Slow-fast systems with fractional environment and dynamics

Abstract

We prove a fractional averaging principle for interacting slow-fast systems. The mode of convergence is in Hölder norm in probability. The main technical result is a quenched ergodic theorem on the conditioned fractional dynamics. We also establish geometric ergodicity for a class of fractional-driven stochastic differential equations, improving a recent result of Panloup and Richard. MSC2010: 60G22, 60H10, 37A25. Keywords: Fractional Brownian motion, averaging, slow-fast system, quenched ergodic theorem, rate of convergence to equilibrium.