Oscillating heat kernels on ultrametric spaces

  • Alexander Bendikov

    University of Wroclaw, Poland
  • Wojciech Cygan

    University of Wroclaw, Poland and Technische Universität Graz, Austria
  • Wolfgang Woess

    Technische Universität Graz, Austria
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Abstract

Let be a proper ultrametric space. Given a measure on and a function defined on the collection of all non-singleton balls of , we consider the associated hierarchical Laplacian . The operator acts in is essentially self-adjoint and has a pure point spectrum. It admits a continuous heat kernel with respect to . We consider the case when has a transitive group of isometries under which the operator is invariant and study the asymptotic behaviour of the function . It is completely monotone, but does not vary regularly. When , the ring of -adic numbers, and , the operator of fractional derivative of order , we show that , where is a continuous non-constant -periodic function. We also study asymptotic behaviour of min and max as the space parameter tends to . When , the infinite symmetric group, and is a hierarchical Laplacian with metric structure analogous to we show that, contrary to the previous case, the completely monotone function oscillates between two functions and such that as .

Cite this article

Alexander Bendikov, Wojciech Cygan, Wolfgang Woess, Oscillating heat kernels on ultrametric spaces. J. Spectr. Theory 9 (2019), no. 1, pp. 195–226

DOI 10.4171/JST/245