Measure-valued branching processes with immigration

AbstractStarting from the cumulant semigroup of a measure-valued branching process, we construct the transition probabilities of some Markov process Y(β)=(Y(β)t, t ϵ R, which we call a measure-valued branching process with discrete immigration of unitβ. The immigration of Y(β) is governed by a Poisson random measure ρ on the time-distribution space and a probability generating function h, both depending on β. It is shown that, under suitable hypotheses, Y(β) approximates to a Markov process Y=(Yt, t ϵ R as β→0+. The latter is the one we call a measure-valued branching process with immigration. The convergence of branching particle systems with immigration is also studied