Isotropic Markov semigroups on ultra-metric spaces

Let $(X,d)$ be a separable ultra-metric space with compact balls. Given a reference measure $\mu $ on $X$ and a distance distribution function $\sigma$ on $[0,\infty)$, a symmetric Markov semigroup $\{P^{t}\}_{t\geqslant 0}$ acting in $L^{2}(X,\mu )$ is constructed. Let $\{\mathcal{X}_{t}\}$ 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 $\mathcal{L}$ and its spectrum, which is pure point. In the particular case when $X=\mathbb{Q}_{p}^{n}$, where $\mathbb{Q}_{p}$ is the field of $p$-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.
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