It's pretty widely documented that a Markov process Y is reducible/irreducible if and only if the embedded Markov chain X is reducible/irreducible.
However I'm not sure this works in reverse. I'm wondering if it's possible to determine the converse without actually calculating the embedded Markov chain: i.e. given the generator matrix for a Markov process Y (and its stationary distribution), is it possible to determine from these facts alone whether an embedded Markov chain X for Y is reducible?
In my example I have for Y my generator matrix:
\begin{bmatrix} -3 & 1 & 2 \\ 1 & -2 & 1 \\ 1 & 0 & -1 \end{bmatrix}
and hence the stationary distribution: $\pi = (1/4, 1/8, 5/8) $