## The Annals of Probability

- Ann. Probab.
- Volume 27, Number 2 (1999), 684-730.

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

János Engländer and Ross G. Pinsky

#### 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 $\beta(x)z-\alpha(x)z^2,x \inD$,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.

#### Article information

**Source**

Ann. Probab., Volume 27, Number 2 (1999), 684-730.

**Dates**

First available in Project Euclid: 29 May 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1022677383

**Digital Object Identifier**

doi:10.1214/aop/1022677383

**Mathematical Reviews number (MathSciNet)**

MR1698955

**Zentralblatt MATH identifier**

0979.60078

**Subjects**

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

**Keywords**

Measure-valued process superprocess diffusion process log-Laplace equation branching h-transform

#### Citation

Engländer, János; Pinsky, Ross G. On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching. Ann. Probab. 27 (1999), no. 2, 684--730. doi:10.1214/aop/1022677383. https://projecteuclid.org/euclid.aop/1022677383