Transitions on a noncompact Cantor set and random walks on its defining tree

Kigami, Jun

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1090-1129.

Nous commençons par introduire des ensembles de Cantor non-compacts, ainsi que leurs arbres associés. Ils peuvent être considerés comme une généralisation naturelle des nombres $p$-adiques. Nous construisons ensuite une classe de processus de saut sur un ensemble de Cantor non-compact, à l’aide d’un couple de valeurs propres et de mesures. De plus, nous obtenons des expressions concrètes pour les noyaux de la chaleurs associés à ces processus de saut et pour les probabilités de transition correspondantes. Sous certaines hypothèses de régularité sur les valeurs propres et les mesures, nous construisons ensuite des métriques intrinsèques sur cet ensemble de Cantor non-compact afin d’obtenir des estimations fines sur les noyaux de la chaleur et les probabilités de transitions. Finalement, nous montrons que les marches aléatoires sur l’arbre définissant l’ensemble de Cantor non-compact induisent une sous-classe des processus de saut discutés dans la seconde partie de l’article.

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of $p$-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions on eigenvalues and measures. Finally transient random walks on the defining tree are shown to induce a subclass of jump processes discussed in the second part.

Classification : 60J75, 60J35, 31C35, 05C81
Mots clés : noncompact Cantor set, $p$-adic numbers, tree, jump process, Dirichlet forms, random walks, Martin boundary

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