Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 4, Pages 307-332
Boundedness of one-dimensional branching Markov processes
1Moscow Institute of Transport Engineers (MIIT), The Russian Ministry of Railways, 15 Obraztsova Str., Moscow 101475, Russia
2University of Cambridge, Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, 16, Mill Lane, Cambridge CB2 1SB, UK
Received 1 November 1996; Revised 1 May 1997
Copyright © 1997 F. I. Karpelevich and Yu. M. Suhov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A general model of a branching Markov process on is considered.
Sufficient and necessary conditions are given for the random variable
to be finite. Here is the position of the th particle, and is the
size of the population at time . For some classes of processes (smooth
branching diffusions with Feller-type boundary points), this results in a
criterion stated in terms of the linear . Here and are the diffusion coefficient and
the drift of the one-particle diffusion, respectively, and and the
intensity of branching and the expected number of offspring at point , respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral
and the product , where
and are as before, is the intensity of jumping at point ,
and is the distribution of the jump from to .