Decomposition and construction of Markov chains

Abstract

In [2, 3] and [51 Chung has considered a technique for decomposing a stochastic transition function (with a conservative initial derivative matrix and a finite passable Martin exit boundary). The decomposition is given in terms of a minimal transition function, entrance and exit laws with respect to the minimal transition function, and certain other parameters. In this paper we consider a similar decomposition in the case of an arbitrary stochastic transition function. The "boundary" of our decomposition consists of a finite number of regular points of the compactified state space corresponding to the given stochastic transition function. The minimal transition function is replaced by the transition function of a process which is killed whenever it hits the boundary. Considerable simplification results from the use of local time and from Neveu's idea of coupled entrance and exit laws (see [9J and [10]). In the last section we give a construction theorem for stochastic transition functions which is suggested by our decomposition.