[Paper Review] Diffusive and rough homogenisation in fractional noise field

Johann Gehringer, Xue-Mei Li|arXiv (Cornell University)|Jun 20, 2020

Advanced Mathematical Modeling in Engineering27 references4 citations

TL;DR

This paper establishes a homogenisation theorem for stochastic differential equations driven by fractional noise with long-range dependence, proving that the effective dynamics emerge as a combination of Stratonovich and Young-type SDEs driven by both Gaussian (Wiener) and non-Gaussian (Hermite) processes. The key contribution is a lifted joint functional central and non-central limit theorem in rough path topology, enabling the characterization of limits beyond classical diffusive behavior.

ABSTRACT

With recently developed tools, we prove a homogenisation theorem for a random ODE with short and long-range dependent fractional noise. The effective dynamics are not necessarily diffusions, they are given by stochastic differential equations driven simultaneously by stochastic processes from both the Gaussian and the non-Gaussian self-similarity universality classes. A key lemma for this is the `lifted' joint functional central and non-central limit theorem in the rough path topology.

Motivation & Objective

• To address homogenisation in multiscale systems with long-range dependent fractional noise, particularly when classical central limit theorems fail.
• To characterize effective dynamics that are not necessarily diffusive, but instead combine Stratonovich and Young integrals driven by processes from both Gaussian and non-Gaussian self-similarity universality classes.
• To extend homogenisation theory beyond the diffusive regime by incorporating non-central limit theorems in rough path spaces.
• To establish a functional limit theorem for the joint convergence of iterated integrals in rough topology, enabling the analysis of non-Markovian, long-range dependent noise.

Proposed method

• The authors use rough path theory to handle the irregularity of fractional noise, particularly in the case of $ H \in (0, \frac{1}{2}) $, where the noise is not semimartingale.
• They derive a lifted joint functional central and non-central limit theorem in the rough path topology, which allows joint convergence of iterated integrals of the noise components.
• The method involves decomposing the noise into short-range (Wiener) and long-range (Hermite) components, with the former contributing via Stratonovich integrals and the latter via Young integrals.
• A key technical step is proving conditional integrability of fractional Ornstein-Uhlenbeck processes and establishing bounds on the iterated integrals in the rough path sense.
• The analysis relies on Hermite rank and self-similarity properties of the noise, with scaling factors $ \alpha_k(\varepsilon) $ tailored to the Hermite rank $ m_k $ of each $ G_k $.
• The solution is constructed in the controlled rough path space $ D_X^{2\alpha} $, ensuring existence and uniqueness of the solution to the limiting rough differential equation.

Experimental results

Research questions

• RQ1 Can homogenisation in systems with long-range dependent fractional noise yield effective dynamics that are not diffusive, but instead involve non-Gaussian self-similar processes?
• RQ2 What is the role of the Hermite rank of the noise transformation $ G_k $ in determining whether the limit is a Wiener process or a higher-order Hermite process?
• RQ3 How can joint functional limit theorems be established in the rough path topology for systems combining central and non-central limit behaviors?
• RQ4 Under what conditions does the joint convergence of iterated integrals of fractional noise hold in rough path topology, especially when the noise is not a semimartingale?
• RQ5 Can the restriction $ H^*(m) < 0 $ in the integral bound (3.8) be lifted using p-variation rough path theory instead of Hölder regularity?

Key findings

• The effective dynamics of the homogenised system are described by a rough differential equation driven by both Wiener processes (via Stratonovich integral) and Hermite processes (via Young integral), resulting in a mixed-type SDE.
• The limit process is not a standard diffusion; instead, it belongs to a broader class of stochastic processes that include both Gaussian and non-Gaussian self-similar components.
• For functions $ G_k $ with Hermite rank $ m_k $, the effective limit is locally a Hermite process of rank $ m_k $ if $ m_k < \frac{1}{2(1-H)} $, otherwise it is a Wiener process.
• The joint functional limit theorem in rough path topology ensures convergence of iterated integrals, which is essential for the well-posedness of the limiting SDE.
• The proof establishes that the conditional integrability of the fractional Ornstein-Uhlenbeck process holds under the given assumptions, enabling the use of rough path techniques.
• An open problem is identified: whether the restriction $ H^*(m) < 0 $ in the integral bound (3.8) can be removed using p-variation rough path theory, suggesting a potential direction for future work.