Harmonic functions on cartesian products of trees with finite graphs

Abstract

Let G be a graph which is the Cartesian product of an infinite, locally finite tree T and a finite, connected graph A . On G , consider a stochastic transition operator P giving rise to a transient random walk and such that positive transitions occur only along the edges of G . We construct a matrix-valued kernel on T , which extends naturally in the second variable to the space of ends Ω of T . This kernel is used to derive a unique integral representation over Ω of all—not necessarily positive—functions on G which are harmonic with respect to P . We explain the relation with the Martin boundary and the positive harmonic functions and, as a particular case, we show what happens when A arises from a finite abelian group and P is compatible with the structure of A .