Oscillating heat kernels on ultrametric spaces

Alexander D. Bendikov, Wojciech Cygan, Wolfgang Woess

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Abstract

Let (X; d) be a proper ultrametric space. Given ameasuremonX and a function B → 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 L 2 (X;m); is essentially self-Adjoint and has a pure point spectrum. It admits a continuous heat kernel 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 → p(t; x; x) = p(t). It is completely monotone, but does not vary regularly. When X = Q p , the ring of p-Adic numbers, and L = Dα, the operator of fractional derivative of order α we show that p(t) = t -1 =αA(logp t), where A(τ) is a continuous non-constant α-periodic function. We also study asymptotic behaviour of minA and maxA as the space parameter p tends to ∞. When X = S∞, the infinite symmetric group, and L is a hierarchical Laplacian with metric structure analogous to Dα; we show that, contrary to the previous case, the completely monotone function p(t) oscillates between two functions (t) and ψ(t) such that (t)=ψ(t)→ 0 as t → 1.

Original languageEnglish
Pages (from-to)195-226
Number of pages32
JournalJournal of spectral theory
Volume9
Issue number1
DOIs
Publication statusPublished - 2019

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Keywords

  • Heat kernel
  • Hierarchical Laplacian
  • Isotropic Markov semigroup
  • Oscillations.
  • Return probabilities
  • Ultrametric space

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Geometry and Topology
  • Mathematical Physics

Fields of Expertise

  • Information, Communication & Computing

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