  This paper is concerned with correlation functions of stochastic systems with
memory, a prominent example being a molecule or colloid moving through a
complex (e.g., viscoelastic) fluid environment. Analytical investigations of
such systems based on non-Markovian stochastic equations are notoriously
difficult. A common approximation is that of a single-exponential memory,
corresponding to the introduction of one auxiliary variable coupled to the
Markovian dynamics of the main variable. As a generalization, we here
investigate a class of "toy" models with altogether three degrees of freedom,
giving rise to more complex forms of memory. Specifically, we consider, mainly
on an analytical basis, the under- and overdamped motion of a colloidal
particle coupled linearly to two auxiliary variables, where the coupling
between variables can be either reciprocal or non-reciprocal. Projecting out
the auxiliary variables, we obtain non-Markovian Langevin equations with
friction kernels and colored noise, whose structure is similar to that of a
generalized Langevin equation. For the present systems, however, the
non-Markovian equations may violate the fluctuation-dissipation relation as
well as detailed balance, indicating that the systems are out of equilibrium.
We then study systematically the connection between the coupling topology of
the underlying Markovian system and various autocorrelation functions.We
demonstrate that already two auxiliary variables can generate surprisingly
complex (e.g., non-monotonic or oscillatory) memory and correlation functions.
Finally, we show that a minimal overdamped model with two auxiliary variables
and suitable non-reciprocal coupling yields correlation functions resembling
those describing hydrodynamic backflow in an optical trap.
