Covariance criterion for a stationary Gaussian Markov process ($C(s+t)C(0)=C(s)C(t)$)

A centered continuous stationary scalar Gaussian process of positive variance can be Markov only if its covariance satisfies the displayed equation for $s,t\ge 0$. Gaussian conditioning makes the future conditional mean $C(t)X_s/C(0)$; the Markov property then forces the equation by conditioning a future observation on the present and past. Conversely this equation makes the Gaussian residual future independent of the past, proving the Markov property. In the nondegenerate decaying case the covariance is a single exponential. A sum $c_1{e}^{-a_1|t|}+c_2{e}^{-a_2|t|}$ with positive coefficients and distinct positive decay rates fails the equation and hence is not scalar Markov.