= Solution
For a function $h$ on the copy-number state space, define the shift operator
$$
(E_i^kh)(x_1,\ldots,x_i,\ldots,x_4)
=h(x_1,\ldots,x_i+k,\ldots,x_4).
$$
With the propensities from part (a), the <forward operator of a Markov jump process> is
$$
\boxed{\begin{aligned}
\mathcal L^*p={}&(E_1^2-1)(a_1p)
+(E_2^{-1}-1)(a_2p)
+(E_2^1-1)(a_3p)\\
&+(E_3^{-1}-1)(a_4p)
+(E_3^1E_4^1-1)(a_5p)
+(E_4^{-1}-1)(a_6p)
+(E_2^1-1)(a_7p).
\end{aligned}}
$$
Equivalently, the <chemical master equation> has the gain-minus-loss form
$$
(\mathcal L^*p)(\mathbf x)
=\sum_{r=1}^7
\left[a_r(\mathbf x-\nu_r)p(\mathbf x-\nu_r)
-a_r(\mathbf x)p(\mathbf x)\right],
$$
where terms outside the state space vanish.
The adjoint <Markov jump-process generator> acts on observables:
$$
\boxed{(\mathcal Lf)(\mathbf x)
=\sum_{r=1}^7a_r(\mathbf x)
\left[f(\mathbf x+\nu_r)-f(\mathbf x)\right].}
$$
Indeed, shifting the summation index in the gain terms gives $\langle f,\mathcal L^*p\rangle=\langle\mathcal Lf,p\rangle$.
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