Some potential theoretic results on an infinite network

Abstract

The greatest harmonic minorant of a superharmonic function is determined as the limit of a sequence of solutions for discrete Dirichlet problems on finite subnetworks. Without using the Green kernel explicitly, a positive superharmonic function is decomposed uniquely as a sum of a potential and a harmonic function. The infimum of a left directed family of harmonic functions is shown to be either $-\infty$ or harmonic. As applications, we study the reduced functions and their properties. We show the existence of the Green kernel with the aid of our reduced function.