Limit theorem for critical catalytic branching random walks

A continuous time branching random walk on the lattice $Z$ is considered in which individuals may produce children at the origin only. Assuming that the underlying Markov random walk is homogeneous and symmetric and the offspring reproduction law is critical, we describe the asymptotic behavior as $t\to\infty$ of the conditional distribution of the two-dimensional vector $(\zeta(t), \mu (t))$ (scaled in an appropriate way), where $\zeta (t)$ and $\mu(t)$ are the numbers of individuals at the origin and outside the origin at moment $t$ given $\zeta(t)>0$.