HARMONIC-FUNCTIONS ON CARTESIAN PRODUCTS OF TREES WITH FINITE GRAPHS

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. © 1991.