We consider a continuous time branching random walk on the lattice
in which individuals may produce children at the origin only.
Assuming that the underlying random walk is symmetric and the
offspring reproduction law is critical we find the asymptotic behavior
of the survival probability of the process at time
as
and the probability that the number of
individuals at the origin at time
is positive. We also prove a
Yaglom type conditional limit theorem for the total number of
particles existing at time