian, and $\\xi\\colon\\,\\Z^d\\times [0,\\infty)\\to\\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a \"reactant\" $u$ under the influence of a \"catalyst\" $\\xi$. In the present paper we focus on the case where $\\xi$ is a system of $n$ independent simple random walks each with step rate $2d\\rho$ and starting from the origin. We study the \\emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\\ $\\xi$ and show that these exponents, as a function of the diffusion constant $\\kappa$ and the rate constant $\\rho$, behave differently depending on the dimension $d$. In particular, we give a description of the intermittent behavior of the system in terms of the annealed Lyapunov exponents, depicting how the total mass of $u$ concentrates as $t\\to\\infty$. Our results are both a generalization and an extension of the work of G\\\"artner and Heydenreich 2006, where only the case $n=1$ was investigated.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-23T14:12:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model",
      "finite number",
      "moving catalysts",
      "lyapunov exponents",
      "independent simple random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.4868C"
    }
  }
}