The Martin boundary and completion of Markov chains

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, and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of, is in a sense an entrance boundary forX_t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X_t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.