Stochastic quasi-geostrophic equation

TL;DR

This paper investigates the stochastic quasi-geostrophic equation on a 2D torus for all parameters between 0 and 1, establishing existence, uniqueness, and ergodic properties of solutions under various conditions.

Contribution

It provides new results on existence, uniqueness, and ergodicity of solutions for the stochastic quasi-geostrophic equation across the full parameter range, including subcritical and general cases.

Findings

Existence of martingale solutions for all lpha in (0,1).

Uniqueness of solutions in the subcritical case lpha > 1/2.

Exponential convergence to a unique invariant measure for lpha > 2/3 with non-degenerate noise.

Abstract

In this note we study the 2d stochastic quasi-geostrophic equation in $\mathbb{T}^2$ for general parameter $\alpha\in (0,1)$ and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all $\alpha\in (0,1)$. In the subcritical case $\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for $\alpha>2/3$ provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.