The Annals of Probability

Poisson representations of branching Markov and measure-valued branching processes

Thomas G. Kurtz and Eliane R. Rodrigues

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Abstract

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(tΛ, 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.

Article information

Source
Ann. Probab., Volume 39, Number 3 (2011), 939-984.

Dates
First available in Project Euclid: 16 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aop/1300281729

Digital Object Identifier
doi:10.1214/10-AOP574

Mathematical Reviews number (MathSciNet)
MR2789580

Zentralblatt MATH identifier
1232.60053

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60K37: Processes in random environments

Keywords
Branching Markov process Dawson–Watanabe process superprocess measure-valued diffusion particle representation Feller diffusion exchangeability Cox process conditioning random environments

Citation

Kurtz, Thomas G.; Rodrigues, Eliane R. Poisson representations of branching Markov and measure-valued branching processes. Ann. Probab. 39 (2011), no. 3, 939--984. doi:10.1214/10-AOP574. https://projecteuclid.org/euclid.aop/1300281729


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