  Subcritical catalytic branching random walk on d-dimensional lattice is
studied. New theorems concerning the asymptotic behavior of distributions of
local particles numbers are established. To prove the results different
approaches are used including the connection between fractional moments of
random variables and fractional derivatives of their Laplace transforms. In the
previous papers on this subject only supercritical and critical regimes were
investigated assuming finiteness of the first moment of offspring number and
finiteness of the variance of offspring number, respectively. In the present
paper for the offspring number in subcritical regime finiteness of the moment
of order 1+\delta is required where \delta is some positive number.
  Keywords and phrases: branching random walk, subcritical regime, finite
variance of offspring number, infinite variance of offspring number, local
particles numbers, limit theorems, fractional moments, fractional derivatives.
