= Equilibrium distribution of a continuous-time Markov chain
{title2=$\pi P(t)=\pi$}
An <Equilibrium distribution of a continuous-time Markov chain> is a <probability vector>[probability row vector] $\pi$ invariant under its <Transition semigroup of a continuous-time Markov chain>[transition semigroup]:
$$
\pi P_t=\pi\qquad(t\geq0).
$$
It satisfies $\pi Q=0$ when the generator sums are defined. An invariant law is a property of the entire evolution, so formal generator equations must be interpreted with the process's <nonexplosion of a continuous-time Markov chain>[nonexplosion] and domain conditions.
For a finite <irreducible Markov chain>[irreducible] chain, the equilibrium distribution is unique and every transition row converges to it. For a general finite chain, the limiting row from state $i$ is a mixture of the stationary distributions $\pi^C$ of its <closed communicating class>[closed irreducible classes]:
$$
P_t(i,\cdot)\longrightarrow\sum_C\mathbb P_i(T_C<\infty)\,\pi^C.
$$
The chain eventually enters one of these classes and then mixes within it. If every <closed communicating class>[closed class] is a single <absorbing state>, this reduces to absorption <probability>[probabilities] at those states. Merely having some <absorbing state>[absorbing states] does not exclude other closed classes.
<holding time>[Holding times] explain the link to an <jump chain>[embedded jump chain]. For a finite <irreducible Markov chain>[irreducible] chain with strictly positive <holding rate of a continuous-time Markov chain>[exit rates] $q_i$ and jump-chain <Stationary distribution> $\alpha$, the time-stationary distribution is
$$
\pi_i=\frac{\alpha_i/q_i}{\sum_j\alpha_j/q_j}.
$$
Visit frequency is multiplied by <mean> time $1/q_i$ per visit. In the countable setting, these time weights need a finite positive normalizer and a <nonexplosive continuous-time Markov chain>[nonexplosive] process before they define an equilibrium <probability>. For an alternating two-state <jump chain>, $\alpha=(1/2,1/2)$ and rates $a,b$ give $\pi=(b/(a+b),a/(a+b))$: slower <holding rate of a continuous-time Markov chain>[holding rates] increase time occupancy.
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