ИСТИНА

We consider a model of catalytic branching random walk (CBRW) on integer lattice Z^d (d belongs to N) in which branching may occur at the origin only. Previous study of the model showed that CBRW may be classified as supercritical, critical and subcritical. For supercritical and subcritical CBRW, the asymptotic behavior of the total number of particles as well as of the local particles numbers was treated, as time tends to infinity. The total size of particles population was also investigated for subcritical symmetric branching random walk on Z^d. However, the limit behavior of local particles numbers in subcritical CBRW on Z^d remained unknown. Our work completes the picture. Firstly, we establish asymptotic behavior of the probability of particles presence at an arbitrary fixed point of the lattice, as time tends to infinity. Secondly, we prove conditional limit (in time) theorem for the number of particles at any fixed lattice point given that this number is strictly positive. It turns out that these results are similar to those for the model of critical CBRW on Z^d with d=2.