J. Aust. Math. Soc.  82 (2007), 403-428
Extinction and explosion of nonlinear Markov branching processes

Anthony G. Pakes
  School of Mathematics and Statistics
  University of Western Australia
  35 Stirling Highway
  Crawley WA 6009

This paper concerns a generalization of the Markov branching process that preserves the random walk jump chain, but admits arbitrary positive jump rates. Necessary and sufficient conditions are found for regularity, including a generalization of the Harris-Dynkin integral condition when the jump rates are reciprocals of a Hausdorff moment sequence. Behaviour of the expected time to extinction is found, and some asymptotic properties of the explosion time are given for the case where extinction cannot occur. Existence of a unique invariant measure is shown, and conditions found for unique solution of the Forward equations. The ergodicity of a resurrected version is investigated.
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