Sub and supercritical stochastic quasi-geostrophic equation

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.

Published at http://dx.doi.org/10.1214/13-AOP887 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Global L_2-solutions of stochastic Navier-Stokes equations
• On ergodicity of some Markov processes
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• Determinstic and stochastic 2d Navier-Stokes equations with anisotropic viscosity
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