2011/03/30 by Zakhar Kabluchko, Kabluchko, Zakhar
Mathematics · #60G55 #60J80 #FOS: Mathematics #Primary #Probability (math.PR) #Secondary #math.PR #msc:60G55 #msc:60J80
paper · pdf · doi:10.48550/arxiv.1103.5865
17 pages, 1 figure
arxiv created 2011/03/30 · arxiv updated 2011/03/31
We consider a system of independent branching random walks on \R which start off a Poisson point process with intensity of the form eλ(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 (stable) meaning that the point process formed by the particles in the n-th generation converges as n→∞ to a non-trivial point process Πeλχ with intensity eλ. If λϕ'(λ)<0, then the branching population suffers local extinction meaning that the limiting point process is empty. We characterize (generally, non-stationary) point processes on \R which are cluster-invariant with respect to the cluster distribution χ as mixtures of the point processes Πceλχ over c>0 and λ∈ Kst, where Kst=\λ∈\R: ϕ(λ)=0, λϕ'(λ)>0\.