Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 4 / 26I / a / Solution

An irreducible nonexplosive continuous-time Markov chain has an equilibrium probability distribution exactly when it is positive recurrent. The distribution is unique, and gives the long-run fraction of elapsed time in each state. A null recurrent chain has no equilibrium probability distribution. The finite one-state absorbing case is trivially stationary; the return-cycle descriptions concern nontrivial irreducible classes.