Probabilities of events involving the jump of a Markov chain to the state space immediately after the last exit before a given time from a boundary atom are determined, for the most part, by the initial time value of the canonical entrance law corresponding to that atom. Three of these probabilities are calculated in terms of canonical quantities in order to attach a probabilistic meaning to an entrance law decomposition of Reuter's and to improve an analytical condition of Chung's for when the Kolmogorov forward equations are satisfied by the chain's transition matrix.
"Last Exit Times from the Boundary of a Continuous Time Markov Chain." Ann. Probab. 5 (1) 152 - 156, February, 1977. https://doi.org/10.1214/aop/1176995901