Limit Distributions for the Number of Particles in Branching Random Walks

We study branching random walks with continuous time. Particles performing a random walk on $\mathbb{Z}^{2}$, are allowed to be born and die only at the origin. It is assumed that the offspring reproduction law at the branching source is critical and the random walk outside the source is homogeneous and symmetric. Given particles at the origin, we prove a conditional limit theorem for the joint distribution of suitably normalized numbers of particles at the source and outside it as time unboundedly increases. As a consequence, we establish the asymptotic independence of such random variables.