# Probability and Mathematical Physics Seminar

#### Some free boundary problems arising from branching Brownian motion with selection

**Speaker:**
Jim Nolen, Duke University

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, April 5, 2019, 11 a.m.

**Synopsis:**

I will explain some current work on a stochastic interacting particle system, branching Brownian motion with selection, and its hydrodynamic limit, which is a free boundary PDE problem. At each branch event in the branching Brownian motion, a particle is removed from the system according to a fitness function, so that the total number of particles, N, is preserved. It is interesting to understand how this selection process effects the evolution of the ensemble of particles. De Masi, Ferrari, Presutti, Soprano-Loto recently showed that in one space dimension, when the left-most particle is always selected, then as N grows the particle system converges to the solution of a certain parabolic free boundary problem which has traveling wave solutions -- this scenario corresponds to a fitness function which is monotone. In joint work with Julien Berestycki, Éric Brunet, Sarah Penington, we study this problem in higher dimensions with a fitness function that has compact level sets. The hydrodynamic limit (large N limit) is also a parabolic free boundary problem, related to the parabolic obstacle problem. The solution of this PDE problem converges, in the large time limit, to an eigenfunction of the laplacian. With Erin Beckman, we also study the problem in 1-d with non-monotone fitness, which leads to a kind of pulsating traveling wave behavior and a metastability phenomenon depending on the fitness function.