(i) Consider the continuous-time Markov chain $\left(X_{t}\right)_{t \geqslant 0}$ with state-space $\{1,2,3,4\}$ and $Q$-matrix

$$Q=\left(\begin{array}{rrrr} -2 & 0 & 0 & 2 \\ 1 & -3 & 2 & 0 \\ 0 & 2 & -2 & 0 \\ 1 & 5 & 2 & -8 \end{array}\right)$$

Set

$$Y_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 2 & \text { if } X_{t}=4\end{cases}$$

and

$$Z_{t}= \begin{cases}X_{t} & \text { if } X_{t} \in\{1,2,3\} \\ 1 & \text { if } X_{t}=4\end{cases}$$

Determine which, if any, of the processes $\left(Y_{t}\right)_{t \geqslant 0}$ and $\left(Z_{t}\right)_{t \geqslant 0}$ are Markov chains.

(ii) Find an invariant distribution for the chain $\left(X_{t}\right)_{t \geqslant 0}$ given in Part (i). Suppose $X_{0}=1$. Find, for all $t \geqslant 0$, the probability that $X_{t}=1$.