Conservative Coexpanding on Average Diffeomorphisms

We show that the generator of a conservative IID random system whose dynamics expands on average codimension $1$ planes has an essential spectral radius strictly smaller than $1$ on Sobolev spaces of small positive index index. Consequently, such a system has finitely many ergodic components. If there is only one component for each power of the random system, then the system enjoys multiple exponential mixing and the central limit theorem. Moreover, these properties are stable under small perturbations. As an application we show that many small perturbations of random homogeneous systems are exponentially mixing.