# Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing

### Abstract

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in $\Bbb{L}^2_0(\mathbb{T}^2)$. Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a Hörmander-type condition. This requires some interesting nonadapted stochastic analysis.

## Authors

Martin Hairer

Mathematics Institute, The University of Warwick, Coventry CV4 7AL, United Kingdom

Jonathan C. Mattingly

Department of Mathematics, Duke University, Durham, NC 27708, United States