5 Markov lumping as exact refinement
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Let \(K\) be a column-stochastic fine transition matrix, let \(C\) aggregate fine states into coarse packets, and let \(\bar K\) be a coarse transition matrix. Define
Theorem 5.1: Exact lumpability
If \(\Delta=0\), then
for every \(n\ge0\). Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
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Proof of Theorem 5.1
The result is immediate for \(n=0\). If \(CK^n=\bar K^nC\), then
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Theorem 5.2: Finite-horizon defect formula
For arbitrary \(\Delta=CK-\bar KC\),
If \(K\) and \(\bar K\) are contractions in the chosen norm, then
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Proof of Theorem 5.2
The identity follows by telescoping:
Summing from \(j=0\) to \(n-1\) leaves \(CK^n-\bar K^nC\). The norm bound follows termwise.
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