Subcritical catalytic branching random walk with finite or infinite variance of offspring number

Subcritical catalytic branching random walk on the $d$-dimensional integer lattice is studied. New theorems concerning the asymptotic behavior of distributions of local particle 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 under the assumptions of 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 the subcritical regime, the finiteness of the moment of order $1+\delta$ is required where $\delta $ is some positive number.