Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 27I / a

An equilibrium distribution of a continuous-time Markov chain is a probability row vector $\pi$ invariant under the transition semigroup: $\pi P(t)=\pi$ for every $t\ge 0$. Differentiating gives $\pi Q=0$, or ${\sum }_i{\pi }_i{q}_{ij}=0$ for each $j$, when the generator sums are defined. Reducible chains may have several closed classes supporting different equilibrium distributions; convex mixtures are then also equilibria. Irreducibility and positive recurrence give uniqueness for a conservative nonexplosive chain.