Martin Exit Boundary Theory

Abstract

The theory of the Martin exit boundary for a Markov chain was first established by Doob [12] and then generalized by Hunt. Many works appeared thereafter, but in all those works there are always some restrictions imposed on the Markov chain. For example, in [13] all the states of the Markov chain are assumed to be nonrecurrent, and in [14] it is assumed that there exists at least one state which can reach all the other states. So the Martin exit boundary theory for general Markov chains has not been established. Moreover, some subjects involved in the exit boundary theory have not been studied in detail. For example, only definitions of the atomic exit point and the nonatomic exit point are given, but no effective criteria. The aim of the present chapter is: (i) to establish the Martin exit boundary theory for general Markov chains; (ii) to present the criteria for the excessive functions, potential functions, minimal excessive functions, minimal potential functions, and minimal harmonic functions, atomic exit points and nonatomic exit points, and for the existence of atomic exit space and nonatomic exit space; (iii) to present the criteria for Blackwell decompositions of atomic almost closed sets, completely nonatomic almost closed sets and state spaces.