Modul:   MAT971  Seminar über stochastische Prozesse

Extremal process of multidimensional branching Brownian motion

Talk by Dr. Bastien Mallein

Speaker invited by: Prof. Dr. Jean Bertoin

Date: 05.04.23  Time: 17.15 - 18.45  Room: Y27H12

The branching Brownian motion is a particle system in which each particle evolves independently of one another. Each particle moves according to a Brownian motion in dimension $d$, and splits into two daughter particles after an independent exponential time of parameter $1$. The daughter particles then start from their positions independent copies of the same process.

We take interest in the long time asymptotic behaviour of the particles reaching farthest away from the origin. We show that these particles can be found at a distance of order $\sqrt{2} t + \frac{d-4}{2\sqrt{2}} \log t$ from the origin of the process, and that they can be grouped into a Poisson point process of families of close relatives, spreading in directions sampled according to the random measure $Z(\mathrm{d} \theta)$ that plays the role of an analogue of the derivative martingale of the branching Brownian motion.