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

TL;DR

This paper establishes a large deviation principle for nonlinear Schrödinger equations with colored noise, introducing a new abstract criterion and bootstrap method that can also be applied to wave and Navier-Stokes equations.

Contribution

It develops a novel abstract criterion and bootstrap approach for large deviations in complex PDEs, extending applicability beyond Schrödinger equations.

Findings

Proves large deviation principle for NLS with colored noise

Introduces a new bootstrap argument for Lipschitz estimates

Applicable to wave and Navier-Stokes 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.