Journal of Applied Mathematics and Stochastic Analysis 
Volume 10 (1997), Issue 4, Pages 307-332
doi:10.1155/S1048953397000397

Boundedness of one-dimensional branching Markov processes

F. I. Karpelevich1 and Yu. M. Suhov2

 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

Abstract

A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M=supt≥0max1≤k≤N(t)Ξk(t) to be finite. Here Ξk(t) is the position of the kth particle, and N(t) is the size of the population at time t. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODEσ2(x)2f″(x)+a(x)f′(x)=λ(x)(1−k(x))f(x). Here σ(x) and a(x) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ(x) and k(x) the intensity of branching and the expected number of offspring at point x, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ(x)∫π(x,dy)(f(y)−f(x)) and the product λ(x)(1−k(x))f(x), where λ(x) and k(x) are as before, μ(x) is the intensity of jumping at point x, and π(x,dy) is the distribution of the jump from x to y.