Isotropic Markov semigroups on ultra-metric spaces

A. D. Bendikov ; A. A. Grigor'yan ; Ch. Pittet ; W. Woess

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 69 (2014) no. 4, pp. 589-680

Résumé

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. Bibliography: 66 titles.

Détail

Keywords: ultra-metric measure space, metric trees, isotropic Markov semigroups, Markov generators, heat kernels, transition density, $p$-number field, Vladimirov–Taibleson operator, nearest neighbour random walk on a tree, Dirichlet form, harmonic functions with finite energy, traces of harmonic functions with finite energy.