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

Abstract

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.