We consider the incompressible, two‐dimensional Navier‐Stokes equation with periodic boundary conditions under the effect of an additive, white‐in‐time, stochastic forcing. Under mild restrictions on the geometry of the scales forced, we show that any finite‐dimensional projection of the solution possesses a smooth, strictly positive density with respect to Lebesgue measure. In particular, our conditions are viscosity independent. We are mainly interested in forcing that excites a very small number of modes. All of the results rely on proving the nondegeneracy of the infinite‐dimensional Malliavin matrix.
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Mattingly et al. (2006) studied this question.