arXiv:2611.14587v1 Announce Type: new Abstract: What does the occupation law of a finite Markov chain determine about its generator $Q$? From the stationary start, the law of the occupation times at one time determines $Q$ exactly up to time reversal and reversals of memoryless blocks, the sets of states that are all left at one rate with one landing law and are entered in proportion to the stationary law. These reversals commute and give exactly $2^k$ generators, with $k$ read off a tree of memoryless blocks. This leaves only time reversal when $Q$ has no rank-one cut, when there are at most 4 states, and generically. The law at one time determines the law at every time, and a generic start determines $Q$. A renewal block is a set of states that is left at the same rates from each of its states, is entered with one law, and is charged by the start in proportion to that law. From every start the law determines $Q$ exactly up to lumped reversals, which lump disjoint renewal blocks to single states and reverse a memoryless block of the lumped chain. We also find every start that hides time reversal. At the scale of large deviations, the functional of Donsker and Varadhan determines exactly the principal minors of $-Q$.
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