  We study sequences of empirical measures of Euler schemes associated to some
non-Markovian SDEs: SDEs driven by Gaussian processes with stationary
increments. We obtain the functional convergence of this sequence to a
stationary solution to the SDE. Then, we end the paper by some specific
properties of this stationary solution. We show that, in contrast to Markovian
SDEs, its initial random value and the driving Gaussian process are always
dependent. However, under an integral representation assumption, we also obtain
that the past of the solution is independent to the future of the underlying
innovation process of the Gaussian driving process.
