Large deviations principle via Malliavin calculus for the Navier-Stokes system driven by a degenerate white-in-time noise

The purpose of this paper is to establish the Donsker-Varadhan type large deviations principle (LDP) for the two-dimensional stochastic Navier--Stokes system. The main novelty is that the noise is assumed to be highly degenerate in the Fourier space. The proof is carried out by using a criterion for the LDP developed in [Jak\v si\'c et.al.\emph{Nonlinearity},31(2):540-596,2018] in a discrete-time setting and extended in [Martirosyan and Nersesyan. \emph{Ann.Inst.Henri Poincar\'{e} Probab. Stat.}, 54(4):2002-2041,2018] to the continuous-time. One of the main conditions of that criterion is the uniform Feller property for the Feynman-Kac semigroup,which we verify by using Malliavin calculus. This work is based on joint work with Vahagn Nersesyan and Lihu Xu.