I call a Strongly Connected Component (SCC) "ergodic" if we cannot get out of it, and "transient" otherwise.

 

It suffices to compute the probability ρ(C) to end up in each ergodic component C

Although your case is not technically an absorbing Markov chain (because not every state will eventually reach an absorbing state) you can still use absorbing Markov chain concepts and notation to compute those probabilities.

Transform your process to an absorbing Markov chain by removing all of the connections within the ergodic components, so that every state in an ergodic component of the original process is an absorbing state in the transformed process.

Now use the formulas in that wikipedia page to compute the 'canonical form' and the 'fundamental matrix' for the transformed process, and use these to find the absorbing probabilities of the transformed process.

The probability p(C) is the sum of absorbing probabilities of the component's states in the transformed absorbing process.

I call a Strongly Connected Component (SCC) "ergodic" if we cannot get out of it, and "transient" otherwise.

It suffices to compute the probability ρ(C) to end up in each ergodic component C

Although your case is not technically an absorbing Markov chain (because not every state will eventually reach an absorbing state) you can still use absorbing Markov chain concepts and notation to compute those probabilities.

Transform your process to an absorbing Markov chain by removing all of the connections within the ergodic components, so that every state in an ergodic component of the original process is an absorbing state in the transformed process.

Now use the formulas in that wikipedia page to compute the 'canonical form' and the 'fundamental matrix' for the transformed process, and use these to find the absorbing probabilities of the transformed process.

The probability p(C) is the sum of absorbing probabilities of the component's states in the transformed absorbing process.