Particle Representations for Measure-valued Population Processes with Spatially Varying Birth Rates

Representations of measure-valued processes in terms of countable systems of particles are constructed for models with spatially varying birth and death rates. In previous constructions for models with birth and death rates not depending on location or type, the particles were assigned integer-valued \levels", the joint distributions of the particle types were exchangeable, and the measure-valued process K was given by K(t) = P(t) Z(t), where P was the \total mass" process and Z(t) was the de Finetti measure for the exchangeable particle types at time t. In the present construction , particles are assigned real-valued levels and for each time t the joint distribution of locations and levels is conditionally Poisson distributed with mean measure K(t) m. The representation gives an explicit construction of the boundary measure in Dynkin's probabilistic solution of the nonlinear partial diierential equation (x)v(x) ? Bv(x) = (x), x 2 D, v(x) = f(x), x 2 @D. The representation also provides a way of generalizing Perkins's models for measure-valued processes in which the individual particle motion depends on the distribution of the population. Questions of uniqueness, however, remain open for most of the models in this larger class.