that the random walk is irreducible, that is, for each $z\in\mathbb{Z}^{d}$ there exist $z_{1},\dots,z_{k}\in\mathbb{Z}^{d}$, such that $z=\sum_{i=1}^{k}z_{i}$ and $a(z_{i})\ne0$ for $i=1,\dots,k$. It means that each point $ y\in\mathbb{Z}^d $ is reachable.

We assume that for any small time $h$ the particle jumps from the point $x$ to the point $y\ne x$ with probability $$\label{E1}
p(h,x,y)=a(x,y)h+o(h),$$ or remains in place with probability $$\label{E2}
p(h,x,x)=1+a(x,x)h+o(h).$$

This implies that $p(t,x,y)$, with $y\in\mathbb{Z}^{d}$ treating as a parameter, satisfies the system of backward Kolmogorov equations: $$\label{E:Kolm}
\partial_tp(t,x,y) = \sum_{x'}a(x-x')p(t,x',y),\quad
p(0,x,y)=\delta_y(x),$$ where $\delta_y(\cdot)$ is the discrete Kronecker $\delta$-function on $\mathbb{Z}^d$.

By introducing the linear operator $\mathscr{A}$ on $l^{p}(\mathbb{Z}^{d})$ for every $p$, $1 \leq p \leq \infty$, as follows $$(\mathscr{A}u)(x):=\sum_{x'}a(x,x')u(x')\equiv \sum_{x'}a(x-x')u(x'), \quad u(\cdot)\in l^{p}(\mathbb{Z}^{d}),~x\in\mathbb{Z}^{d},$$ we can rewrite the Kolmogorov’s equations  in the operator form: $$\partial_tp(t,x,y) = (\mathscr{A}p(t,\cdot,y))(x),\quad
p(0,x,y)=\delta_y(x),$$ where $y$ is treated as a parameter.

The branching mechanism is defined by a continuous-time Bienayme-Galton-Watson process (see, e.g., [@Sev]) which is determined by the following infinitesimal generating function $$\label{pf}
f(u):=\sum_{n=0}^{\infty}b_nu^n,\quad 0\le u\le 1,$$ where $b_n\ge0$ for $n\neq 1$, $b_1<0$ and $\sum_nb_n=0$.

By $p(h, n)$ we denote the probability to produce $n\neq 1$ offspring particles by the progenitor particle situated at a point $x\in \mathbb{Z}^{d}$ in a small time 