Rate of convergence to equilibrium for discrete-time stochastic dynamics with memory

Abstract

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). When the 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 (Ann. Probab. 33 (2005) 703–758). In this fractional setting, we also emphasize the significant dependence of the rate of convergence to equilibrium on the local behaviour of the covariance function of the Gaussian noise.