Averaging principle for rough slow-fast systems of level 3
Yuzuru Inahama
TL;DR
This work extends the averaging principle to slow–fast systems of rough differential equations driven by random rough paths of level $3$, addressing the gap left by prior level-$2$ results. It develops a level-$3$ controlled path framework and an anisotropic rough path construction to model the driving signal, proving a strong Khas'minskii-type averaging result under suitable dissipativity and Lipschitz conditions. The authors construct driving rough paths by combining Brownian and fractional Gaussian components, establish geometric and integrability properties, and apply a deterministic–probabilistic analysis to show that the slow component $X^{\varepsilon}$ converges in $L^p$ to the averaged process $\bar X$. The framework accommodates fractional Brownian RP with $H\in(1/4,1/3]$, yielding a robust level-$3$ averaging principle relevant for rough stochastic dynamics and potential applications in multi-scale systems with rough drivers.
Abstract
The averaging principle for slow-fast systems of various kind of stochastic (partial) differential equations has been extensively studied. An analogous result was shown for slow-fast systems of rough differential equations driven by random rough paths a few years ago and the study of ``rough slow-fast systems" seems to be gaining momentum now. In all known results, however, the driving rough paths are of level 2. In this paper we formulate rough slow-fast systems driven by random rough paths of level 3 and prove the strong averaging principle of Khas'minskii-type.