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The stochastic 2D Navier-Stokes equations on the torus driven by degeneratenoise are studied. We characterize the smallest closed invariant subspace forthis model and show that the dynamics restricted to that subspace is ergodic.In particular, our results yield a purely geometric characterization of a classof noises for which the equation is ergodic in $\L^2_0(\TT^2)$. Unlike previousworks, this class is independent of the viscosity and the strength of thenoise. The two main tools of our analysis are the \textit{asymptotic strongFeller} property, introduced in this work, and an approximate integration byparts formula. The first, when combined with a weak type of irreducibility, isshown to ensure that the dynamics is ergodic. The second is used to show thatthe first holds under a H{\"o}rmander-type condition. This requires someinteresting nonadapted stochastic analysis.

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