Summary
In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX t in such a way as to obtain a dual Markov chain\(\tilde X_t \), and considering the two chains together. Various relations between the Martin exit boundaries\(B_0^* \) and\(\tilde B_0^* \) of these processes are studied. The exit boundary\(B_0^* \) of\(\tilde X_t \), is in a sense an entrance boundary forX t and vice versa. After a natural identification of certain points in\(B_0^* \) and\(\tilde B_0^* \) one can topologizeI ∪\(B_0^* \) ∪\(\tilde B_0^* \) in such a way thatboth X t and\(\tilde X_t \) have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.
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Research supported in part at Stanford University, Stanford, California under AFOSR 0049.
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Walsh, J. The Martin boundary and completion of Markov chains. Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 169–188 (1970). https://doi.org/10.1007/BF01111415
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DOI: https://doi.org/10.1007/BF01111415