### 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 |

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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

### Cite this

*Journal of spectral theory*,

*9*(1), 195-226. https://doi.org/10.4171/JST/245