= Nonnegative superharmonic function for a Markov chain
{title2=$g\ge Pg\ge0$}
For a <transition matrix> $P$, a finite-valued function on its state space is nonnegative superharmonic when $g\ge Pg\ge0$. Thus $g\ge0$ and its expected value cannot increase after one step. Replacing transitions on a target set by identity rows preserves $g\ge\widetilde Pg$. Iterating this inequality gives $g\ge\widetilde P^{\,n}g$, a useful bound for <hitting probabilities> without a separate optional-stopping theorem.
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