Exponential decay of correlations for random interval diffeomorphisms

We consider a finite number of orientation preserving $C^2$ interval diffeomorphisms and apply them randomly in such a way that the expected Lyapunov exponents at the boundary points are positive. We prove the exponential decay of correlations for Lipschitz observables with respect to the unique stationary measure supported on the interior of the interval. The key step is to show the exponential synchronization in average.