{
  "id": "1208.0330",
  "version": "v2",
  "published": "2012-08-01T19:58:41.000Z",
  "updated": "2013-03-01T13:36:23.000Z",
  "title": "The parabolic Anderson model in a dynamic random environment: basic properties of the quenched Lyapunov exponent",
  "authors": [
    "Dirk Erhard",
    "Frank den Hollander",
    "Grégory Maillard"
  ],
  "comment": "50 pages. The comments of the referee are incorporated into the paper. A missing counting estimate was added in the proofs of Lemma 3.6 and Lemma 4.7",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study 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 the u-field and the \\xi-field are \\R-valued, \\kappa \\in [0,\\infty) is the diffusion constant, and $\\Delta$ is the discrete Laplacian. 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. Our goal is to prove a number of basic properties of the solution u under assumptions on $\\xi$ that are as weak as possible. Throughout the paper 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. Under a mild assumption on the tails of the distribution of \\xi, we show that the solution to the parabolic Anderson equation exists and is unique for all \\kappa\\in [0,\\infty). Our main object of interest is the quenched Lyapunov exponent \\lambda_0(\\kappa)=\\lim_{t\\to\\infty}\\frac{1}{t}\\log u(0,t). Under certain weak space-time mixing conditions on \\xi, we show the following properties: (1)\\lambda_0(\\kappa) does not depend on the initial condition u_0; (2)\\lambda_0(\\kappa)<\\infty for all \\kappa\\in [0,\\infty); (3)\\kappa \\mapsto \\lambda_0(\\kappa) is continuous on [0,\\infty) but not Lipschitz at 0. We further conjecture: (4)\\lim_{\\kappa\\to\\infty}[\\lambda_p(\\kappa)-\\lambda_0(\\kappa)]=0 for all p\\in\\N, where \\lambda_p (\\kappa)=\\lim_{t\\to\\infty}\\frac{1}{pt}\\log\\E([u(0,t)]^p) is the p-th annealed Lyapunov exponent. Finally, we prove that our weak space-time mixing conditions on \\xi are satisfied for several classes of interacting particle systems.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-03-01T13:36:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H25",
      "82C44",
      "60F10",
      "35B40"
    ],
    "keywords": [
      "quenched lyapunov exponent",
      "dynamic random environment",
      "parabolic anderson model",
      "basic properties",
      "parabolic anderson equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.0330E"
    }
  }
}