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 in the entries of the drift
and volatility matrices. 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 resulting from the Lyapunov equation and those resulting from
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 on 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.
