Solution
= Solution
Let $\xi=x-\langle x\rangle$ and $C_{ij}=\langle\xi_i\xi_j\rangle$. In the <linear noise approximation>, $A_i=a_{ik}\xi_k$ and $B_{ij}=b_{ij}$. At the stationary mean, the preceding moment identity gives $\dot C_{ij}=a_{ik}C_{kj}+a_{jk}C_{ik}+b_{ij}$. Hence the stationary <covariance matrix> solves the <Continuous Lyapunov equation>
$$
\boxed{aC+Ca^T+b=0,}
$$
which is precisely the requested component formula because $C$ is symmetric.