  The overdamped dynamics of a particle is in general affected by its
interaction with the surrounding medium, especially out of equilibrium, and
when the latter develops spatial and temporal correlations. Here we consider
the case in which the medium is modeled by a scalar Gaussian field with
relaxational dynamics, and the particle is dragged at constant velocity through
the medium by a moving harmonic trap. This mimics the setting of an active
microrheology experiment conducted in a near-critical medium. When the particle
is displaced from its average position in the nonequilibrium steady state, its
subsequent relaxation is shown to feature damped oscillations. This is similar
to what has been recently predicted and observed in viscoelastic fluids, but
differs from what happens in the absence of driving or for an overdamped
Markovian dynamics, in which cases oscillations cannot occur. We characterize
these oscillating modes in terms of the parameters of the underlying mesoscopic
model for the particle and the medium, confirming our analytical predictions
via numerical simulations.
