  Gaussian Processes (GPs) and Linear Dynamical Systems (LDSs) are essential
time series and dynamic system modeling tools. GPs can handle complex,
nonlinear dynamics but are computationally demanding, while LDSs offer
efficient computation but lack the expressive power of GPs. To combine their
benefits, we introduce a universal method that allows an LDS to mirror
stationary temporal GPs. This state-space representation, known as the
Markovian Gaussian Process (Markovian GP), leverages the flexibility of kernel
functions while maintaining efficient linear computation. Unlike existing
GP-LDS conversion methods, which require separability for most multi-output
kernels, our approach works universally for single- and multi-output stationary
temporal kernels. We evaluate our method by computing covariance, performing
regression tasks, and applying it to a neuroscience application, demonstrating
that our method provides an accurate state-space representation for stationary
temporal GPs.
