Poisson representations of branching Markov and measure-valued branching processes

TL;DR

This paper introduces a novel Poisson representation for branching Markov and measure-valued processes with spatially varying rates, where particle levels evolve over time, simplifying analysis and providing new proofs for key results.

Contribution

The paper develops a new Poisson-based particle system representation with time-evolving levels, offering simplified analysis and proofs for branching processes and their limits.

Findings

Provides a unified Poisson representation for branching processes.

Simplifies calculations for extinction and nonextinction conditioning.

Offers alternative proofs for convergence and diffusion results.

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 $K(t)\times\varLambda$, 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.