= Hitting probability is the minimal nonnegative harmonic extension
{title2=$h_i^A=\mathbb P_i(H^A<\infty)$}
The <hitting probability> of a set is one on that set and transition-harmonic elsewhere by the <Markov property>. Finite-horizon hitting probabilities begin at the boundary indicator and increase through the same positive transition recursion. Any nonnegative solution dominates every iterate; monotone passage to infinite time shows that it dominates the hitting probability. This proves minimality without requiring almost-sure hitting.
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