Conditional Limit Distributions of Critical Branching Brownian Motions

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.