t, x, y) := Ee^{-z\eta_{x, t}(y)},$$ where $z \geq 0$, the index $\infty$ indicates the generating functions for the whole particle population at $y\in \mathbb{Z}^{d}$, and the index $1$ indicates the generating functions for subpopulations of particles both over $\mathbb{Z}^{d}$ and at $y\in \mathbb{Z}^{d}$ generated by the parent particle with the initial location at $x \in \mathbb{Z}^{d}$.

Now we focus on the derivation of the backward Kolmogorov differential equations for the subpopulations.

\[L1\] For each $ 0\leqslant z \leqslant \infty $ the generating functions $F_{1}(z; t, x) $ and $ F_{1}(z; t, x, y) $ are continuously differentiable with respect to $t$ uniformly in $ x,y\in \mathbb{Z}^d$, satisfy the inequalities $ 0\leqslant F_{1}(z; t, x), F_{1}(z; t, x, y) \leqslant 1 $ and the backward Kolmogorov differential equations $$\begin{aligned}
\label{E1.1}
\partial_{t}F_{1}(z; t, x) &= (\mathscr{A}F_{1}(z; t, \cdot))(x) + \delta_{0}(x)f(F_{1}(z; t, x)),\\
\label{E1.2}
\partial_{t}F_{1}(z; t, x, y) &= (\mathscr{A}F_{1}(z; t, \cdot, y))(x) + \delta_{0}(x)f(F_{1}(z; t, x, y)),\end{aligned}$$ with the initial conditions $ F_{1}(z; 0, y) = e^{-z} $ and $ F_{1}(z; 0, x, y) = e^{-z\delta_{x}(y)}$, respectively.

The proof of is by analogy with [@YarBRW:r Lemma 1.2.1]. We confine ourselves to proving the statement of the lemma related to $F_{1}(z; t, 