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.

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