Stochastic Euler-Poincar\'e reduction for central extension

This paper explores the application of central extensions of Lie groups and Lie algebras to derive the viscous quasi-geostrophic (QGS) equations, with and without Rayleigh friction term, on the torus as critical points of a stochastic Lagrangian. We begin by introducing central extensions and proving the integrability of the Roger Lie algebra cocycle $\omega_\alpha$, which is used to model the QGS on the torus. Incorporating stochastic perturbations, we formulate two specific semi-martingales on the central extension and study the stochastic Euler-Poincar\'e reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\widehat{\mathfrak{g}} = \mathfrak{g} \rtimes_{\omega_\alpha} \mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.