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April 1999 On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching
János Engländer, Ross G. Pinsky
Ann. Probab. 27(2): 684-730 (April 1999). DOI: 10.1214/aop/1022677383

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.

Citation

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János Engländer. Ross G. Pinsky. "On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching." Ann. Probab. 27 (2) 684 - 730, April 1999. https://doi.org/10.1214/aop/1022677383

Information

Published: April 1999
First available in Project Euclid: 29 May 2002

zbMATH: 0979.60078
MathSciNet: MR1698955
Digital Object Identifier: 10.1214/aop/1022677383

Subjects:
Primary: 60J60, 60J80

Rights: Copyright © 1999 Institute of Mathematical Statistics

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Vol.27 • No. 2 • April 1999
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