Recurrence characterized by constant nonnegative superharmonic functions

An irreducible Markov chain is recurrent exactly when every nonnegative superharmonic function for a Markov chain is constant. In the recurrent case, absorb the chain at $j$. The inequality $g_i\ge ({\tilde{P}}^{n}g{)}_i\ge g_j{\mathbb{P}}_i({\tau }_j\le n)$ and almost-sure hitting give $g_i\ge g_j$; exchange $i,j$. In the transient case, the hitting probability of a state $j$ is one there and satisfies the superharmonic inequality, but its one-step average at $j$ is the strictly smaller return probability. It is a nonconstant example.