I will discuss a stochastic interacting particle system in R^d, involving 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. We study the large N and large t limits. We prove a hydrodynamic limit for this particle system as N diverges; the limit is a parabolic free boundary problem with nonlocal constraint on the free boundary. In the large time limit, the PDE solution approches a certain eigenfunction. We also show that the so-called strong selection principle holds: the large N and large t limits commute for the particle system. Some related problems and results will be discussed. This is joint work with Julien Berestycki, Éric Brunet, Sarah Penington.
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