ИСТИНА

We consider two models of continuous time branching random walks (BRW) on ${\bf Z}^{2}$ where particles may produce offsprings at the origin only. The first model called symmetric BRW (SBRW) was studied in by Yarovaya in 2007. The main feature of the model is symmetry and homogeneity of the random walk generator. Another model named catalytic BRW (CBRW) was proposed by V.A.Vatutin, V.A.Topchii and E.B.Yarovaya in 2003. It differs from the previous one by the introduction of an additional parameter governing the relation between walk and branching at the source. To formulate the main result some notation is needed. Let $\zeta(t)$ and $\mu(t)$ stand for the numbers of the particles at the origin and outside the origin at time $t$ respectively. The limit conditional joint distribution of $\zeta(t)$ and $\mu(t)$ was investigated in \cite{VT} for critical CBRW on ${\bf Z}$. Our work is devoted to study of the limit conditional joint distribution of these variables for both critical SBRW and critical CBRW on ${\bf Z}^{2}$. Under the same conditions as in \cite{TVP} we establish that $$\lim\limits_{t\to\infty}{\sf E}_{0}\left\{\left.\exp \left\{-\lambda_{1}\zeta(t)-\frac{\lambda_{2}\mu(t)}{c_{m}\ln{t}}\right\}\right|\zeta(t)>0\right\}$$ $$=\lim\limits_{t\to\infty}{\sf E}_{0}\left\{e^{-\lambda_{1}\zeta(t)}|\zeta(t)>0\right\}\lim\limits_{t\to\infty}{\sf E}_{0}\left\{\left.\exp\left\{-\frac{\lambda_{2}\mu(t)}{c_{m}\ln{t}} \right\}\right|\zeta(t)>0\right\},\;\;\lambda_{1},\lambda_{2}\in{\bf R}_{+}.$$ Index $0$ in ${\sf E}_{0}$ indicates that BRW starts at the origin at the initial time and the positive constant $c_{m}=\lim\nolimits_{t\to\infty}{t{\sf E}_{0}\zeta(t)}$ (this limit exists according to papers by E.Vl.Bulinskaya in 2010 and E.B.Yarovaya in 2007). This result entails asymptotic independence of $\zeta(t)$ and $\mu(t)$ conditioned by $\zeta(t)>0$. Moreover, the explicit form of the limit distribution of $\mu(t)$ given $\zeta(t)>0$ is obtained. The corresponding conditional limit distribution of $\zeta(t)$ was found by E.Vl.Bulinskaya in 2010.