Classification of Random Walks and Green’s Theorem on Infinite Homogeneous Trees

In the context of random walks whose states are the vertices of an infinite tree, a classification of random walks is given as transient or recurrent. On the infinite homogeneous trees with the assumption that the transition probability between any two neighboring states are the same, a form of the classical Green’s formula is derived. As a consequence, two versions of the mean-value property for median functions are obtained.