The Lyapunov equation is a linear matrix equation characterizing the cross-sectional steady-state covariance matrix of a Gaussian Markov process. We show a new version of the trek rule for this equation, which links the graphical structure of the drift of the process to the entries of the steady-state covariance matrix. In general, the trek rule is a power series expansion of the covariance matrix, though for acyclic models 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. These results illustrate notable differences between covariance models entailed by the Lyapunov equation and those entailed by linear additive noise models. To further explore differences and similarities between these two model classes, we use the trek rule to derive a new lower bound for the marginal variances in the acyclic case. This sheds light on the phenomenon, well known for the linear additive noise model, that the variances in the acyclic case tend to increase along a topological ordering of the variables.

翻译：Lyapunov方程是一种线性矩阵方程，用于刻画高斯马尔可夫过程的截面稳态协方差矩阵。我们为该方程提出了一个全新版本的trek规则，该规则将过程漂移的图结构与稳态协方差矩阵的各个元素联系起来。一般而言，trek规则是协方差矩阵的幂级数展开式，但在无环模型中，它会简化为漂移矩阵非对角线元素的多项式。利用trek规则，我们可以针对漂移矩阵的某些特殊情况，给出协方差矩阵元素的相对显式公式。这些结果揭示了由Lyapunov方程所蕴含的协方差模型与由线性加性噪声模型所蕴含的协方差模型之间的显著差异。为了进一步探究这两类模型之间的差异与相似性，我们运用trek规则推导出了无环情形下边际方差的一个新下界。这阐明了在线性加性噪声模型中广为人知的现象：在无环情形下，方差倾向于沿着变量的拓扑排序方向递增。