Events Archive

15:15 Uhr: Louigi Addario-Berry (McGill University)

Title: The front location for branching Brownian motion with decay of mass

Abstract: Consider a standard branching Brownian motion whose particles
have varying mass. At time t, if a total mass m of particles have
distance less than one from a fixed particle x, then the mass of
particle x decays at rate m. The total mass increases via branching
events: on branching, a particle of mass m creates two identical mass-m
particles.

One may define the front of this system as the point beyond which there
is a total mass less than one (or beyond which the expected mass is less
than one). This model possesses much less independence than standard
BBM, and martingales are hard to come by. However, using careful
tracking of particle trajectories and a PDE approximation to the
particle system, we are able to prove an almost sure law of large
numbers for the front speed. We also show that, almost surely, there are
arbitrarily large times at which the front lags distance ~ c t^{1/3}
behind the typical BBM front. At a high level, our argument for the
latter may be described as a proof by contradiction combined with fine
estimates on the probability Brownian motion stays in a narrow tube of
varying width.

This is joint work with Sarah Penington and Julien Berestycki.




16:45 Uhr: Julien Berestycki (Universtiy of Oxford)

Titel: The hydrodynamic limit of two variants of Branching Brownian motion.

Abstract: In this talk, I'll consider two variants of branching Brownian
motion (BBM): with decay of mass (as in Louigi's talk) and with selection.
In the BBM with selection, the number of particles is fixed at some
number N and is kept constant by killing the leftmost particle at each
branching event. Both models are motivated by considerations from
ecology and evolutionary biology.

A particle system has a hydrodynamic limit when, as the number of
particles tends to infinity, the behaviour of the system becomes well
approximated by the solution of a partial differential equation. In this
case I will show that the behaviour of the BBM with decay of mass is
governed by the non-local version of the celebrated Fisker-KPP equation
while the BBM with selection tends to the solution of a new free
boundary problem also in the Fisher-KPP class that we study.

This is based on joint work with Louigi Addario-Berry and Sarah
Penington on the one hand and Eric Brunet and Sarah Penington on the other.