Large deviation principle for slow-fast rough differential equations via controlled rough paths
Xiaoyu Yang, Yong Xu
TL;DR
This work establishes a large deviation principle for the slow component of a slow-fast rough differential equation driven by a mixed fractional Brownian motion with $H\in(1/3,1/2)$. The authors leverage the controlled rough path framework and a variational representation to reduce the LDP to a weak convergence problem for controlled slow-fast dynamics, treating the fast component through its invariant measure at fixed slow variables. The rate function is characterized by a skeleton map $\mathcal{G}^0$ driven by the averaged fast dynamics, with the value $I(\xi)$ given by a variational form involving the Cameron–Martin norm, linking the large deviations to controlled trajectories of the averaged system. This advances LDP results for slow-fast systems into the rough-path regime with rough drivers, establishing ergodicity-based averaging and continuity of the solution map as central tools for the analysis.
Abstract
We prove a large deviation principle for the slow-fast rough differential equations under the controlled rough path framework. The driver rough paths are lifted from the mixed fractional Brownian motion with Hurst parameter $H\in (1/3,1/2)$. Our approach is based on the continuity of the solution mapping and the variational framework for mixed fractional Brownian motion. By utilizing the variational representation, our problem is transformed into a qualitative property of the controlled system. In particular, the fast rough differential equation coincides with Itô SDE almost surely, which possesses a unique invariant probability measure with frozen slow component. We then demonstrate the weak convergence of the controlled slow component by averaging with respect to the invariant measure of the fast equation and exploiting the continuity of the solution mapping.