A flea jumps on the vertices of a triangle $A B C$; its position is described by a continuous time Markov chain with a $Q$-matrix

$$Q=\left(\begin{array}{ccc} -1 & 1 & 0 \\ 0 & -1 & 1 \\ 1 & 0 & -1 \end{array}\right) \quad \begin{aligned} &A \\ &B \\ &C \end{aligned}$$

(a) Draw a diagram representing the possible transitions of the flea together with the rates of each of these transitions. Find the eigenvalues of $Q$ and express the transition probabilities $p_{x y}(t), x, y=A, B, C$, in terms of these eigenvalues.

[Hint: $\operatorname{det}(Q-\mu \mathbf{I})=(-1-\mu)^{3}+1$. Specifying the equilibrium distribution may help.]

Hence specify the probabilities $\mathbb{P}\left(N_{t}=i \bmod 3\right)$ where $\left(N_{t}, t \geqslant 0\right)$ is a Poisson process of rate $1 .$

(b) A second flea jumps on the vertices of the triangle $A B C$ as a Markov chain with Q-matrix

$$Q^{\prime}=\left(\begin{array}{ccc} -\rho & 0 & \rho \\ \rho & -\rho & 0 \\ 0 & \rho & -\rho \end{array}\right) \quad \begin{aligned} &A \\ &B \\ &C \end{aligned}$$

where $\rho>0$ is a given real number. Let the position of the second flea at time $t$ be denoted by $Y_{t}$. We assume that $\left(Y_{t}, t \geqslant 0\right)$ is independent of $\left(X_{t}, t \geqslant 0\right)$. Let $p(t)=\mathbb{P}\left(X_{t}=Y_{t}\right)$. Show that $\lim _{t \rightarrow \infty} p(t)$ exists and is independent of the starting points of $X$ and $Y$. Compute this limit.