Ultrametric and Tree Potential

Abstract

In this article we study which infinite matrices are potential matrices. We tackle this problem in the ultrametric framework by studying infinite tree matrices and ultrametric matrices. For each tree matrix, we show the existence of an associated symmetric random walk and study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. For ultrametric matrices, we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension.