Julien Berestycki (University of Oxford)

Branching Brownian motion with absorption

What does the genealogy of a population under selection look like? This question is crucial for ecology and evolutionary biology and yet it is not fully understood. Recently, Brunet and Derrida have conjectured that for a whole class of models of such populations, we can expect the genealogy to be described by a universal scaling limit: the Bollthausen-Sznitman coalescent. 
The purpose of this talk is to present several recent results which put this prediction on a rigorous footing. The model we chose is that of a one-dimensional branching Brownian motion in which particles are absorbed at the origin.  A particle’s position is interpreted as the fitness of an individual and the killing at zero correspond to the removal from the population of individuals whose fitness is too low. 
We assume that when a particle branches, the offspring distribution is supercritical, but the particles are given a drift μ towards the origin. Depending on the value of μ the process can be (sub/super)-critical. 
I will particularly focus on the critical case, for which I will present results concerning the extinction time and Yaglom-type limits for the behavior of the process conditioned to survive for an unusually long time, which both improve upon results of Kesten (1978).  An important tool in the proofs of these results is the convergence of branching Brownian motion with absorption to a continuous state branching process.

Based on joint works with N. Berestycki, J Schweinsberg and P. Maillard, J. Schweinsberg.