Изотропные Марковские Полугруппы На Ультраметрических Пространствах

Abstract

Let (X, d) be a separable ultra-metric space with compact balls. Given a reference measure µ on X and a distance distribution function σ on [0 , ∞), we construct a symmetric Markov semigroup {P t } t≥0 acting in L 2 (X, µ). Let {X t } be 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 L and its spectrum which is pure point. In the particular case when X = Q n p , where Q p is the field of p-adic numbers, our construction recovers the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian. 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.