  We study paradigmatic examples of a tracer trapped in a harmonic potential
and coupled to a bath particle of various types. Restricting to linear
couplings between tracer and bath allows for analytical treatment. We discuss
the resulting stochastic Langevin equation for the tracer and its relaxation
function, for which this equation is averaged under an initial tracer position.
For equilibrium, the tracer-bath force on average vanishes, a well known
consequence of Boltzmann statistics. If tracer and bath are subject to
different temperatures, the conditioned tracer-bath force is finite, and can be
as large in magnitude as the force between tracer and trapping potential. For a
bath particle with intrinsic memory, e.g., an active particle, even the noise
felt by the bath particle takes a finite average under conditioning of the
tracer. If the noise of the bath particle is non-Gaussian, the relaxation
function of the tracer can be non-monotonic as a function of time. We display
how these phenomena are reflected in the Langevin equation of the tracer.
