Coordinate invariance in stochastic dynamical systems

Abstract

Stochastic dynamical systems have been used to model a broad range of atmospheric and oceanic phenomena. Previous work has focused on the stochastic differential equation formulation of these systems, has largely remained in a single coordinate system, and has highlighted the role of non-normality of the deterministic dynamics. Here, the coordinate independent properties of stochastic dynamical systems are studied. The properties previously attributed to non-normality, which can be removed by a coordinate transformation, are more fundamentally seen to be coordinate-dependent manifestations of violations of detailed balance. Systems violating detailed balance can both amplify and rectify the random forcing. New coordinate-invariant measures of noise amplification are introduced and shown to achieve their lower bound when detailed balance is satisfied. Rectification results in a coherent phase space velocity which gives rise to a structured nonzero flux of all physically important quantities such as energy and momentum. The qualitative and quantitative features of these fluxes provide new predictions which can be used to further validate previously proposed stochastic models of geophysical systems.