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Explosive Markov branching processes: entrance laws and limiting behaviour

The supercritical Markov branching process is examined in the case where the minimal version of the process has strictly substochastic transition laws. This provides a nice example of the general construction theory for discrete-state Markov processes. Entrance laws corresponding to the minimal process are characterised. Limit properties of the processes constructed from these entrance laws are examined. All such processes which are honest and cannot hit zero are ergodic. Otherwise these processes are λ-positive and limit theorems conditional on not having left the positive states are given. A connection is made with recent work on the general construction problem when a λ-subinvariant measure is given. The case where immigration is allowed is mentioned.