The maximum of branching Brownian motion in \(\mathbb{R}^d\) (Q6104003)

The paper considers the branching Brownian motion with binary branching and branching rate 1. A particle starts from the origin of \(\mathbb{R}^d\) and performs a standard Brownian motion. After an exponentially distributed time (independent of the motion of the particle), it gives birth to two particles, and dies. The process now repeats itself: all particles alive at time \(t\) perform independent Brownian motion, with their own (independent) exponential clocks determining their branching. The authors show that in branching Brownian motion in \(\mathbb{R}^d\), \(d \geqslant 2\), the law of the maximum distance of a particle from the origin at time \(t\), converges as \(t \to \infty \) to the law of a randomly shifted Gumbel random variable.