Rank Dependent Branching-Selection Particle Systems

We consider a large family of branching-selection particle systems. The branching rate of each particle depends on its rank and is given by a function $b$ defined on the unit interval. There is also a killing measure $D$ supported on the unit interval as well. At branching times, a particle is chosen among all particles to the left of the branching one by sampling its rank according to $D$. The measure $D$ is allowed to have total mass less than one, which corresponds to a positive probability of no killing. Between branching times, particles perform independent Brownian Motions in the real line. This setting includes several well known models like Branching Brownian Motion (BBM), $N$-BBM, rank dependent BBM, and many others. We conjecture a scaling limit for this class of processes and prove such a limit for a related class of branching-selection particle system. This family is rich enough to allow us to use the behavior of solutions of the limiting equation to prove the asymptotic velocity of the rightmost particle under minimal conditions on $b$ and $D$. The behavior turns out to be universal and depends only on $b(1)$ and the total mass of $D$. If the total mass is one, the number of particles in the system $N$ is conserved and the velocities $v_N$ converge to $\sqrt{2 b(1)}$. When the total mass of $D$ is less than one, the number of particles in the system grows up in time exponentially fast and the asymptotic velocity of the rightmost one is $\sqrt{2 b(1)}$ independently of the number of initial particles.