I would be curious to know if there's still open question about discrete markovian chains
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You might find some of the links at What are some open research problems in Stochastic Processes?. – Gareth Oct 29 '13 at 10:31
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Chapter 23 of Markov Chains and Mixing Times by Levin, Peres, and Wilmer is called "Open Problems". The book is available online here.