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On the construction and support properties of measure-valued diffusions on \(D\subseteq \mathbb{R}^d\) with spatially dependent branching. (English) Zbl 0979.60078

Summary: 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 \(L={1\over 2}\nabla\cdot a\nabla+ b\cdot\nabla\) 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.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J60 Diffusion processes
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