  In light of recent work on particles fluctuating in linear viscoelastic
fluids, we study a linear stochastic partial-integro-differential equation with
memory that is driven by a stationary noise on a bounded, smooth domain. Using
the framework of generalized stationary solutions introduced
in~\cite{mckinley2018anomalous}, we provide sufficient conditions on the
differential operator and the noise to obtain the existence as well as H\"older
regularity of the stationary solutions for the concerned equation. As an
application of the regularity results, we compare to analogous classical
results for the stochastic heat equation. When the 1d stochastic heat equation
is driven by white noise, solutions are continuous with space and time
regularity that is H\"older $(1/2-\ep)$ and $(1/4-\ep)$ respectively. When
driven by colored-in-space noise, solutions can have a range of regularity
properties depending on the structure of the noise. Here, we show that the
particular form of colored-in-time memory that arises in viscoelastic diffusion
applications, satisfying what is called the Fluctuation--Dissipation
relationship, yields sample paths that are H\"older $(1/2-\ep)$ and $(1/2-\ep)$
in space and time.
