The Annals of Probability

Sub and supercritical stochastic quasi-geostrophic equation

Michael Röckner, Rongchan Zhu, and Xiangchan Zhu

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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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Ann. Probab., Volume 43, Number 3 (2015), 1202-1273.

First available in Project Euclid: 5 May 2015

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Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H30: Applications of stochastic analysis (to PDE, etc.) 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Stochastic quasi-geostrophic equation well posedness martingale problem Markov property strong Feller property Markov selections ergodicity for the subcritical case degenerate noise


Röckner, Michael; Zhu, Rongchan; Zhu, Xiangchan. Sub and supercritical stochastic quasi-geostrophic equation. Ann. Probab. 43 (2015), no. 3, 1202--1273. doi:10.1214/13-AOP887.

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