Malliavin Calculus for the Stochastic 2D Navier Stokes Equation

Details Malliavin Calculus for the Stochastic 2D Navier Stokes Equation Malliavin Calculus for the Stochastic 2D Navier Stokes Equation

2004-07-13

arxiv.org/abs/math/0407215v2

Mathematics

Probability

42 Pages. Corrected version. Fixed error in the simple existence lemma (The smoothness proof still also gave existence.) Fixed

Scientific paper

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 density with respect to Lebesgue measure. We also show that under natural assumptions the density of such a projection is everywhere strictly positive. In particular, our conditions are viscosity independent. We are mainly interested in forcing which excites a very small number of modes. All of the results rely on the nondegeneracy of the infinite dimensional Malliavin matrix.

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