Ancestral lineages for a branching annihilating random walk

We study ancestral lineages of individuals of a stationary discrete-time branching annihilating random walk (BARW) on the $d$-dimensional lattice ${\mathbb{Z}}^d$. 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. 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.