Local Particles Numbers in Critical Branching Random Walk

Critical catalytic branching random walk on an integer lattice ℤd is investigated for all d∈ℕ. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of the presence of particles at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman–Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on ℤd.