= Solution
For $i\notin A$, condition on the first step. The <Markov property> makes the chance of eventual entry after reaching $j$ equal to $h_j$, including $h_j=1$ when $j\in A$. Total probability gives
$$
\boxed{h_i=\sum_{j\in S}p_{ij}h_j\quad(i\notin A).}
$$
The countable sum is justified by its nonnegative terms. This first-step <identity element>[identity] alone need not uniquely determine a <hitting probability> on an infinite state space; one must choose the minimal nonnegative solution or supply the appropriate boundary behavior.
Back to article page
