{
  "id": "2506.20146",
  "version": "v1",
  "published": "2025-06-25T05:51:49.000Z",
  "updated": "2025-06-25T05:51:49.000Z",
  "title": "Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics",
  "authors": [
    "Xi Geng",
    "Weijun Xu"
  ],
  "comment": "51 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We establish the second-order moment asymptotics for a parabolic Anderson model $\\partial_{t}u=(\\Delta+\\xi)u$ in the hyperbolic space with a regular, stationary Gaussian potential $\\xi$. It turns out that the growth and fluctuation asymptotics both are identical to the Euclidean situation. As a result, the solution exhibits the same moment intermittency property as in the Euclidean case. An interesting point here is that the fluctuation exponent is determined by a variational problem induced by the Euclidean (rather than hyperbolic) Laplacian. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-25T05:51:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "parabolic anderson model",
      "hyperbolic space",
      "annealed asymptotics",
      "curvature dilation effect",
      "stationary gaussian potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}