Oscillating heat kernels on ultrametric spaces

Alexander D. Bendikov, Wojciech Cygan, Wolfgang Woess

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

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.

Originalspracheenglisch
Seiten (von - bis)195-226
Seitenumfang32
FachzeitschriftJournal of spectral theory
Jahrgang9
Ausgabenummer1
DOIs
PublikationsstatusVeröffentlicht - 2019

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    ASJC Scopus subject areas

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

    Fields of Expertise

    • Information, Communication & Computing

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