The Martin boundary and completion of Markov chains

Abstract

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.