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

Abstract

We consider the martingale problem associated to the Navier-Stokes in dimension $2$ or $3$. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension $2$, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension $3$, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.