SUB AND SUPERCRITICAL STOCHASTIC QUASI-GEOSTROPHIC EQUATION

• 成果类型: Article
• 署名作者: Roeckner, Michael; Zhu, Rongchan; Zhu, Xiangchan
• 署名单位: University of Bielefeld; Beijing Institute of Technology; Beijing Jiaotong University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP887
• 发表日期: 2015
• 页码: 1202-1273
• 关键词: navier-stokes equations; coupling approach; markov selections; itos formula; ergodicity; martingale; THEOREM; uniqueness; driven; pdes

摘要

In this paper, we study the 2D stochastic quasi-geostrophic equation on T-2 for general parameter alpha epsilon (0, 1) and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all alpha epsilon (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.