Abstract
Branching Brownian Motion describes a system of particles which diffuse in
space and split into offsprings according to a certain random mechanism. In
virtue of the groundbreaking work by M. Bramson on the convergence of solutions
of the Fisher-KPP equation to traveling waves, the law of the rightmost
particle in the limit of large times is rather well understood. In this work,
we address the full statistics of the extremal particles (first-, second-,
third- etc. largest). In particular, we prove that in the large $t-$limit, such
particles descend with overwhelming probability from ancestors having split
either within a distance of order one from time $0$, or within a distance of
order one from time $t$. The approach relies on characterizing, up to a certain
level of precision, the paths of the extremal particles. As a byproduct, a
heuristic picture of Branching Brownian Motion ``at the edge'' emerges, which
sheds light on the still unknown limiting extremal process.
Users
Please
log in to take part in the discussion (add own reviews or comments).