Harmonic Function Theory on Infinite Networks

Abstract

Similar to the classification of Riemannian manifolds as hyperbolic and parabolic, an infinite network X is said to be hyperbolic if the Green function is defined on X, otherwise X is said to be parabolic. Different criteria like Minimum Principle, the harmonic measure of the point at infinity and the sections determined by a vertex in X are introduced to effect the classification. The first part of this chapter studies the properties of superharmonic functions in hyperbolic networks: the existence of non-constant bounded or positive harmonic functions on X, domination principle and balayage. The second part carries out, in parabolic networks, the construction of superharmonic functions in X with point harmonic singularity, similar to the logarithmic potentials in the plane; such functions do not naturally appear in the context of random walks and electrical networks. Balayage, Maximum Principle and the representation of harmonic functions outside a finite set are considered in parabolic networks, leading to the concepts of flux at infinity and pseudo-potentials which play an important role in developing a potential theory on parabolic networks.