Does there exist a Markov Chain with $4$ states and a unique invariant measure $\pi$ such that $\pi_{1}= \pi_{3} = 0$ and $\pi_{2}, \pi_{4} > 0$?
I couldn't think of an example off the top of my head with this property. If it actually doesn't exist, how to prove such a Markov chain cannot exist?
Sure, just make $1$ and $3$ transient while 2 and 4 are both recurrent, for example $P=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 \\ 0 & 1 & 0 & 0 \\ 0 & 1/2 & 0 & 1/2 \end{bmatrix}$. Note that if you assume the invariant measure is unique and has these properties then these transience/recurrence behaviors are also necessary.