Solution
= Solution
The <damped free-transport evolution> is $S(t)h(x,v)=e^{-t}h(x-tv,v)$. Substituting it into the <Duhamel principle> gives
$$
\boxed{f(t,x,v)=e^{-t}f_0(x-tv,v)+\int_0^t e^{-(t-s)}g\bigl(s,x-(t-s)v,v\bigr)\,ds}.
$$
The spatial shifts all follow the same backward <characteristic curve> ending at $(x,v)$ at time $t$; the velocity is constant.