Minimal points of a supercritical branching random walk on the lattice $\mathbf N_0^r$, and multitype Galton–Watson branching processes

We consider a supercritical Galton–Watson branching process with mean value of offspring of one particle equal to $A>1$. The initial particle is placed at the point $\boldsymbol0\in N_0^r$, where $N_0=\{0,1,2,\dots\}$. If a particle is at a point $\mathbf z\in N_0^r$, then its direct descendants are placed at points $\mathbf z+\mathbf x\in N_0^r$ with probabilities
$$ p(\mathbf x),\qquad \sum_{\mathbf x\in N_0^r}p(\mathbf x)=1, $$
independently of each other. We suppose that $Ap(\boldsymbol0)>1$. Let $\mu_t(\mathbf x)$ be that number of particles of the $t$th generation at the point $\mathbf x\in N_0^r$. The random set $S\subseteq N_0^r$ is defined in the following way: $\mathbf x\in S$ if and only if $\lim_{t\to\infty}\mu_t(\mathbf x)=\infty$. A point $\mathbf z\in S$ is called minimal if $\mathbf x\notin S$ for all $\mathbf x\le\mathbf z$, $\mathbf x\ne\mathbf z$, We denote by $S_0$ the set of minimal points. We show how to calculate the probabilities $\mathsf P\{\mathbf z\in S_0\}$, $\mathsf P\{\mathbf z_1\in S_0,\mathbf z_2\in S_0\}$, and the like by using some auxiliary branching processes with finite number of types.
The research was supported by the Russian Foundation for Basic Research, grants 99–0100012, 96–15–96092, and INTAS–RFBR, grant 95–0099.