Ancestral lineages for a branching annihilating random walk

Abstract

We study the ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the $d$-dimensional lattice $Z^d$. Each individual produces a Poissonian number of offspring with mean $\mu$ which then jump independently to a uniformly chosen site with a fixed distance $R$ of their parent. Should two or more particles jump to the same site, all particles at that site get annihilated. By interpreting the ancestral lineage of such an individual as a random walk in a dynamical random environment, we obtain a law of large numbers and a functional central limit theorem for the ancestral lineage whenever the model parameters satisfy $\mu \in (1,e^2)$ and $R=R\left(\mu \right)$ is large enough.