Abstract: The linear Lyapunov equation of a covariance matrix parametrizes theequilibrium covariance matrix of a stochastic process. This parametrization canbe interpreted as a new graphical model class, and we show how the model classbehaves under marginalization and introduce a method for structure learning via$\ell_1$-penalized loss minimization. Our proposed method is demonstrated tooutperform alternative structure learning algorithms in a simulation study, andwe illustrate its application for protein phosphorylation network reconstruction.