Question
Consider a Markov process $\{X(t)\}_{t \geq 0}$ with state space $$S = \{0, 1, 2, 3\}$$ and generator matrix $$Q = \begin{pmatrix} -q_0 & 2 & 0 & 0\\ 2 & -q_1 & 4 & 0\\ 0 & 4 & -q_2 & 2\\ 0 & 0 & 1 & -q_3 \end{pmatrix}.$$
Find a stationary distribution $\pi_R$ of the embedded chain. Is $\pi_R$ unique? Explain your reasoning.
My working
In earlier parts of this problem, I have determined a stationary distribution associated with $Q$ to be $$\pi_Q = \left(\frac 1 5, \frac 1 5, \frac 1 5, \frac 2 5\right).$$ I have also determined the transition matrix of the embedded chain to be $$R = \begin{pmatrix} 0 & 1 & 0 & 0\\ \frac 1 3 & 0 & \frac 2 3 & 0\\ 0 & \frac 2 3 & 0 & \frac 1 3\\ 0 & 0 & 1 & 0 \end{pmatrix},$$ so that $$\pi_R = \left(\frac 1 8, \frac 3 8, \frac 3 8, \frac 1 8\right).$$
However, I am unsure whether $\pi_R$ is unique and why. I am thinking that $\pi_R$ is indeed unique and in particular, $\pi_R$ and $\pi_Q$ are related in some way. Would I be correct? Any advice will be greatly appreciated :)