Параметры публикации

Тип публикации: 

Статья в журнале/сборнике


Isotropic Markov semigroups on ultra-metric spaces

Наименование источника: 

  • Russian Mathematical Surveys

Обозначение и номер тома: 

Т. 69, № 4


  • Лондон


  • Turpion Limited

Год издания: 



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 {Pt}t≥0 acting in L2(X, μ). Let {Xt} 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 = Qnp , where Qp 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 the Markov process {Xt} and Kigami’s process on the boundary of a tree, which is induced by a random walk on the tree. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

Библиографическая ссылка: 

Bendikov A., Григорьян А.А., Pittet C., Woess W. Isotropic Markov semigroups on ultra-metric spaces // Russian Mathematical Surveys. 2014. Т. 69, № 4. С. 3-102.