Suppose $(X_n)_{n\geq0}$ is a markov chain on a countable set $S$, and $P_x (\tau_y < \infty) = 1$ for all $x,y \in S$. Let $h : S \rightarrow [0,\infty)$ be harmonic. Is it true that $ E_x [h(X_{\tau_y})] = h(y) $ ?
From the definition of expectation,
$$ E_x [h(X_{\tau_y})] = \sum_{s \in S} P_x (X_{\tau_y} = s)h(s) $$
I am not sure how to proceed from here. Any help would be appreciated.