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)×Λ, where Λ 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.
"Poisson representations of branching Markov and measure-valued branching processes." Ann. Probab. 39 (3) 939 - 984, May 2011. https://doi.org/10.1214/10-AOP574