= Solution
A conservative <Q-matrix> has $q_{ij}\geq0$ for $i\ne j$, finite exit rates $-q_{ii}=\sum_{j\ne i}q_{ij}$, and hence row sums zero. It specifies the infinitesimal <transition probabilities> $P_{ij}(h)=\delta_{ij}+hq_{ij}+o(h)$. For a finite state space,
$$
\boxed{P(t)=e^{tQ},\quad P(0)=I,\quad P'(t)=QP(t)=P(t)Q.}
$$
The first <derivative> at zero is the generator. On infinite state spaces one must additionally address nonexplosion and the domains needed for the forward and backward equations.
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