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

TL;DR

This paper establishes a large deviations principle for the 2D stochastic Navier-Stokes equations driven by highly degenerate noise, using Malliavin calculus to verify key properties of the associated semigroup.

Contribution

It introduces a novel approach to prove the LDP for Navier-Stokes systems with degenerate noise by extending existing criteria and employing Malliavin calculus.

Findings

Proved the LDP for the 2D stochastic Navier-Stokes system with degenerate noise.

Verified the uniform Feller property using Malliavin calculus.

Extended existing LDP criteria to continuous-time stochastic PDEs.

Abstract

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 arXiv:1410.6188 in a discrete-time setting and extended in arXiv:1505.03686 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.