A Trek Rule for the Lyapunov Equation

In this note we show a new version of the trek rule for the continuous Lyapunov equation. This linear matrix equation characterizes the cross-sectional steady-state covariance matrix of a Gaussian Markov process, and the trek rule links the graphical structure of the drift of the process to the entries of this covariance matrix. In general, the trek rule is a power series expansion of the covariance matrix, while for the special case where the drift is acyclic, it simplifies to a polynomial in the off-diagonal entries of the drift matrix. Using the trek rule we can give relatively explicit formulas for the entries of the covariance matrix for some special cases of the drift matrix. Furthermore, we use the trek rule to derive a new lower bound for the variances in the acyclic case.