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.

翻译：本文提出连续李雅普诺夫方程路径法则的新形式。该线性矩阵方程刻画了高斯马尔可夫过程的截面稳态协方差矩阵，而路径法则将过程漂移项的图结构与协方差矩阵元素相联系。一般而言，路径法则体现为协方差矩阵的幂级数展开；当漂移项具有无环特性时，该法则可简化为漂移矩阵非对角元素的多项式表达式。借助路径法则，我们针对特定漂移矩阵结构给出了协方差矩阵元素的显式计算公式。此外，基于该法则我们推导出无环情形下方差的新下界。