In this paper, we consider the 2D Navier–Stokes system driven by a white-in-time noise. We show that the occupation measures of the trajectories satisfy a large deviations principle, provided that the noise acts directly on all Fourier modes. The proofs are obtained by developing an approach introduced previously for discrete-time random dynamical systems, based on a Kifer-type criterion and a multiplicative ergodic theorem.
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[MN18a] Local large deviations principle for occupation measures of the stochastic damped nonlinear wave equation, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 54 (2018) no. 4, pp. 2002-2041 | DOI | MR | Zbl
[MN18b] Multiplicative ergodic theorem for a non-irreducible random dynamical system (2018) https://www.archives-ouvertes.fr/hal-01695046v1