# Steady state probabilities for a continuous-time Markov chain

I have a finite state and time-homogeneous continuous-time Markov chain (CTMC) which is not irreducible. Will steady state probabilities exist for this CTMC? How to prove this?

• "CTMC" = Continuous Time Markov Chain, presumably. (Using unexplained acronyms is a good way to lose many potential replies.) – whuber Dec 6 '11 at 15:41
• Please update our question and be more specific, in particular, about what you mean by "steady-state probabilities". Also, if you can include any other known structure about the chain you are interested in, it might help. – cardinal Dec 7 '11 at 1:24

Consider a two-state discrete-time MC with transition matrix $P(1,1) = P(2,2)=1, P(1,2)=P(2,1) = 0$. Clearly it is aperiodic but not irreducible. Any steady-state distribution $\pi$ satisfies $\pi = \pi P$, is nonnegative, and sums to one. Obviously this is satisfied for any $\pi$ that is nonnegative and sums to one. So every distribution on $\{1,2\}$ is a steady-state distribution for this example. (Hence, clearly, a unique one does not exist.)