  Treebolic space HT(q,p) is a key example of a strip complex in the sense of
Bendikov, Saloff-Coste, Salvatori, and Woess [Adv. Math. 226 (2011), 992-1055].
It is an analog of the Sol geometry, namely, it is a horocylic product of the
hyperbolic upper half plane with a "stretching" parameter q and the homogeneous
tree T with vertex degree p+1 < 2, the latter seen as a one-complex. In a
previous paper [arXiv:1212.6151, Rev. Mat. Iberoamericana, in print] we have
explored the metric structure and isometry group of that space. Relying on the
analysis on strip complexes, a family of natural Laplacians with "vertical
drift" and the escape to infinity of the associated Brownian motion were
considered.
  Here, we undertake a potential theoretic study, investigating the positive
harmonic functions associated with those Laplacians. The methodological
subtleties stem from the singularites of treebolic space at its bifurcation
lines. We first study harmonic functions on simply connected sets with
"rectangular" shape that are unions of strips. We derive a Poisson
representation and obtain a solution of the Dirichlet problem on sets of that
type. This provides properties of the density of the induced random walk on the
collection of all bifuraction lines. Subsequently, we prove that each positive
harmonic function with respect to that random walk has a unique extension which
is harmonic with respect to the Laplacian on treebolic space. Finally, we
derive a decomposition theorem for positive harmonic functions on the entire
space that leads to a characterisation of the weak Liouville property. We
determine all minimal harmonic functions in those cases where our Laplacian
arises from lifting a (smooth) hyperbolic Laplacian with drift from the
hyperbolic plane to treebolic space.
