Survival and complete convergence for a branching annihilating random walk

Abstract

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 $\hat{I}¼$ 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 $\hat{I}¼>1$ the process survives when $R$ is sufficiently large. For fixed $R$ we show that the process dies out if $\hat{I}¼$ is too small or too large. Furthermore, we exhibit an interval of $\hat{I}¼$-values for which the process survives and possesses a unique non-trivial ergodic equilibrium for $R$ sufficiently large. We also prove complete convergence for that case.