Given two Markov chains with respective state-spaces $S$ and $T$, say that a function $\phi$ from $S$ to $T$ is holomorphic iff for all states $t \in T$, every real-valued function $f$ on $T$ that is locally harmonic at $t$ pulls back to a function on $S$ that is locally harmonic at the pre-image of $t$.

Is there any literature on this property, or on other properties that imply it?