## The Annals of Probability

### Ergodicity of Critical Spatial Branching Processes in Low Dimensions

#### Abstract

We consider two critical spatial branching processes on $\mathbb{R}^d$: critical branching Brownian motion, and the Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension-dependent. It is known that in low dimensions, $d \leq 2$, the unique invariant measure with finite intensity is $\delta_0$, the unit point mass on the empty state. In high dimensions, $d \geq 3$, there is a one-parameter family of nondegenerate invariant measures. We prove here that for $d \leq 2, \delta_0$ is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation $\partial u/\partial t = (1/2) \Delta u - bu^2$.

#### Article information

Source
Ann. Probab., Volume 21, Number 4 (1993), 1946-1957.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176989006

Digital Object Identifier
doi:10.1214/aop/1176989006

Mathematical Reviews number (MathSciNet)
MR1245296

Zentralblatt MATH identifier
0788.60119

JSTOR