Some aspects of the Fisher-KPP equation and the branching Brownian motion
Éric Brunet (LPS)

The Fisher-Kolmogorov, Petrovski, Piscounov equation (FKPP)
is a deterministic partial differential equation. It describes
the evolution of an invasion front from a stable phase into
an unstable phase. Branching Brownian motion (BBM) is a
stochastic Markov process where particles diffuse and
duplicate. Both the FKPP equation and the BBM can be seen as
modelling the evolution of a population, but the former is
deterministic and with saturation, while the latter is
stochastic and without saturation. They are however directly
related to each other by McKean’s duality.

In this dissertation, after a brief review of classical and
essential results concerning the FKPP equation and the BBM, I
present some of the contributions my collaborators and I have
made to this field.

A first set of results concerns the asymptotic position of the
FKPP front ; on two well-chosen models in the FKPP class, I
present two different ways to recover the classical results of
Bramson and the prediction by Ebert and van Saarloos. I also
make a prediction for the next order term.

A second set of results concerns the limiting distribution of
the rightmost particles in the BBM. As we found out, they are
distributed according to a so-called “randomly shifted
σ-decorated exponential Poisson point process”, which we define
and characterize. These results were mostly obtained by using the
duality between the BBM and the FKPP equation.

A last set of results concerns the behaviour of noisy FKPP fronts
in the limit of a weak noise. I present a phenomenological theory
which allows to compute, to leading order, all the cumulants of
the position. Furthermore, in models for which it makes sense,
the genealogical tree of the population is given by a rescaled
Bolthausen-Sznitman coalescent.