Brownian motion on treebolic space: positive harmonic functions
  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. 
Alexander Bendikov, Laurent Saloff-Coste, Maura Salvatori, and   Wolfgang Woess
math.PR math.DG math.GR