Stochastic quasi-geostrophic equation
Michael R\"ockner, Rongchan Zhu, Xiangchan Zhu

TL;DR
This paper investigates the stochastic quasi-geostrophic equation on a 2D torus for all parameters between 0 and 1, establishing existence, uniqueness, and ergodic properties of solutions under various conditions.
Contribution
It provides new results on existence, uniqueness, and ergodicity of solutions for the stochastic quasi-geostrophic equation across the full parameter range, including subcritical and general cases.
Findings
Existence of martingale solutions for all lpha in (0,1).
Uniqueness of solutions in the subcritical case lpha > 1/2.
Exponential convergence to a unique invariant measure for lpha > 2/3 with non-degenerate noise.
Abstract
In this note we study the 2d stochastic quasi-geostrophic equation in for general parameter and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all . In the subcritical case , we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for 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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Taxonomy
TopicsStochastic processes and financial applications · Stochastic processes and statistical mechanics · Navier-Stokes equation solutions
