ИСТИНА

We deal with two different models of a continuous time branching random walk (BRW) on $\mathbb{Z}^{d}$, $d\in\mathbb{N},$ in which particles may produce offsprings only at the origin. The first model is called symmetric BRW (SBRW). It was investigated in details by E.B.Yarovaya in 2007. The second model differs from the previous one by introduction of an additional parameter enabling artificial prevalence of branching or walk at the source. This modification is called catalytic BRW (CBRW) and it was proposed by V.A.Vatutin, V.A.Topchii and E.B.Yarovaya in 2003. This work is devoted to the study of a conditional limit distribution of number $\mu_{t}(0)$ of particles at the source of branching at time $t$ as $t\to\infty$ provided that $\mu_{t}(0)>0$. It was proven by V.A.Vatutin and V.A.Topchii in 2004 for critical CBRW on $\mathbb{Z}$ that $$\lim\limits_{t\to\infty}{{\sf E}_{0}\left\{\left.\exp\left\{-\frac{\lambda\,\mu_{t}(0)}{{\sf E}_{0}{(\mu_{t}(0)|\mu_{t}(0)>0)}}\right\}\right|\mu_{t}(0)>0\right\}}=\frac{1}{1+\lambda}$$ where index $_{0}$ indicates that the process starts at the origin at the initial time. As was shown subsequently by Bulinskaya in 2010, this limit law is not valid for critical CBRW on $\mathbb{Z}^{2}$ and the conditional limit distribution in that case is discrete. However, we establish here that the above limit law holds for both critical SBRW and critical CBRW on $\mathbb{Z}^{d},\;d=3$ and $d\geq5$. To prove these results we essentially employ the known asymptotic behavior of non-extinction probability at the origin and use the technique developed by V.A.Vatutin and Topchii in 2004. Thus, the exponential law is a limit distribution of the number of particles at the origin for both critical SBRW and critical CBRW on $\mathbb{Z}^{d}$ for any $d\in\mathbb{N}$ except $d=2$ and $d=4$.