{
  "id": "1607.00682",
  "version": "v1",
  "published": "2016-07-03T21:01:11.000Z",
  "updated": "2016-07-03T21:01:11.000Z",
  "title": "Large time asymptotics for the parabolic Anderson model driven by space and time correlated noise",
  "authors": [
    "Jingyu Huang",
    "Khoa Lê",
    "David Nualart"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1509.00897",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we study the linear stochastic heat equation on $\\mathbb{R}^\\ell$, driven by a Gaussian noise which is colored in time and space. The spatial covariance satisfies general assumptions and includes examples such as the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\\in (\\frac 14, \\frac 12]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman-Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents and lower and upper exponential growth indices in terms of a variational quantity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-03T21:01:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model driven",
      "large time asymptotics",
      "time correlated noise",
      "spatial covariance satisfies general assumptions",
      "upper exponential growth indices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}