the of BRW model with a single initial particle and the BRW model with infinitely many particles.

\[M:osn\] The function $M_{\infty,1}$ satisfies the Cauchy problem $$\label{M1.1}
\partial_{t}M_{\infty,1}(y) = (\mathscr{A}^{*}M_{\infty,1}(t,\cdot))(y) + \delta_{0}(y)M_{\infty,1}(y), \quad M_{\infty,1}(t,\cdot)=1,$$ where $$(\mathscr{A}^{*}u)(x):=\sum_{x'}a(x',x)u(x'), \quad u(\cdot)\in l^{p}(\mathbb{Z}^{d}),~x\in\mathbb{Z}^{d}.$$

\[T:osn\]

\[R:opA\]Let us note that throughout the paper $a(x,y)\equiv a(y-x)$, and then $\mathscr{A}^{*}=\mathscr{A}$.

Let us estimate $M_{\infty,1}(t+h,y)$ using the formula of total probability. The variation of the number of particles at the point $y$ over the time interval $[t,t+h]$ up to terms of order $o(h)$ is defined by one of the following possibilities: (a) particle jump from some point $y'$, $y'\neq y$, to $y$, (b) particle jump from the point $y$ to another point and, finally, (c) branching of particles at the point $y$, subject to the condition $y=0$. Hence, taking into account the equality $\sum_{n\neq 1}b_{n}=-b_{1}$, we get $$\begin{gathered}
M_{\infty,1}(t+h,y)-M_{\infty,1}(t,y)=E[\eta_{t+h}(y)-\eta_{t}(y)]=
\\=E \Bigg[\sum_{y'\ne y} \eta_{t}(y')a(y',y)
h+\eta_{t}(y)a(y,y)h+
\delta_{0}(y)\sum_{n\ne1}\eta_{t}(y)b_{n}(n-1) h+o(h)\Bigg]=
\\=\sum_{y'} M_{\infty,1}(t,y')a(y',y) h+
\delta_{0}(y)M_{\infty,1}(t,y)f'(1)h+o(h)=
\\=\left[\mathscr{A}^{*}M_{\infty,1}(t,y)+\delta_{0}(y)\beta M_{\infty,1}(t,y)
\right]h+o(h).\end{gathered}$$ and passing to the limit for $h\to0$ we obtain . Since the legitimacy of the corresponding passage to the limit is explaied along the same lines as the corresponding place in the proof of Lemma \[L1\], it is omitted here.

To prove identity  let us introduce notations 