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May 2015 Sub and supercritical stochastic quasi-geostrophic equation
Michael Röckner, Rongchan Zhu, Xiangchan Zhu
Ann. Probab. 43(3): 1202-1273 (May 2015). DOI: 10.1214/13-AOP887

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.

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Michael Röckner. Rongchan Zhu. Xiangchan Zhu. "Sub and supercritical stochastic quasi-geostrophic equation." Ann. Probab. 43 (3) 1202 - 1273, May 2015. https://doi.org/10.1214/13-AOP887

Information

Published: May 2015
First available in Project Euclid: 5 May 2015

zbMATH: 1322.60121
MathSciNet: MR3342662
Digital Object Identifier: 10.1214/13-AOP887

Subjects:
Primary: 35R60, 60H15, 60H30

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 3 • May 2015
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