Sub and supercritical stochastic quasi-geostrophic equation

Röckner, Michael ^[3] ; Zhu, Rongchan ^[1] ; Zhu, Xiangchan ^[2]
1. [1] Beijing Institute of Technology
  
  Beijing Institute of Technology
  
  China
2. [2] Beijing Jiaotong University
  
  Beijing Jiaotong University
  
  China
3. [3] University of Bielefeld
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 3, 2015, págs. 1202-1273
Idioma: inglés
DOI: 10.1214/13-AOP887
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Resumen
- In this paper, we study the 2D stochastic quasi-geostrophic equation on T2 for general parameter α∈(0,1) and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all α∈(0,1). In the subcritical case α>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 α>2/3 in addition exponential ergodicity is proved.