Denumerable Markov processes with bounded generators: a routine for calculating pij (∞)

Abstract

1. Let the (honest) Markov process with transition functions (pij (0)) have transition rates (qij ) and suppose that, for some M, so that the matrix Q = (qij ) determines a bounded operator on the Banach space l 1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and Ω F will be a closed restriction of Q with dense domain, so that Ω F = Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt = (pij (t)) is given by where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P ∞ = (lim t→∞ pij (t)).