  K. It\^{o} characterised in \cite{ito} zero-mean stationary Gauss
Markov-processes evolving on a class of infinite-dimensional spaces. In this
work we extend the work of It\^{o} in the case of Hilbert spaces: Gauss-Markov
families that are time-homogenous are identified as solutions to linear
stochastic differential equations with singular coefficients. Choosing an
appropriate locally convex topology on the space of weakly sequentially
continuous functions we also characterize the transition semigroup, the
generator and its core thus providing an infinite-dimensional extension of the
classical result of Courr\`ege \cite{courrege} in the case of Gauss-Markov
semigroups.
