Stochastic Euler-Poincaré reduction for central extension

Abstract

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 $\ddot{I}{‰}_{\hat{I}}\pm$, 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-Poincaré reduction. Specifically, we add stochastic perturbations to the $\mathfrak{g}$ part of the extended Lie algebra $\hat{\mathfrak{g}}=\mathfrak{g}{⋊}_{\ddot{I}{‰}_{\hat{I}}\pm }\mathbb{R}$ and prove that the resulting critical points of the stochastic right-invariant Lagrangian solve the viscous QGS equation, with and without Rayleigh friction term.

30 pages