Isotropic Markov semigroups on ultra-metric spaces

TL;DR

This paper constructs and analyzes symmetric Markov semigroups on ultra-metric spaces, including p-adic numbers, providing bounds, spectral analysis, and connections to quantum mechanics and jump processes.

Contribution

It introduces a new framework for Markov processes on ultra-metric spaces, extending fractional derivatives and Laplacians to these settings with novel results.

Findings

Derived bounds for transition densities and Green functions.

Established spectral properties of the Markov generator.

Connected the processes to p-adic quantum mechanics and jump processes.

Abstract

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure \mu\ 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,\mu). 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.