 Nonnegative superharmonic function for a Markov chain

For a transition matrix $P$, a finite-valued function on its state space is nonnegative superharmonic when $g\ge Pg\ge 0$. Thus $g\ge 0$ and its expected value cannot increase after one step. Replacing transitions on a target set by identity rows preserves $g\ge \tilde{P}g$. Iterating this inequality gives $g\ge {\tilde{P}}^{n}g$, a useful bound for hitting probabilities without a separate optional-stopping theorem.