Sub and supercritical stochastic quasi-geostrophic equation

TL;DR

This paper investigates the stochastic quasi-geostrophic equation on a 2D torus, establishing existence, uniqueness, ergodicity, and long-term statistical properties of solutions across different parameter regimes and noise conditions.

Contribution

It provides new results on existence, uniqueness, ergodicity, and statistical laws for the stochastic quasi-geostrophic equation for all parameters in (0,1), including degenerate noise cases.

Findings

Existence of weak solutions for all b1a0b1a0(0,1)

Uniqueness of strong solutions for b1>a01/2

Ergodicity and law of large numbers in the subcritical case

Abstract

In this paper, we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^2$ for general parameter $\alpha\in(0,1)$ and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all $\alpha\in(0,1)$. In the subcritical case $\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise. The law of large numbers for the solution of the stochastic quasi-geostrophic equations in the subcritical case is also established. In the case of nondegenerate noise and $\alpha>2/3$ in addition exponential ergodicity is proved.