  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.
