## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 1 (1988), 200-221.

### On the Supports of Measure-Valued Critical Branching Brownian Motion

#### Abstract

Let $(X_t)_{t \geq 0}$ denote the measure-valued critical branching Brownian motion. When the support of the initial state, $X_0$, is bounded, temporally global results are given concerning the range, i.e., the size of the supports of $(X_t)_{t \geq 0}$, and the hitting (i.e., charging) probabilities of distant balls are evaluated asymptotically; they depend strongly on the dimension, $d$, of the underlying Euclidean space $\mathbb{R}^d$. In contrast, in the case $d = 1$ and $X_0 = \lambda$ (Lebesgue measure), it is shown that (spatially) local extinction occurs. Also extensions are indicated for the case of an infinite variance branching mechanism; these results are also dimensionally dependent.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 1 (1988), 200-221.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991895

**Digital Object Identifier**

doi:10.1214/aop/1176991895

**Mathematical Reviews number (MathSciNet)**

MR920265

**Zentralblatt MATH identifier**

0635.60094

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G57: Random measures

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 34B15: Nonlinear boundary value problems

**Keywords**

Measure-valued branching diffusion support asymptotics range hitting probability local extinction singular elliptic boundary value problems

#### Citation

Iscoe, I. On the Supports of Measure-Valued Critical Branching Brownian Motion. Ann. Probab. 16 (1988), no. 1, 200--221. doi:10.1214/aop/1176991895. https://projecteuclid.org/euclid.aop/1176991895

#### See also

- Addendum: I. Iscoe. Addendum: On the Supports of Measure-Valued Critical Branching Brownian Motion. Ann. Probab., Volume 17, Number 2 (1989), 813--813.Project Euclid: euclid.aop/1176991429