Trees as Brelot spaces.
A Brelot space is a connected locally compact, noncompact Hausdorff space together with the choice of a sheaf of functions on this space, which are called harmonic. The authors prove that by considering functions on a tree to be functions on the edges as well as on the vertices, a tree becomes a Brelot space. This leads to plenty of results on potential theory. By restricting the functions to the vertices only, certain new advances of the potential theory on trees are obtained in the usual sense. This concerns also trees whose nearest-neighbor transition probabilities are defined by both transient and recurrent random walks.
scientific article; zbMATH DE number 1981416
scientific article