POISSON REPRESENTATIONS OF BRANCHING MARKOV AND MEASURE-VALUED BRANCHING PROCESSES

• 成果类型: Article
• 署名作者: Kurtz, Thomas G.; Rodrigues, Eliane R.
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison; Universidad Nacional Autonoma de Mexico
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP574
• 发表日期: 2011
• 页码: 939-984
• 关键词: density-independent catastrophes; LIMIT-THEOREMS; particle representations; WEAK-CONVERGENCE; extinction; immigration; superprocess; DIFFUSIONS; SEQUENCES; BEHAVIOR

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 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) x Lambda. where Lambda 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.