The General Stationary Gaussian Markov Process

We find the class, ${\mathcal{C}}_k,k\ge 0$, of all zero mean stationary Gaussian processes, $Y(t),t\in \mathbb{R}$ with $k$ derivatives, for which \begin{equation} Z(t) \equiv (Y^{(0)}(t), Y^{(1)}(t), \ldots, Y^{(k)}(t) ), ~ t \ge 0 \end{equation} \noindent is a $(k+1)$-vector Markov process. (here, $Y^{(0)}(t)=Y(t)$).