Absorbing Markov and branching processes with instantaneous resurrection

A Markov branching process with instantaneous immigration from the zero state can be constructed so as to be honest and have the non-negative integers as state-space, but the construction requires the branching part to be explosive. We show that a realistic model can be constructed without this restriction if the state-space is restricted to the natural numbers. Moreover this construction is the weak limit, in the sense of finite dimensional laws, of the Yamazato model as the zero state holding-time parameter tends to infinity. This idea of immediate resurrection from an absorbing subset is extended to any minimal discrete-state Markov process, and even to a larger class. Our emphasis is on existence and uniqueness of the transition functions of the resurrected process, and classification of its states.