Ancestral lineages for a branching annihilating random walk

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 is an element of(1, e(2)) and R = R(mu) is large enough.