Large deviation principle for the three dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise

TL;DR

This paper establishes the large deviation principle for three-dimensional stochastic planetary geostrophic equations modeling large-scale ocean circulation under small multiplicative noise, using weak convergence methods.

Contribution

It proves well-posedness of weak solutions and applies the weak convergent method to derive the large deviation principle for the system.

Findings

Large deviation principle is valid in the small noise limit.

Well-posedness of weak solutions is established.

Methodology combines monotonicity, Itô, Burkholder-Davis-Gundy, and weak convergence techniques.

Abstract

We demonstrate the large deviation principle in the small noise limit for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation. In this paper, we first prove the well-posedness of weak solutions to this system by the method of monotonicity. As we know, a recently developed method, weak convergent method, has been employed in studying the large deviations and this method is essentially based on the main result of \cite{ba2} which discloses the variational representation of exponential integrals with respect to the Brownian noise. The It\^{o} inequality and Burkholder-Davis-Gundy inequality are the main tools in our proofs, and the weak convergence method introduced by Budhiraja, Dupuis and Ganguly in \cite{ba3} is also used to establish the large deviation principle.