Large and moderate deviations principles and central limit theorem for the stochastic 3D primitive equations with gradient dependent noise

TL;DR

This paper proves large and moderate deviations principles and an almost sure CLT for 3D stochastic primitive equations with gradient-dependent noise, extending previous results and correcting technical issues.

Contribution

It introduces new large and moderate deviations principles for 3D stochastic primitive equations with more general noise, and refines the CLT analysis.

Findings

Established LDP and MDP for the equations.

Corrected a technical issue in previous work.

Extended results to more general noise models.

Abstract

We establish the large deviations principle (LDP) and the moderate deviations principle (MDP) and an almost sure version of the central limit theorem (CLT) for the stochastic 3D viscous primitive equations driven by a multiplicative white noise allowing dependence on spatial gradient of solutions with initial data in $H^2$. The LDP is established using the weak convergence approach of Budjihara and Dupuis and uniform version of the stochastic Gronwall lemma. The result corrects a minor technical issue in Z. Dong, J. Zhai, and R. Zhang: Large deviations principles for 3D stochastic primitive equations, J. Differential Equations, 263(5):3110-3146, 2017, and establishes the result for a more general noise. The MDP is established using a similar argument.