{
  "id": "2301.05121",
  "version": "v1",
  "published": "2023-01-12T16:20:30.000Z",
  "updated": "2023-01-12T16:20:30.000Z",
  "title": "Singular SPDEs on Homogeneous Lie Groups",
  "authors": [
    "Avi Mayorcas",
    "Harprit Singh"
  ],
  "comment": "69 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form $$\\partial_t u = \\mathfrak{L} u+ F(u, \\xi)\\ ,$$ where the differential operator $\\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space $\\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\\mathbb{G}$ such that the differential operator $\\partial_t -\\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on $\\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator $\\partial_t -\\Delta_v - v\\cdot \\nabla_x$ and heat-type operators associated to sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations $$\\partial_t u = \\sum_{i} X^2_i u + u (\\xi-c)$$ on the compact quotient of an arbitrary Carnot group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-12T16:20:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60L30",
      "60H17",
      "35H10",
      "35K70"
    ],
    "keywords": [
      "singular spdes",
      "parabolic anderson type equations",
      "translation invariant hypoelliptic operator",
      "differential operator",
      "arbitrary carnot group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 69,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}