Persistence criteria for a class of critical branching particle systems in continuous time
This paper is concerned with the long-time behavior of a branching particle system in \({\mathbb{R}}^ d\) with spatially homogeneous initial Poisson random field. The particles follow \(\alpha\)-symmetric stable processes and the offspring distributions are chosen from a class of distributions in the domain of attraction of a stable law with index \(1+\beta\). The central result of this paper is the determination of a formula (based on Kallenberg's backward technique (1977)) for the Palm distributions of these branching particle systems [the analogous result for superprocesses is given by the reviewer and \textit{E. A. Perkins}, Mem. Am. Math. Soc. 454 (1991)]. The Palm formula is then used to establish that the branching particle system is persistent (i.e. has a stationary distribution preserving the initial intensity) if and only if \(d>\alpha /\beta\).
long-time behavior of a branching particle system
scientific article; zbMATH DE number 4211224