In this paper, we investigate large-scale linear systems driven by a fractional Brownian motion (fBm) with Hurst parameter $H\in [1/2, 1)$. We interpret these equations either in the sense of Young ($H>1/2$) or Stratonovich ($H=1/2$). Especially fractional Young differential equations are well suited for modeling real-world phenomena as they capture memory effects. Although it is very complex to solve them in high dimensions, model reduction schemes for Young or Stratonovich settings have not yet been studied much. To address this gap, we analyze important features of fundamental solutions associated to the underlying systems. We prove a weak type of semigroup property which is the foundation of studying system Gramians. From the introduced Gramians, dominant subspace can be identified which is shown in this paper as well. The difficulty for fractional drivers with $H>1/2$ is that there is no link of the corresponding Gramians to algebraic equations making the computation very difficult. Therefore, we further propose empirical Gramians that can be learned from simulation data. Subsequently, we introduce projection-based reduced order models (ROMs) using the dominant subspace information. We point out that such projections are not always optimal for Stratonovich equations as stability might not be preserved and since the error might be larger than expected. Therefore, an improved ROM is proposed for $H=1/2$. We validate our techniques conducting numerical experiments on some large-scale stochastic differential equations driven by fBm resulting from spatial discretizations of fractional stochastic PDEs. Overall, our study provides useful insights into the applicability and effectiveness of reduced order methods for stochastic systems with fractional noise, which can potentially aid in the development of more efficient computational strategies for practical applications.

翻译：本文研究由赫斯特参数 $H\in [1/2, 1)$ 的分数布朗运动（fBm）驱动的大规模线性系统。我们分别以Young积分（$H>1/2$）或Stratonovich积分（$H=1/2$）意义解释这些方程。特别地，分数Young微分方程因能捕捉记忆效应而适用于模拟实际物理现象。尽管高维求解极具复杂性，但针对Young或Stratonovich系统的模型降阶方法尚未得到充分研究。为填补这一空白，我们分析了与底层系统相关的基本解的重要特性，证明了弱形式的半群性质——这是研究系统格拉姆矩阵的基础。本文进一步展示了如何通过引入的格拉姆矩阵识别主导子空间。对于分数阶驱动项（$H>1/2$），其难点在于相应格拉姆矩阵与代数方程间缺乏关联，导致计算极为困难。因此，我们提出可从模拟数据中学习的经验性格拉姆矩阵。随后，利用主导子空间信息引入基于投影的降阶模型（ROMs）。需要指出的是，此类投影对Stratonovich方程并非总是最优——稳定性可能无法保持，且误差可能超出预期。为此，针对$H=1/2$情形提出改进型ROM。我们通过空间离散化分数阶随机偏微分方程获得的大规模分数布朗运动驱动随机微分方程数值实验验证了所提方法的有效性。总体而言，本研究为分数噪声驱动随机系统的降阶方法适用性和有效性提供了有益见解，有望推动实际应用中更高效计算策略的发展。