  Using a powerful combination of projection-operator method and path-space
response theory, we derive the fluctuation dynamics of a slow inertial probe
coupled to a steady nonequilibrium medium under the assumption of time-scale
separation. The nonequilibrium can be realized by external nongradient driving
on the medium particles or by their (athermal) active self-propulsion. The
resulting friction on the probe is explicit as a time-correlation for medium
observables and is decomposed into two terms, one entropic and proportional to
the noise amplitude as in the Einstein relation for equilibrium media, and a
frenetic term that can take both signs. As illustration, we give the exact
expressions for the friction and noise of a probe in a rotating run-and-tumble
medium and in a sheared overdamped medium. In both examples, we find an
interesting transition to absolute negative probe friction as the
nonequilibrium medium exhibits sufficient and persistent rotational current.
