STOCHASTIC QUASI-GEOSTROPHIC EQUATION
In this note we study the 2D stochastic quasi-geostrophic equation in T-2 for general parameter alpha is an element of (0, 1) and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all alpha is an element of (0,1). In the subcritical case alpha > 1/2, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for alpha > 2/3 provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.
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World Scientific