Effective Diffusions with Intertwined Structures

The aim of the article is to explore multi-scale stochastic differential equations (SDEs) on manifolds. In these equations the variables of interest evolve at their natural speeds and interact with other variables that evolve at a faster pace. The primary objective is to identify the effective motion. This is an autonomous equation, its solutions approximate the evolution of the slowly moving variables as the scale of speed separation approaches zero. While multiscale SDEs with linear state spaces have been a popular research topic for several decades, their study on manifolds is a more recent development. In this work, we illustrate the derivation of multi-scale equations by incorporating geometric information, such as symmetries and conserved quantities. Subsequently, we delve into these examples to investigate their effective dynamics.