  The main objective of the paper is to study the long-time behavior of general
discrete dynamics driven by an ergodic stationary Gaussian noise. In our main
result, we prove existence and uniqueness of the invariant distribution and
exhibit some upper-bounds on the rate of convergence to equilibrium in terms of
the asymptotic behavior of the covariance function of the Gaussian noise (or
equivalently to its moving average representation). Then, we apply our general
results to fractional dynamics (including the Euler Scheme associated to
fractional driven Stochastic Differential Equations). Whenthe Hurst parameter H
belongs to (0, 1/2) we retrieve, with a slightly more explicit approach due to
the discrete-time setting, the rate exhibited by Hairer in a continuous time
setting. In this fractional setting, we also emphasize the significant
dependence of the rate of convergence to equilibriumon the local behaviour of
the covariance function of the Gaussian noise.
