  Computational multi-scale methods capitalize on a large time-scale separation
to efficiently simulate slow dynamics over long time intervals. For stochastic
systems, one often aims at resolving the statistics of the slowest dynamics.
This paper looks at the efficiency of a micro-macro acceleration method that
couples short bursts of stochastic path simulation with extrapolation of
spatial averages forward in time. To have explicit derivations, we elicit an
amenable linear test equation containing multiple time scales. We make
derivations and perform numerical experiments in the Gaussian setting, where
only the evolution of mean and variance matters. The analysis shows that, for
this test model, the stability threshold on the extrapolation step is largely
independent of the time-scale separation. In consequence, the micro-macro
acceleration method increases the admissible time steps far beyond those for
which a direct time discretization becomes unstable.
