Poisson representations of branching Markov and measure-valued branching processes

Details Poisson representations of branching Markov and measure-valued branching processes Poisson representations of branching Markov and measure-valued branching processes

2011-04-08

arxiv.org/abs/1104.1496v1

Annals of Probability 2011, Vol. 39, No. 3, 939-984

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/10-AOP574 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/10-AOP574

Representations of branching Markov processes and their measure-valued limits in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. Each particle has a location and a "level," but unlike earlier constructions, the levels change with time. In fact, death of a particle occurs only when the level of the particle crosses a specified level $r$, or for the limiting models, hits infinity. For branching Markov processes, at each time $t$, conditioned on the state of the process, the levels are independent and uniformly distributed on $[0,r]$. For the limiting measure-valued process, at each time $t$, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure $K(t)\times\varLambda$, where $\varLambda$ denotes Lebesgue measure, and $K$ is the desired measure-valued process. The representation simplifies or gives alternative proofs for a variety of calculations and results including conditioning on extinction or nonextinction, Harris's convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments.

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