Multiple Markov Gaussian processes

We get a necessary and sufficient condition on the density of the spectral measure for stationary Gaussian processes with a discrete set of parameters to be Markov of order $k$. We introduce a natural definition of the Markov property of order $r\in \mathbb R_+,$ in the case of continuous parameter. Moreover we give an extension of multiple Markov Gaussian processes with discrete parameters to Gaussian semiflows with some weaker property than the multiple Gaussian property.