International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1997 / Article

Open Access

Volume 10 |Article ID 210796 | 26 pages | https://doi.org/10.1155/S1048953397000397

Boundedness of one-dimensional branching Markov processes

Received01 Nov 1996
Revised01 May 1997

Abstract

A general model of a branching Markov process on is considered. Sufficient and necessary conditions are given for the random variable M=supt0max1kN(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)(1k(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)(1k(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.

Copyright © 1997 Hindawi Publishing Corporation. 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.


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