A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbortransition probability that yields a recurrent random walk andshow that, although such trees have no positive potentials,many of the standard results of potential theory can be transferredto this setting. We accomplish this by defining a non-negativefunction H, harmonic outside the root e and vanishing only ate, and a substitute notion of potential which we call H-potential.We define the flux of a superharmonic function outside a finiteset of vertices, give some simple formulas for calculating theflux and derive a global Riesz decomposition theorem for superharmonicfunctions with a harmonic minorant outside a finite set. Wediscuss the connection of the H-potentials with other notionsof potentials for recurrent Markov chains in the literature.