Exponential Mixing and Limit Theorems of Quasi-periodically Forced 2D Stochastic Navier–Stokes Equations in the Hypoelliptic Setting

Abstract

We consider the incompressible 2D Navier–Stokes equations on the torus driven by a deterministic time quasi-periodic force and a noise that is white in time and degenerate in Fourier space. We show that the asymptotic statistical behavior is characterized by a quasi-periodic invariant measure that exponentially attracts the law of all solutions. The result is true for any value of the viscosity \(\nu >0\) and does not depend on the strength of the external forces. By utilizing this quasi-periodic invariant measure, we establish a quantitative version of the strong law of large numbers and central limit theorem for the continuous time inhomogeneous solution processes with explicit convergence rates. It turns out that the convergence rate in the central limit theorem depends on the time inhomogeneity through the Diophantine approximation property on the quasi-periodic frequency of the quasi-periodic force. The scheme of analyzing the statistical behavior of the time inhomogeneous solution process by the quasi-periodic invariant measure is new and could be extended to other inhomogeneous Markov processes.