Controllability implies mixing II. Convergence in the dual-Lipschitz metric

This paper continues our study of the interconnection between controllability and mixing properties of random dynamical systems. We begin with an abstract result showing that the approximate controllability to a point and a local stabilisation property imply the uniqueness of a stationary measure and exponential mixing in the dual-Lipschitz metric. This result is then applied to the 2D Navier-Stokes system driven by a random force acting through the boundary. A by-product of our analysis is the local exponential stabilisation of the boundary-driven Navier-Stokes system by a regular boundary control.

Automated summary

This paper explores the relationship between controllability and mixing properties of random dynamical systems, showing that approximate controllability and local stabilisation properties are key factors in achieving the uniqueness of a stationary measure and exponential mixing in specific systems. The results are then applied to the boundary-driven Navier-Stokes system, leading to local exponential stabilisation through regular boundary control.