Donsker-Varadhan large deviation principle for locally damped and randomly forced NLS equations

Abstract: We study large deviations from the invariant measure for nonlinear Schr\"odinger equations with colored noises on determining modes. The proof is based on a new abstract criterion, inspired by [V. Jak\v{s}i\'{c} et al., Comm. Pure Appl. Math., 68 (2015), 2108-2143]. To address the difficulty caused by fixed squeezing rate, we introduce a bootstrap argument to derive Lipschitz estimates for Feynman-Kac semigroups. This criterion is also applicable to wave equations and Navier-Stokes system.