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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 arxiv.org/pdf/0707.1309.pdf . –  James Propp Jul 13 '13 at 12:54
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