Survival and complete convergence for a branching annihilating random walk

We study a discrete-time branching annihilating random walk (BARW) on the d-dimensional lattice. Each particle produces a Poissonian number of offspring with mean μ which independently move to a uniformly chosen site within a fixed distance R from their parent’s position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any μ>1 the process survives when R is sufficiently large. For fixed R we show that the process dies out if μ is too small or too large. Furthermore, we exhibit an interval of μ-values for which the process survives and possesses a unique nontrivial ergodic equilibrium for R sufficiently large. We also prove complete convergence for that case.