{
  "id": "2012.12512",
  "version": "v1",
  "published": "2020-12-23T06:45:49.000Z",
  "updated": "2020-12-23T06:45:49.000Z",
  "title": "Phase Analysis for a family of Stochastic Reaction-Diffusion Equations",
  "authors": [
    "Davar Khoshnevisan",
    "Kunwoo Kim",
    "Carl Mueller",
    "Shang-Yuan Shiu"
  ],
  "comment": "69 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a reaction-diffusion equation of the type \\[ \\partial_t\\psi = \\partial^2_x\\psi + V(\\psi) + \\lambda\\sigma(\\psi)\\dot{W} \\qquad\\text{on $(0\\,,\\infty)\\times\\mathbb{T}$}, \\] subject to a \"nice\" initial value and periodic boundary, where $\\mathbb{T}=[-1\\,,1]$ and $\\dot{W}$ denotes space-time white noise. The reaction term $V:\\mathbb{R}\\to\\mathbb{R}$ belongs to a large family of functions that includes Fisher--KPP nonlinearities [$V(x)=x(1-x)$] as well as Allen-Cahn potentials [$V(x)=x(1-x)(1+x)$], the multiplicative nonlinearity $\\sigma:\\mathbb{R}\\to\\mathbb{R}$ is non random and Lipschitz continuous, and $\\lambda>0$ is a non-random number that measures the strength of the effect of the noise $\\dot{W}$. The principal finding of this paper is that: (i) When $\\lambda$ is sufficiently large, the above equation has a unique invariant measure; and (ii) When $\\lambda$ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-23T06:45:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H15",
      "35R60"
    ],
    "keywords": [
      "stochastic reaction-diffusion equations",
      "phase analysis",
      "denotes space-time white noise",
      "unique invariant measure",
      "non-trivial line segment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}