The General Stationary Gaussian Markov Process

We find the class, {\cal{C}}_k, k \ge 0, of all zero mean stationary Gaussian processes, Y(t), ~t \in \reals 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)).