{
  "id": "1304.2274",
  "version": "v2",
  "published": "2013-04-08T17:21:26.000Z",
  "updated": "2013-07-11T20:53:32.000Z",
  "title": "The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent",
  "authors": [
    "Dirk Erhard",
    "Frank den Hollander",
    "Gregory Maillard"
  ],
  "comment": "35 pages, 4 figures, the main result in this version is stronger than in the previous version",
  "categories": [
    "math.PR"
  ],
  "abstract": "We continue our study of the parabolic Anderson equation $\\partial u(x,t)/\\partial t = \\kappa\\Delta u(x,t) + \\xi(x,t)u(x,t)$, $x\\in\\Z^d$, $t\\geq 0$, where $\\kappa \\in [0,\\infty)$ is the diffusion constant, $\\Delta$ is the discrete Laplacian, and $\\xi$ plays the role of a \\emph{dynamic random environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$, $x\\in\\Z^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\\kappa$, split into two at rate $\\xi \\vee 0$, and die at rate $(-\\xi) \\vee 0$. We assume that $\\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies $\\E(|\\xi(0,0)|)<\\infty$, where $\\E$ denotes expectation w.r.t.\\ $\\xi$. Our main object of interest is the quenched Lyapunov exponent $\\lambda_0 (\\kappa) = \\lim_{t\\to\\infty} \\frac{1}{t}\\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $\\xi$-a.s., is finite and continuous on $[0,\\infty)$, is globally Lipschitz on $(0,\\infty)$, is not Lipschitz at 0, and satisfies $\\lambda_0(0) = \\E(\\xi(0,0))$ and $\\lambda_0(\\kappa) > \\E(\\xi(0,0))$ for $\\kappa \\in (0,\\infty)$.In the present paper we show that $\\lim_{\\kappa\\to\\infty} \\lambda_0(\\kappa) =\\E(\\xi(0,0))$ under an additional space-time mixing condition on $\\xi$. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of $\\xi$ that fulfill our assumption for which the annealed Lyapunov exponent $\\lambda_1(\\kappa) = \\lim_{t\\to\\infty} \\frac{1}{t}\\log \\E(u(0,t))$ is infinite on $[0,\\infty)$, a situation that is referred to as strongly catalytic behavior.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-07-11T20:53:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60H25",
      "82C44",
      "35B40",
      "60F10"
    ],
    "keywords": [
      "parabolic anderson model",
      "quenched lyapunov exponent",
      "dynamic random environment",
      "space-time ergodicity",
      "independent simple random walks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.2274E"
    }
  }
}