On the Construction and Support Properties of Measure-Valued Diffusions on D ⊆ Rᵈ with Spatially Dependent Branching

In this paper, we construct a measure-valued diffusion on D⊆ Rᵈ whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator L = 1/2 ∇ · a∇ + b · ∇ on D ⊆ Rᵈ and whose spatially dependent branching term is of the form β(x)z-α(x)z²,x ∈ D,where β satisfies a very general condition and α> 0. In the case that α and β are bounded from above, we show that the measure-valued process can also be obtained as a limit of approximating branching particle systems. We give criteria for extinction/survival, recurrence/transience of the support, compactness of the support, compactness of the range, and local extinction for the measure-valued diffusion. We also present a number of examples which reveal that the behavior of the measure-valued diffusion may be dramatically different from that of the approximating particle systems.