Tangential Curves and Fatou's Theorem on Trees

Abstract

The paper shows that the distinction between tangential sequences and tangential curves, which is known to be relevant for the boundary behaviour of harmonic functions in Euclidean half spaces, is also meaningful and relevant in the discrete setting of the potential theory on a tree associated to a very regular, nearest neighbour random walk. In particular, the paper first defines the class of tangential curves on a tree, and then proves that, for any family of tangential curves which have the same order of tangency and are associated (and converging) to points in the boundary, there is a bounded harmonic function which, for (almost) every point in the boundary, has positive oscillation along the corresponding curve. This result does not hold if, in place of tangential curves, tangential sequences are admitted. The tree is not assumed to be homogeneous. In particular, the results do not depend on the existence of isometries.