all the moments grow exponentially. As is known [@YarBRW:r] in this case the growth rates of moments is closely related to the largest positive eigenvalue $\lambda_{0}$ of the operator $\mathscr{H}=\mathscr{A}+ \beta\Delta_{0}$.

\[T3\] Let $\sum_{x\in \mathbb{Z}^{d}} a(0,x)|x|^{2}<\infty$ then for all $ n \in \mathbb{N}$ and $ t \rightarrow \infty$ $$M_{n}(t, x, y)  \sim C_{n}(x, y)u_{n}(t), \quad M_{n}(t, x) \sim C_{n}(x)v_{n}(t),$$ where $ C_{n}(y, x) > 0,  C_{n}(x) > 0$, and the function $ v_{n}(t) $ is expressed by:

a\) for $ \beta > \beta_{c} $, $$u_{n}(t) = e^{n\lambda_0 t}, \quad  v_{n}(t) = e^{n\lambda_0 t};$$ b) for $ \beta = \beta_{c} $: $$\begin{aligned}
d&=1: &u_{n}(t)&= t^{(n-1)/2}(\ln t)^{n-1}, &v_{n}(t)&= t^{(n-1)/2};\\
d&=2: &u_{n}(t)&= t^{-1}, &v_{n}(t)&= (\ln t)^{n-1};\\  
d&=3: &u_{n}(t)&= t^{-1/2}(\ln t)^{n-1}, &v_{n}(t)&= t^{n-1/2};\\   
d&=4:  &u_{n}(t)&= t^{n-1}(\ln t)^{1-2n}, &v_{n}(t)&= t^{2n-1}(\ln t)^{1-2n};\\ 
d&\geq 5: &u_{n}(t)&= t^{2n-1}, &v_{n}(t)&= t^{2n-1};\end{aligned}$$ c) for $ \beta < \beta_{c} $: $$\begin{aligned}
d&=1:  &u_{n}(t) &= t^{-3/2}, &v_{n}(t) &= t^{-1/2}; \\
d&=2:  &u_{n}(t) &= (t \ln^{2} t)^{-1}, &v_{n}(t)&= (\ln t)^{-1};\\
d&\geq 3:  &u_{n}(t) &= t^{-d/2}, &v_{n}(t)&\equiv 1 .\end{aligned}$$

The duality of BRW models {#BRWIF}
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The goal of this section is to prove the following relation $$\label{E:Meqm}
M_{\infty, 1}(t, y) \equiv m_{1}(t, y),$$ which can be treated as a kind of duality between 