Summary
The countable state space of a Markov chain whose stationary transition probabilities satisfy the continuity condition (1.5) is compactifled to get a state space on which the corresponding processes can be made right continuous with left limits, and strongly Markovian. There is a form of quasi left continuity, modified by the possible presence of branch points. Excessive functions are investigated.
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Doob, J.L. Compactification of the discrete state space of a Markov process. Z. Wahrscheinlichkeitstheorie verw Gebiete 10, 236–251 (1968). https://doi.org/10.1007/BF00536277
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DOI: https://doi.org/10.1007/BF00536277