The Small Scales of the Stochastic Navier–Stokes Equations Under Rough Forcing

We prove that the small scale structures of the stochastically forced Navier–Stokes equations approach those of the naturally associated Ornstein–Uhlenbeck process as the scales get smaller. Precisely, we prove that the rescaled kth spatial Fourier mode converges weakly on path space to an associated Ornstein–Uhlenbeck process as |k|→ ∞. In addition, we prove that the Navier–Stokes equations and the naturally associated Ornstein–Uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper-viscosity. This gives a simple proof of unique ergodicity for the hyperviscous Navier–Stokes system. We show how different strengthened hyperviscosity produce varying levels of equivalence.