The Annals of Probability

On the Supports of Measure-Valued Critical Branching Brownian Motion

I. Iscoe

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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


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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.