POISSON REPRESENTATIONS OF BRANCHING MARKOV ANDMEASURE-VALUED BRANCHING PROCESSES

摘要

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 lo-cation 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 branch-ing 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 limit-ing measure-valued process, at each time t, the joint distribution of locations and levels is conditionally Poisson distributed with mean measure K (t) x Λ, where A 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 nonextinc-tion, Harris's convergence theorem for supercritical branching processes, and diffusion approximations for processes in random environments