Asymptotics and criticality for a space-dependent branching process

Abstract

We investigate a non-conservative semigroup determined by a branching process tracing the evolution of particles moving in a domain in . When a particle is killed at the boundary, a new generation of particles with mean number is born at a random point in the domain. Between branching, the particles are driven by a diffusion process with Dirichlet boundary conditions. According to the sign of , we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles , with depending explicitly on . We prove the Yaglom limit , where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point . The main application is in particle systems, where the normalization of the semigroup by its total mass gives the hydrodynamic limit of the Bak-Sneppen branching diffusions (BSBD). Since ν is the asymptotic profile under equilibrium, and the family of quasi-stationary distributions ν is indexed by , the model provides an explicit example of self-organized criticality.