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 language | English |
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Pages (from-to) | 195-226 |
Number of pages | 32 |
Journal | Journal of Spectral Theory |
Volume | 9 |
Issue number | 1 |
DOIs | |
Publication status | Published - 2019 |
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