ИСТИНА

We consider the modification of a continuous time branching random walk on $\mathbb{Z}^{2}$ in which particles may produce offsprings only at the origin. The population is initiated at time $t=0$ by a single particle located at the origin. This modification of branching random walk was proposed and studied for $\mathbb{Z}$ by V.A.Vatutin, V.A.Topchii and E.B.Yarovaya in 2003. It differs from other known models by introduction of an additional parameter enabling artificial intensification of prevalence of branching or walk at the source. Assuming that the random walk being outside the origin is symmetric and the offspring reproduction law is critical, at first we studied the asymptotic behavior of the survival probability of the process at time $t$ and the probability that the number of particles at the origin at time $t$ is positive, as $t\rightarrow\infty$. The main aim of the present work is to prove conditional limit theorems for the total number of particles $\eta(t)$ existing on $\mathbb{Z}^{2}$ at time t and for the number of particles outside the origin $\mu(t)$ using these results. Namely, for any $s\in[0,1]$ one has $$\lim\limits_{t\rightarrow\infty}{\mathbf{E}\{s^{\mu(t)}|\eta(t)>0\}} =1-\sqrt{1-s},$$ $$\lim\limits_{t\rightarrow\infty}{\mathbf{E}\{s^{\eta(t)}|\eta(t)>0\}} =1-\sqrt{1-s}.$$