Multiscale systems, homogenization, and rough paths

TL;DR

This paper surveys recent advances in the convergence of fast-slow deterministic systems to stochastic differential equations, emphasizing mild assumptions and incorporating recent developments in rough path theory.

Contribution

It revisits and enhances the analysis of Kelly-Melbourne by integrating recent progress in p-variation and càdlàg rough path theory.

Findings

Improved understanding of convergence under mild assumptions

Enhanced analysis using p-variation techniques

Incorporation of càdlàg rough path theory

Abstract

In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very mild. We survey the origins of this theory and then revisit and improve the analysis of Kelly-Melbourne [Ann. Probab. Volume 44, Number 1 (2016), 479-520], taking into account recent progress in $p$-variation and c\`adl\`ag rough path theory.