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

Abstract

1. Let the (honest) Markov process with transition functions (p_ij(0)) have transition rates (q_ij) and suppose that, for some M, so that the matrix Q = (q_ij) determines a bounded operator on the Banach space l₁ 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 P_t = (p_ij(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→∞p_ij(t)).