= Solution
The identity for the <killed deterministic flow> follows directly from the <chain rule> and the <fundamental theorem of calculus>, independently of the construction issue in part (b). For a continuously differentiable $f$, put $z(t)=\varphi(s,x,t)$ and $H(t)=\int_s^t\alpha(\varphi(s,x,u))\,du$. Then
$$
\begin{aligned}
\partial_t(P^\alpha_{s,t}f)(x)
&=e^{-H(t)}\bigl[f'(z(t))\partial_tz(t)-\alpha(z(t))f(z(t))\bigr]\\
&=e^{-H(t)}\bigl[a(t,z(t))f'(z(t))-\alpha(z(t))f(z(t))\bigr]\\
&=(P^\alpha_{s,t}L_t^\alpha f)(x).
\end{aligned}
$$
Hence, on such <test functions>,
$$
\boxed{\partial_tP^\alpha_{s,t}=P^\alpha_{s,t}L_t^\alpha,\qquad
L_t^\alpha f=a(t,\cdot)f'-\alpha f.}
$$
The time subscript on $L_t^\alpha$ is necessary because the <vector field> may depend on time; it is suppressed on the right-hand side of the source's displayed operator identity. The <killed deterministic flow> transports an observable along the <flow of a vector field> and multiplies it by the no-jump <survival probability>.
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