Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations with vanishing noise correlation

TL;DR

This paper establishes a large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations with small, correlated noise, providing insights into the probabilistic behavior of the system under vanishing stochastic forcing.

Contribution

It introduces a large deviations framework for invariant measures of 2D stochastic Navier-Stokes equations with vanishing noise correlation and magnitude, under a specific scaling regime.

Findings

Large deviations principle proved for solutions and invariant measures

Results hold as noise magnitude and correlation scale vanish simultaneously

Provides a probabilistic understanding of the system's behavior under small noise

Abstract

We study the two-dimensional incompressible Navier-Stokes equation on the torus, driven by Gaussian noise that is white in time and colored in space. We consider the case where the magnitude of the random forcing $\sqrt{\e}$ and its correlation scale $\delta(\e)$ are both small. We prove a large deviations principle for the solutions, as well as for the family of invariant measures, as $\e$ and $\delta(\e)$ are simultaneously sent to $0$, under a suitable scaling.