Non-tangential, radial and stochastic asymptotic properties of harmonic functions on trees

TL;DR

This paper establishes that non-tangential asymptotic properties of harmonic functions on trees are almost surely equivalent to radial and stochastic properties, extending known results in the field.

Contribution

It proves the equivalence of non-tangential and radial/stochastic properties for harmonic functions on trees, generalizing previous convergence and boundedness results.

Findings

Non-tangential properties are almost surely equivalent to radial and stochastic properties.

The results apply to harmonic functions with bounded transition probabilities on trees.

Extends classical convergence results to non-tangential contexts.

Abstract

For a harmonic function on a tree with random walk whose transition probabilities are bounded between two constants in (0,1/2), it is known that the radial and stochastic properties of convergence, boundedness and finiteness of energy are all a.s. equivalent. We prove here that the analogous non-tangential properties are a.e. equivalent to the above ones.