{
  "id": "1610.03292",
  "version": "v1",
  "published": "2016-10-11T12:05:31.000Z",
  "updated": "2016-10-11T12:05:31.000Z",
  "title": "Oscillating heat kernels on ultrametric spaces",
  "authors": [
    "Alexander Bendikov",
    "Wojciech Cygan",
    "Wolfgang Woess"
  ],
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "Let $(X,d)$ be a proper ultrametric space. Given a measure $m$ on $X$ and a function $B \\mapsto C(B)$ defined on the collection of all non-singleton balls $B$ of $X$, we consider the associated hierarchical Laplacian $L=L_{C}\\,$. The operator $L$ acts in $\\mathcal{L}^{2}(X,m),$ is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel $\\mathfrak{p}(t,x,y)$ with respect to $m$. We consider the case when $X$ has a transitive group of isometries under which the operator $L$ is invariant and study the asymptotic behaviour of the function $t\\mapsto \\mathfrak{p}(t,x,x)=\\mathfrak{p}(t)$. It is completely monotone, but does not vary regularly. When $X=\\mathbb{Q}_{p}\\,$, the ring of $p$-adic numbers, and $L=\\mathcal{D}^{\\alpha} $, the operator of \\ fractional derivative of order $\\alpha,$ we show that $\\mathfrak{p}(t)=t^{-1/\\alpha}\\mathcal{A}% (\\log_{p}t)$, where $\\mathcal{A}(\\tau)$ is a continuous non-constant $\\alpha$-periodic function. We also study asymptotic behaviour of $\\min\\mathcal{A}$ and $\\max\\mathcal{A}$ as the space parameter $p$ tends to $\\infty$. When $X=S_{\\infty}\\,$, the infinite symmetric group, and $L$ is a hierarchical Laplacian with metric structure analogous to $\\mathcal{D}^{\\alpha},$ we show that, contrary to the previous case, the completely monotone function $\\mathfrak{p}(t)$ oscillates between two functions $\\psi(t)$ and $\\Psi(t)$ such that $\\psi(t)/\\Psi(t)\\to 0$ as $t \\to \\infty\\,$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-11T12:05:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J35",
      "12H25",
      "20K25",
      "47S10",
      "60J25"
    ],
    "keywords": [
      "oscillating heat kernels",
      "infinite symmetric group",
      "hierarchical laplacian",
      "proper ultrametric space",
      "pure point spectrum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}