Instantaneous states of Markov processes
Gerald A. Smith · Transactions of the American Mathematical Society · 1964
k=1 pij(t) is interpreted to be the probability that the process will be in state j at time s + t given that it is in state i at time s. An additional condition which is usually imposed on the pij(t), and which we will impose, is that lim to0 pij(t) = 1. This implies (see Chung [1]) the continuity of all the pij(t), and the existence of all p!'(t), which may be - oo for i = j, t = 0. Letting p!-(O) = qij, i 0 j, and pi(O) -qi, we have O? qu < oo and O < qi < ?? In the case qi = oo we say i is an instantaneous state. An equivalent definition (see Chung) of an instantaneous state is a state which the process cannot remain in for any positive time interval. In ?2 an example is presented of a Markov process with lim suprto p'(t) = + x0 for an instantaneous state i, answering negatively the question of whether lim t.0 p!i(t) -oo always if i is instantaneous.