Let $X$ be a Markov chain on $S$, and $T_C = \min\{n \ge 1 : X_n \in C\}$ be the first hitting time of $C$ for $C \subset S$. Let $A$, $B$ be two disjoint subsets of $S$ with $T_{A \cup B} < \infty$ almost surely, and let $h$ be the unique bounded harmonic function on $S \setminus (A \cup B)$ with $h(x) = 1$ for $x \in A$ and $h(x) = 0$ for $x \in B$. Show that $X$ conditioned on $X_{T_{A \cup B}} \in A$ is a Markov chain with transition probabilities \(q(x, y) = \frac{ p(x, y) h(y) }{ h(x) } .\)

Attribution: This problem is modified from worksheets provided by Jim Pitman (thanks!).