Averaging principle for fast-slow system driven by mixed fractional Brownian rough path

TL;DR

This paper establishes an averaging principle for fast-slow rough differential equations driven by mixed fractional Brownian rough paths, demonstrating strong convergence of the slow component to an averaged solution.

Contribution

It introduces a novel averaging principle within rough path theory for systems driven by mixed fractional Brownian motion, combining fractional calculus and classical discretization methods.

Findings

Proves strong convergence of the slow component to the averaged solution.

Develops a new framework for averaging in rough path systems with mixed fractional Brownian motion.

Extends rough path theory to include systems with both Brownian and fractional Brownian drivers.

Abstract

This paper is devoted to studying the averaging principle for fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index $H ~(1/3 < H\leq 1/2)$. Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the $L^1$-sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new.