## The Annals of Probability

### Conditional Limit Distributions of Critical Branching Brownian Motions

Tzong-Yow Lee

#### Abstract

A critical branching Brownian motion in $R^d$ is studied where the initial state is either a single particle or a homogeneous field with finite or infinite density. Conditioned on survival in a bounded subset $B$ of $R^d$ at a large time $t$, some normalized limits of the number of particles in a bounded subset $A$ are obtained. When the initial state is a single particle, the normalization factor is a power of $t$ in low dimensions, a power of $\log t$ in the critical dimension and a constant in high dimensions. Extensions to the other initial states and/or more general critical offspring distributions are discussed. Both factors affect the critical dimension. The results are motivated by probabilistic consideration and are proved with the aid of analytic technique of differential equations.

#### Article information

Source
Ann. Probab., Volume 19, Number 1 (1991), 289-311.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176990545

Digital Object Identifier
doi:10.1214/aop/1176990545

Mathematical Reviews number (MathSciNet)
MR1085337

Zentralblatt MATH identifier
0739.60019

JSTOR