{
  "id": "1601.01652",
  "version": "v1",
  "published": "2016-01-07T20:00:49.000Z",
  "updated": "2016-01-07T20:00:49.000Z",
  "title": "Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\\geq 3$",
  "authors": [
    "Chiranjib Mukherjee",
    "Alexander Shamov",
    "Ofer Zeitouni"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\\eps,t}= \\frac 12 \\Delta u_{\\eps,t} d t+ \\beta \\eps^{\\frac{d-2}{2}}\\, \\, u_{\\eps, t} \\, d B_{\\eps,t} , \\;\\;u_{\\eps,0}=1,$$ in dimension $d\\geq 3$, where $B_{\\eps,t}$ is a spatially smoothed (at scale $\\eps$) space-time white noise, and $\\beta>0$ is a parameter. We show the existence of a $\\bar\\beta\\in (0,\\infty)$ so that the solution exhibits weak disorder when $\\beta<\\bar\\beta$ and strong disorder when $\\beta > \\bar\\beta$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-07T20:00:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "continuous directed polymer",
      "strong disorder",
      "multiplicative noise stochastic heat equation",
      "smoothed multiplicative noise stochastic heat",
      "space-time white noise"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}