Renewal Theory on the Affine Group of an Oriented Tree

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 when g tends to infinity. In particular we show that under suitable regularity hypotheses this kernel can be continuously extended to the tree's boundary and we determine the limit measures.