  We consider a driven tracer particle (TP) in a bath of hard-core particles
undergoing continuous exchanges with a reservoir. We develop an analytical
framework which allows us to go beyond the standard force-velocity relation
used for this minimal model of active microrheology and quantitatively analyze,
for any density of the bath particles, the fluctuations of the TP position and
their correlations with the occupation number of the bath particles. We obtain
an exact Einstein-type relation which links these fluctuations in the absence
of a driving force and the bath particles density profiles in the linear
driving regime. For the one-dimensional case we also provide an approximate but
very accurate explicit expression for the variance of the TP position and show
that it can be a non-monotoneous function of the bath particles density:
counter-intuitively, an increase of the density may increase the dispersion of
the TP position. We show that this non trivial behavior, which could in
principle be observed in active microrheology experiments, is induced by subtle
cross-correlations quantified by our approach.
