The article explores ways to model changing covariance matrices using linear stochastic systems. By applying concepts from optimal mass transport and the Schrödinger bridge problem, the researchers studied different paths of covariance induced by various regularizations on system matrices. They developed differential equations for these paths and found closed-form expressions for scalar-valued covariance. One path extends geodesics based on the Fisher-Rao metric to situations with stochastic input, while another path involves linear system matrices with rotating eigenspace in noiseless scenarios. The researchers compared these paths using examples.