Mixing for the primitive equations under bounded non-degenerate noise

TL;DR

This paper proves that the stochastic 3D primitive equations of atmospheric mechanics, under bounded, non-degenerate, and periodic noise, exhibit mixing behavior with a unique stationary measure, ensuring convergence of all trajectories.

Contribution

It establishes the mixing property and uniqueness of the stationary measure for the primitive equations under specific stochastic forcing conditions.

Findings

The Markov chain associated with the equations is mixing.

Existence of a unique stationary measure is proven.

All trajectories converge to this stationary measure.

Abstract

We study the stochastic 3D primitive equations of the atmospheric mechanics. We consider them under a bounded and non-degenerate noise, which is statistically periodic in time with period $1$. In such a case we prove that the associated integer-time Markov chain is mixing, which means that there exists a unique stationary measure to which converge the laws all trajectories of this Markov chain.