Absorbing Markov and branching processes with instantaneous resurrection
Modifying the standard Markov branching process so that zero is no longer an absorbing state may be done in several ways. \textit{M. Yamazato} [J. Math. Soc. Japan 27, 479-496 (1975; Zbl 0303.60081)] introduced an exponential holding time at zero followed by a jump to a random state, the process then behaving as before. \textit{A. Chen} and \textit{E. Renshaw} [Probab. Theory Relat. Fields 87, No. 2, 209-240 (1990; Zbl 0695.60080)] studied a process with instantaneous visits to zero and in which the branching part is necessarily explosive; this process is mathematically properly defined but difficult to envisage as a realistic model. Stimulated by this criticism, the present author considers the limit of Yamazato's model as the exponential holding parameter tends to infinity; in this third model the state zero effectively disappears and the modified behaviour now takes place in state 1. The author compares various properties of all three models, such as the recurrence of states. He also extends the notion of instantaneous resurrection from an absorbing barrier to more general Markov processes.
scientific article; zbMATH DE number 488398