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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?

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Can you say, what does local harmonicity means in terms of Markov Chains? – Ilya Jul 12 '13 at 19:12
f is locally harmonic at x iff f(x) is the sum of f(y)P(x,y) (for y in the state space), I believe. – Calvin Condon Jul 12 '13 at 19:52
@Ilya: Calvin's interpretation is what I had in mind. (Thanks, Calvin!) – James Propp Jul 12 '13 at 20:16
James, did you really mean to define holomorphic this way? the immediate "trivial" example I was thinking of was when S is the watched MC for some subset of the state space, but this doesn't generally satisfy your condition since a function which is locally harmonic just at $t$ can have any value outside the neighborhood of $t$, while local harmonicity at $s=f^{-1}(t)$ might say something about completely different states. – Ori Gurel-Gurevich Jul 13 '13 at 5:28
@Ori: I'm pretty sure this is the definition I want. In the case where the Markov chains are (unbiased) random walks on (undirected) graphs, my definition is satisfied when the mapping is "harmonic" in the sense of Urakawa, as described in . – James Propp Jul 13 '13 at 12:54

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