Persistence and equilibria of branching populations with exponential intensity

Abstract

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form e_λ(du) = e^-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Π_eλ^χ with intensity e_λ. If λφ'(λ) < 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Π_ceλ^χ over c > 0 and λ ∈ K_st, where K_st = {λ ∈ R: φ(λ) = 0, λφ'(λ) > 0}.