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Poisson representations of branching Markov and measure-valued branching processes. (English) Zbl 1232.60053

The authors give some representations of branching Markov processes and their measure-valued limits in terms of countable particle systems. A particle in a system has a location and a level. The results improve ealier ones of the first author and his collaborators. In particular, for models with branching rates independent of the particle location, a similar approach was carried out in the earlier work [P. Donnelly and T. G. Kurtz, “Particle representations for measure-valued population models”, Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)]. The result was extended to general branching rates and critical or subcritical offspring distributions in [T. G. Kurtz, “Particle representations for measure-valued population processes with spatially varying birth rates”, CMS Conf. Proc. 26, 299–317 (2000; Zbl 0960.60069)]. The representation given in the present paper applies to both subcritical and supercritical processes as well as to processes with heavy-tailed offspring distributions. It also applies to models with infinite mass. The levels of the particles in the model considered here change with time. In fact, before taking the limit, death of a particle occurs only when the level of the particle crosses a specified level \(r\geq 0\). For the limiting models, death occurs only when the level goes to infinity. As in the earlier work, the justification for the representation is a consequence of a Markov mapping theorem. For branching Markov processes, at each time \(t\geq 0\), 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 \(t\geq 0\), the joint distribution of locations and levels is conditionally Poisson distributed with mean measure \(K(t)\times \Lambda\), where \(\Lambda\) denotes the Lebesgue measure, and \(K(t)\) is the desired measure-valued process. The representation simplifies or gives new proofs for a variety of results including conditioning on extinction or nonextinction, Harris’s convergence theorem and diffusion approximations. In particular, the Feller diffusion approximation for nearly critical branching processes is obtained as a consequence of the construction.

MSC:

60J68 Superprocesses
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J60 Diffusion processes
60J25 Continuous-time Markov processes on general state spaces
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