On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching

Abstract

In this paper, we construct a measure-valued diffusion on $D\subseteq \mathbb{R^d}$ whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator $$L = 1/2 \nabla \cdot a\nabla + b \cdot \nabla \text{ on } D \subseteq \mathbb{R}^d$$ and whose spatially dependent branching term is of the form $\beta(x)z-\alpha(x)z^2,x \in D$,where $\beta$ satisfies a very general condition and $\alpha> 0$. In the case that $\alpha$ and $\beta$ 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.