Renewal theory on the oriented tree

Abstract

The affine group of a tree is the group of the isometries of a homogeneous tree that fix an end of its boundary. Consider a probability measure on this group and the associated random walk. The main goal of this paper is to determine the accumulation points of the potential kernel at the infinity. In particular we show that under suitable regularity hypotheses this kernel can be continuously extended to the tree boundary and we determine the limit measures.