A Riesz decomposition theorem on harmonic spaces without positive potentials

Potential theory on trees and mutliplication operators

“…In this section we describe the work in [18]. We let Ω be a Brelot space in which constants are harmonic and there are no potentials.…”

“…Some of the discrete ideas discussed above in section 3 can be formulated in this setting. We describe here some of the results in [18] where standard and H-potential are defined, and a Riesz decomposition theorem for admissible superharmonic functions is proved.…”

“…Definition 5.1. ( [18], Definition 2.1) Let K ⊂ Ω be a nonempty compact set that is not polar (that is, there does not exist a superharmonic function on Ω which is identically ∞ on K). A function H harmonic off K is called a standard for Ω associated with K if the following is true: given any function h which is harmonic off a compact set, there exist a unique function h Ω harmonic on Ω and a unique real number α such that b = h − αH − h Ω is bounded off a compact set and lim inf x→∞ b(x) = 0, where the lim inf is taken with respect to the Alexandrov one-point compactification of Ω.…”

“…In this section we describe the work in [18] where trees are viewed as Brelot spaces, and the task is to classify the possible Brelot structures one can have.…”

“…In §5, we describe the work in [18] where we show that our theories of flux and H-potential can be formulated in a general Brelot space without positive potentials, and so we obtain a Riesz decomposition theorem for admissible superharmonic functions in this general setting. We show that the theories described here and in §3 agree in the special case that we extend linearly along the edges of the tree.…”