M. Hairer, J. C. Mattingly
Ergodicity of the 2D Navier-Stokes Equations with Degenerate Stochastic Forcing
(327K, PDF)
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 L^2_0(T^2). Unlike in 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 H rmander-type condition. This
requires some interesting non-adapted stochastic analysis.