  Critical catalytic branching random walk on d-dimensional integer lattice is
investigated for all d. The branching may occur at the origin only and the
start point is arbitrary. The asymptotic behavior, as time grows to infinity,
is determined for the mean local particles numbers. The same problem is solved
for the probability of particles presence at a fixed lattice point. Moreover,
the Yaglom type limit theorem is established for the local number of particles.
Our analysis involves construction of an auxiliary Bellman-Harris branching
process with six types of particles. The proofs employ the asymptotic
properties of the (improper) c.d.f. of hitting times with taboo. The latter
notion was recently introduced by the author for a non-branching random walk on
an integer lattice.
