 Solution

At the stationary mean,The $(1,1)$ entry of the Continuous Lyapunov equation gives $-2\gamma C_{11}+2\lambda =0$, so $C_{11}=\lambda /\gamma$. Its $(1,2)$ entry gives $\gamma C_{11}-(\gamma +\beta )C_{12}-\lambda =0$, hence $C_{12}=0$. Finally $(2,2)$ gives $2\gamma C_{12}-2\beta C_{22}+2\lambda =0$. ThusBoth variances equal their means; the absence of stationary covariance does not mean that the maturation jumps are independent events in the two populations.