Excessive functions and the exit-space of a Markov process

The Martin theory enables to describe the set of all positive superharmonic functions in a domain of an Euclidean space. A class of excessive functions (the natural generalization of positive superharmonic functions) corresponds to any Markov process. There exist close relations between the excessive functions and the so-called exit-space of a Markov process. These topics were investigated by Doob, Hunt, Watanabe (for discrete Markov chains) and by Kunita and Watanabe (for some types of continuous-time processes). We develop a general theory which contains as special cases the results of the authors above.