Isotropic Markov semigroups on ultra-metric spaces

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, ∞), a symmetric Markov semigroup {P t}t>0 acting in L2(X, μ) is
constructed. Let {Xt} 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 L and its spectrum, which is pure point. In the par-
ticular case when X = Qn p , where Qp 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.