Quenched invariance principle with a rate for random dynamical systems

In this paper, we consider the quenched invariance principle for random Young towers driven by an ergodic system. In particular, we obtain the Wassertein convergence rate in the quenched invariance principle. As a key ingredient, we derive a new martingale-coboundary decomposition for the random tower map, which provides a good control over sums of squares of the approximating martingale. We apply our results to a class of random dynamical systems that admit a random Young tower, such as independent and identically distributed (i.i.d.) translations of Viana maps, intermittent maps of the interval and small random perturbations of Anosov maps with an ergodic driving system.