## RMK Frankfurt SoSe 2018

Rhein-Main-Kolloquium

Date: 22.06.2018

Time: 15:15–17:45 h

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.

### Number

65

### Speakers

- Julien Berestycki, University of Oxford
- Louigi Addario-Berry, McGill University Montreal

### Place

- Goethe-Universität Frankfurt, Raum 711 (groß)
- Institut für Mathematik,
Robert-Mayer-Str. 10, 60486 Frankfurt
Campus Bockenheim, Robert-Mayer-Str. 10, Raum 711 (groß), 7. Stock

### Organizing partners

Technische Universität Darmstadt, Johannes Gutenberg-Universität Mainz