Persistence of critical multitype particle and measure branching processes
The paper considers multitype branching particle systems with a special branching law which corresponds to mutations of types and monotype branchings, and their corresponding high density limits known as measure branching processes, Dawson-Watanabe (DW) processes, or superprocesses. More precisely, the branching particle system under consideration consists of particles of types \(i=1,\dots,k\); particles of type \(i\) move in \(\mathbb{R}^ d\) for an exponential type-dependent lifetime according to a spherically symmetric stable process with exponent \(\alpha_ i\in(0,2]\), they can eventually mutate into any other type, and they can produce particles of their own type according to a critical branching law which belongs to the domain of attraction of a stable law with exponent \(1+\beta_ i\), \(\beta_ i\in(0,1]\). The question of persistence of the system is investigated by a method due to Kallenberg, the main tool for applying this method being a representation of the Palm distributions of the branching particle system in terms of a ``backward tree'' (Theorem A). The persistence criterion for the multitype branching particle system (Theorem B) says that, if at time 0 the particles are distributed according to a Poisson random measure on \(\mathbb{R}^ d\times\{1,\dots,k\}\) with a product intensity measure \(\lambda\) whose ``position component'' is Lebesgue measure and whose ``type proportions'' are stationary in time with respect to the process of mutations, then the system converges in distribution towards an equilibrium with intensity \(\lambda\) if and only if \(d>\min(\alpha_ 1,\dots,\alpha_ k)/\min(\beta_ 1,\dots,\beta_ k)\). In case of non- persistence the system goes to local extinction in the large time limit, i.e., the number of particles in any bounded subset of \(\mathbb{R}^ d\) tends to 0 in probability. This result means that persistence holds when the mobility of the system is large enough to fill the holes produced by the tendency to extinction of the critical branching. For the multitype DW process the persistence criterion is the same as for the particle system (Theorem C), due to the fact that the particle system and the DW process are related through a Cox representation.
scientific article; zbMATH DE number 112841
scientific article