Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise

We consider the incompressible 2D Navier-Stokes equations on the torus, driven by a deterministic time periodic force and a noise that is white in time and degenerate in Fourier space. The main result is twofold. Firstly, we establish a Ruelle-Perron-Frobenius type theorem for the time inhomogeneous Feynman-Kac evolution operators with regular potentials associated with the stochastic Navier-Stokes system. The theorem characterizes asymptotic behaviors of the Feynman-Kac operators in terms of the periodic family of principal eigenvalues and corresponding unique eigenvectors. The proof involves a time inhomogeneous version of Ruelle's lower bound technique. Secondly, utilizing this Ruelle-Perron-Frobenius type theorem and a Kifer's criterion, we establish a Donsker-Varadhan type large deviation principle with a nontrivial good rate function for the occupation measures of the time inhomogeneous solution processes.