Hitting probability is the minimal nonnegative harmonic extension ($h_i^A={\mathbb{P}}_i(H^A<\infty )$)

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.