= 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 $g_i\ge(\widetilde P^{\,n}g)_i\ge g_j\mathbb P_i(\tau_j\le n)$ 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.
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