Isotropic Markov semigroups on ultra-metric spaces

Abstract

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure μ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,μ). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

References in corpus (5)

• Traces of symmetric Markov processes and their characterizations
• On the spectrum of the hierarchical Laplacian
• Spectral properties of a class of random walks on locally finite groups
• Taibleson Operators, p-adic Parabolic Equations and Ultrametric Diffusion
• On the duality between jump processes on ultrametric spaces and random walks on trees

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• Minkowski dimension and explicit tube formulas for $p$-adic fractal strings
• Convergences and projection Markov property of Markov processes on ultrametric spaces