On the martingale problem associated to the 2D and 3D stochastic Navier-Stokes equations

We consider a Markov semigroup (P(t))(t >= 0) associated to the 2D and 3D Navier Stokes equations. In the two-dimensional case P t is unique, whereas in the three-dimensional case (where uniqueness is not known) it is constructed as in [4] and [7]. For d = 2, we specify a core, identify the abstract generator of (P(t))(t >= 0) with the differential Kolmogorov operator L on this core and prove existence and uniqueness for the corresponding martingale problem. In dimension 3, we are not able to prove a similar result and we explain the difficulties encountered. Nonetheless, we specify a core for the generator of the transformed semigroup. (S(t))(t >= 0); obtained by adding a suitable potential and then using the Feynman-Kac formula. Then we identify the abstract generator. (S(t))(t >= 0) with a differential operator N on this core and prove uniqueness for the stopped martingale problem.