{
  "id": "1208.0306",
  "version": "v1",
  "published": "2012-08-01T18:15:53.000Z",
  "updated": "2012-08-01T18:15:53.000Z",
  "title": "Moment asymptotics for branching random walks in random environment",
  "authors": [
    "Onur Gün",
    "Wolfgang König",
    "Ozren Sekulović"
  ],
  "comment": "18 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider the long-time behaviour of a branching random walk in random environment on the lattice $\\Z^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments $< m_n^p > $, i.e., the $p$-th moments over the medium of the $n$-th moment over the migration and killing/branching, of the local and global population sizes. For $n=1$, this is well-understood \\cite{GM98}, as $m_1$ is closely connected with the parabolic Anderson model. For some special distributions, \\cite{A00} extended this to $n\\geq2$, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for $m_n$. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that $< m_n^p >$ and $< m_1^{np} >$ are asymptotically equal, up to an error $\\e^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac-type formula for $m_n$, which we establish using the spine techniques developed in \\cite{HR11}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-01T18:15:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J80",
      "60J55",
      "60F10",
      "60K37"
    ],
    "keywords": [
      "branching random walk",
      "random environment",
      "moment asymptotics",
      "th moment",
      "direct feynman-kac-type formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.0306G"
    }
  }
}