Invariant distribution of a continuous-time Markov chain

For a finite-state chain with transition semigroup $P(t)=e^{tQ}$, a probability row vector $\pi$ is invariant when $\pi P(t)=\pi$ for every $t\ge 0$. Equivalently,

Every closed communicating class of a finite Markov chain has its own invariant probability distribution. All invariant distributions are convex combinations of those class distributions. In particular, a finite irreducible chain has a unique invariant distribution, while a reducible chain may have many.

An explosive chain can have a summable solution of $\pi Q=0$ despite being transient. Nonexplosion is needed for the usual equivalence between invariant distributions and positive recurrence.