Isotropic Markov semigroups on ultra-metric spaces

Abstract

Let be a separable ultra-metric space with compact balls. Given a reference measure on and a distance distribution function on , a symmetric Markov semigroup acting in is constructed. Let be the corresponding Markov process. The authors obtain upper and lower bounds for 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 , where is the field of -adic numbers, the construction recovers the Taibleson Laplacian (spectral multiplier), and one can also apply the theory to the study of the Vladimirov Laplacian. Even in this well-established setting, several of the results are new. The paper also describes the relation between the processes involved and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, examples illustrating the interplay between the fractional derivatives and random walks are provided. Bibliography: 66 titles.