1 Three states
Let $\sigma_i^2>0, i=1,2,3$. Set $\Sigma:=\operatorname{diag}\left(\sigma_1^2, \sigma_2^2, \sigma_3^2\right) . \mu:=\operatorname{diag}\left(\mu_1, \mu_2, \mu_3\right) . Q=\left(q_{i j}\right)_{3 \times 3}$. The $Q$ matrix has the following properties (i) $q_{i i}<0, q_{i j}>0, i=1,2,3$; (ii) $\sum_{j=1}^3 q_{i j}=0, i=1,2,3$. It follows that there exists a positive vector $\pi:=\left(\pi_1, \pi_2, \pi_3\right)$ such that $$ \begin{aligned} & \pi Q=0, \\ & \sum_{i=1}^3 \pi_i=1 . \end{aligned} $$
Define $f(x):=(f(x, 1), f(x, 2), f(x, 3))^T$ Questions 1: Find the general solution to $$ \frac{1}{2} \Sigma f^{\prime \prime}+\mu f^{\prime}+Q f=(0,0,0)^T . $$
Hints. Guess the solution has the following $f=e^{\lambda x} \xi$, where $\lambda$ are eigenvalues and eigenvectors to be determined. Questions 2: Find the solution to $$ \frac{1}{2} \Sigma f^{\prime \prime}+\mu f^{\prime}+Q f=(-1,-1,-1)^T $$ with boundary conditions $f(a, 1)=0, f(b, 1)=1, f(a, 2)=0, f(b, 2)=1, f(a, 3)=0, f(b, 3)=$ 1.