= Solution
A <Q-matrix> satisfies $q_{ij}\geq0$ for $i\ne j$ and $q_{ii}=-\sum_{j\ne i}q_{ij}$; on a countable state space the exit rate $-q_{ii}$ is required to be finite. For a finite-state <continuous-time Markov chain> with <transition semigroup of a continuous-time Markov chain>[transition semigroup] $P(t)$,
$$
Q=P'(0)=\lim_{t\downarrow0}\frac{P(t)-I}{t},
\qquad P(t)=e^{tQ}.
$$
The backward and forward <Kolmogorov equations> are respectively
$$
\boxed{P'(t)=QP(t),\qquad P'(t)=P(t)Q.}
$$
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