Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

We introduce and analyse a two-sided branching-selection particle system which generalises the well-known $N$-particle branching Brownian motion ($N$-BBM) model, which we call the $(N,p)$-BBM, where either the leftmost or rightmost particle is deleted at each branching event according to a parameter $p\in(0,1)$. We establish that, as $N\to\infty$, the empirical distribution of the $(N,p)$-BBM converges to a deterministic hydrodynamic limit described by a free boundary problem on a finite interval with two moving boundaries, and Neumann and Dirichlet boundary conditions parametrized by $p$. Again, this generalises the one-sided free boundary problem which characterises the hydrodynamic limit of the $N$-BBM. Existence and regularity of the free boundary problem is also proved, by appealing to a connection with inverse first passage problems. We further prove that the asymptotic velocity $v_{N,p}$ of the $(N,p)$-BBM converges, as $N\to\infty$, to $v_p$, the unique travelling wave speed of the limiting free boundary problem. These results generalize previous one-sided models and connect to broader classes of free boundary problems found in evolutionary dynamics and flame propagation.