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Let $T$ be the shift transformation. Let $P$ be invariant for $T$ and also define a discrete state space Markov Chain. Let $C_{1},\ldots,C_{n}$ be the connected components of the Markov Chain. It follows from Dobrushin's Decomposition Theorem that there exist $(P_{i})_{i \leq n}$ ergodic measures that define discrete state Markov Chains in each of the $C_{i}$'s and a measure $\mu$ such that:

$$P = \sum_{i}{P_{i}\mu(i)}$$

Are there known generalizations of this decomposition when $P$ is an invariant measure and $P$ defines a continuous state space Markov Chain? That is, do there exist ergodic $P_{\lambda}$ which define continuous state Markov Chains and a measure $\mu$ such that:

$$P = \int{P_{\lambda} \mu(d\lambda)}$$

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Your claim regarding the discrete case is an application of the ergodic decomposition theorem (decomposition into ergodic components). It also holds for a continuous state space Markov process but I'm not sure the process is Markov under each conditional laws. –  Stéphane Laurent Nov 4 '12 at 11:26
    
From what I know, the Ergodic Decompostition Theorem guarantees that any invariant measure $\mu$ on a finite dimensional space can be written as a convex combination of ergodic measures. Why does that imply that the ergodic measures can be taken as Markovian? –  madprob Nov 5 '12 at 1:47
    
In the discrete case the ergodic components correspond to the recurrence classes of the Markov chain. –  Stéphane Laurent Nov 5 '12 at 7:42
    
I don't understand yet. I think there exist ergodic measures for the shift operator which are not Markovian. Using the Ergodic Decomposition Theorem, I don't see how to guarantee that the $P_{i}$ are Markovian (as required in the statement). –  madprob Dec 6 '12 at 21:05
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