Asymptotics and criticality for a space-dependent branching process

Abstract

AbstractWe investigate a non-conservative semigroup (St)t≥0 determined by a branching process tracing the evolution of particles moving in a domain in Rd. When a particle is killed at the boundary, a new generation of particles with mean number K¯ 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 K¯−1, we distinguish super/sub-critical regimes and determine the exact exponential rate for the total number of particles n(t)∼exp⁡(α∗t), with α∗ depending explicitly on K¯. We prove the Yaglom limit St/n(t)→ν, where the quasi-stationary distribution ν is determined by the resolvent of the Dirichlet kernel at the point α∗. 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 K¯, the model provides an explicit example of self-organized criticality.Keywords: SemigroupYaglom limitbranching processessupercriticalqsdBak-SneppenFleming-ViotDirichlet kernelKey Words and Phrases: Primary: 60J3560J80Secondary: 47D0760K35 Disclosure statementNo potential conflict of interest was reported by the author(s).