x, 0)g_{n}(M_{1}(q,0, y),\ldots, M_{n-1}(q, 0, y))dq,\end{gathered}$$ where $$\begin{aligned}
g_{n}(M_{1}, M_{2},\ldots, M_{n-1}) &=  \sum_{r=1}^{n} \frac{\beta^{(r)}}{r!} \cdot \sum_{ \stackrel{i_{1},\cdots,i_{r}>0}{i_{1}+\cdots+i_{r}=n}}
\frac{n!}{i_{1}!\cdots i_{n}!} M_{i_{1}} \cdots M_{i_{r}}.\end{aligned}$$

The proof of the lemma is by analogy with [@YarBRW:r]. Using variation of parameters in  and , we obtain the required equations.

In conclusion of this section let us recall the model of BRW defined in [@YarBRW:r]. Let at time $t=0$ there be only one particle at $x\in \mathbb{Z}^{d}$, from which the evolution of BRW starts. Denote by $\mu_{x,t}$ the total number of particles on the lattice at time $t$ (the population of particles on $\mathbb{Z}^{d}$), and by $\mu_{x,t}(y)$ the number of particles at $\mu_{x,t}(y)$. They satisfy the initial conditions $ \mu_{x,0}=1 $ and $\mu_{x,0}(y)=\delta_{x}(y)$, respectively. Following [@YarBRW:r] let us consider the Laplace generating functions: $$F (z; t, x) := Ee^{-z\mu_{x, t}}, \quad
F (z; t, x, y) := Ee^{-z\mu_{x, t}(y)}.$$

\[L:osn\] For each $ 0\leqslant z \leqslant \infty $ the generating functions $F_{1}(z; t, x) $ and $ F (z; t, x) $ are continuously differentiable with respect to $t$ uniformly in $ x\in \mathbb{Z}^d$, satisfy the inequalities $ 0\leqslant F_{1}(z; t, x), F (z; t, x) \leqslant 1 $ and the backward Kolmogorov differential equation ; 