Leírás
We are interested in a general class of recursive discrete dynamics: $X_{n+1}=F(X_n,\Delta_{n+1})$ where $(\Delta_{n})$ is an ergodic stationary Gaussian sequence. We can think for instance of fractional
Brownian motion increments. First, we will see how it is possible to define invariant distribution in this a priori non-Markovian setting. Then, after proving existence of such a measure, we will get a uniqueness result and a rate of convergence to this invariant distribution in total variation distance.
The proof is based on a coupling method (with a step specific to this non-Markovian framework) first implemented by M. Hairer in a continuous-time setting. This method was also used by J.Fontbona and F.Panloup as well as A.Deya, F.Panloup and S.Tindel to extend M.Hairer's results.