On the Markov transformation of Gaussian processes

Given a Gaussian process $(X_t{)}_{t\in \mathbb{R}}$, we construct a Gaussian \emph{Markov} process with the same one-dimensional marginals using sequences of transformations of $(X_t{)}_{t\in \mathbb{R}}$ “made Markov” at finitely many times. We prove that there exists at least such a Markov transform of $(X_t{)}_{t\in \mathbb{R}}$. In the case the instantaneous decorrelation rate of $(X_t{)}_{t\in \mathbb{R}}$ is continuous, we prove that the Markov transform is uniquely determined and characterized through the same instantaneous decorrelation rate.