Fluctuations from a random fractional averaging limit

TL;DR

This paper proves a fluctuation limit theorem for a multiscale stochastic system driven by fractional Brownian motion, revealing how deviations from the averaged behavior converge to a fractional SDE with complex Gaussian influences.

Contribution

It introduces a novel fluctuation result for multiscale systems with fractional Brownian motion, overcoming challenges in joint convergence and developing new analytical techniques.

Findings

Deviation scaled by epsilon^{1/2-H} converges to a fractional SDE.

Established joint convergence of averaging and homogenization limits.

Developed innovative methods combining cumulant techniques and rough differential equations.

Abstract

We consider a system of multiscale stochastic differential equations whose slow component is drivenby a fractional Brownian motion with Hurst parameter H greater than 1/2. Under ergodic assumptions ensuring the applicability of the fractional averaging and fractional homogenization theorems of Hairer and Li (arXiv:1902.11251, arXiv:2109.06948), we establish a fluctuation result. The deviation of the slow motion, scaled by epsilon^{1/2-H}, from its effective, time-dependent random limit converges, as the time-separation scale epsilon tends to zero, to the solution of a stochastic differential equation driven by a fractional Brownian motion and influenced by an additional space--time Gaussian field. Since the averaging principle and the fractional homogenization hold in different modes of convergence, obtaining the required joint convergence is a delicate matter. Moreover, neither the continuity of the Ito--Lyons solution map nor the martingale method is directly applicable for our purposes, so the proof requires several innovations. To establish the fluctuation theorem, we combine cumulant methods with a residue lemma and formulate the enlarged system as a rough differential equation in a suitable space.