  Colloids held by optical or magnetic tweezers have been used to explore the
local rheological properties of a complex medium and to extract work from
fluctuations with some appropriate protocols. However, a general theoretical
understanding of the interplay between the confinement and the interaction with
the environment is still lacking. Here, we explore the statistical properties
of the position of a probe confined in a harmonic trap moving at constant
velocity and interacting with a bath of colloidal particles maintained at a
different temperature. Interactions among particles are accounted for by a
systematic perturbation, whose range of validity is tested against direct
simulations of the full dynamics. Overall, our results provide a way to predict
the effect of the driving and the environment on the probe, and can potentially
be used to investigate the properties of colloidal heat engines with many-body
interactions.
