$h$ under condition $\eta_{x, 0}(y)=\delta(y-x)$: $$\begin{aligned}
\label{pn}
p(h, n) &= b_nh + o(h)\quad \text{for}\quad  n\neq 1,\\
p(h, 1) &= 1 + b_1h + o(h).\end{aligned}$$

We assume also that there exists the first derivative of the generating function $\beta_{1} := f'(1)< \infty $, that is, the first moment of the direct particle offsprings is finite. Denote in this case $\beta := \beta_{1}$. The assumption that all moments are finite, that is, $\beta_{r} := f^{(r)}(1)< \infty$ for all $r$, is used in the method-based proofs of the limit theorems on behavior of the numbers of particles in BRW (see, e.g., [@YarBRW:r]).

Now, let us combine the branching and walking mechanisms in order to get a branching random walk with the source at $x=0$. Suppose that the particles may be at the source as well as at an arbitrary lattice point. Then at time $t$ the particles in small time $h$ may jump to a point $y\neq x$ with the probability $$\label{E:phxy}
p(h, x, y) = a(x, y)h + o(h)$$ or, as earlier, it will stay at the point $x$ with the probability $$\label{E:phxx}
p(h,x,x)=1 +a(x,x)h + o(h)= 1 +a(0,0)h + o(h),$$ and in this last case (staying at the source $x=0$) the 