Fluctuation from averaging limit under fractional Brownian motion

TL;DR

This paper investigates the fluctuation from the averaging limit in slow-fast systems driven by fractional Brownian motion with H > 1/2, extending classical results to non-Markovian settings using Young-Wiener integrals.

Contribution

It develops a novel weak convergence analysis for fractional Brownian motion-driven systems, incorporating Young-Wiener integrals and Poisson PDE methods, addressing non-Markovian complexities.

Findings

Weak limit includes an additional Gaussian process.

Established tightness using Holder semi-norms and PDE properties.

Extended averaging fluctuation results to fractional Brownian motion with H > 1/2.

Abstract

This work considers a type of slow-fast system, where the slow component is driven by fractional Brownian motion with H > 1/2 and the fast component is a Markovian stationary process. Our solution mapping is defined based on the Young-Wiener sense, which is constructed via the stochastic sewing lemma. Then we aim to show the fluctuation from averaging limit. Unlike the case of standard Brownian motion, this must be extended to a non-Markovian fractional setting that cannot be handled using Ito stochastic analysis, so the weak convergence analysis also becomes very complicated. We consider two cases here. The first one assumes that the driver is a small noise, and the slow one depends fully on fast one. Specially, the integral against fractional Brownian motion coincides with Young integral due to the small parameter. Note that to apply the Poisson PDE method, the Young-Ito formula for the stochastic dynamical system with mixed fractional Brownian motion is required. Moreover, the fast one is assumed to satisfy stricter Holder conditions related to the time scale parameter. The tightness is then derived through the Holder semi-norm of the fast one, the properties of the Poisson PDE solution, and a Gronwall-type result for linear Young differential equation. The weak limit shown includes an extra Gaussian process. The second part is dedicated to the problem under a general fractional Brownian motion. In contrast to the former one, here the fractional Brownian motion integral term will be more difficult to bound due to coupling between the regularity of Poisson PDE solution and dependence on the unbounded Holder seminorm of the fast one. This issue is resolved delicately via the Young-Wiener-Ito integral, the Holder regularity of slow one and ergodicity of the fast one.