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

作者：Nersesyan, Vahagn; Peng, Xuhui^*; Xu, Lihu

来源：Journal of Differential Equations, 2023, 362: 230-249.

DOI：10.1016/j.jde.2023.03.004

• Navier-Stokes system
• Large deviations
• Degenerate noise
• Malliavin calculus
• Feynman-Kac semigroup
• Uniform Feller property

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 [17] in a discrete-time setting and extended in [26] 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.