A global Riesz decomposition theorem on trees without positive potentials

Abstract

We study the potential theory of trees with nearest-neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non-negative function H, harmonic outside the root e and vanishing only at e, and a substitute notion of potential which we call H-potential. We define the flux of a superharmonic function outside a finite set of vertices, give some simple formulas for calculating the flux and derive a global Riesz decomposition theorem for superharmonic functions with a harmonic minorant outside a finite set. We discuss the connection of the H-potentials with other notions of potentials for recurrent Markov chains in the literature.