## 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, ∞), 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.

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.

Original language | English |
---|---|

Pages (from-to) | 589-680 |

Journal | Russian Mathematical Surveys |

Volume | 69 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 |

## Fields of Expertise

- Information, Communication & Computing

## Treatment code (Nähere Zuordnung)

- Basic - Fundamental (Grundlagenforschung)