  The motion of a colloidal probe in a viscoelastic fluid is described by
friction or mobility, depending on whether the probe is moving with a velocity
or feeling a force. While the Einstein relation describes an inverse
relationship valid for Newtonian solvents, both concepts are generalized to
time-dependent memory kernels in viscoelastic fluids. We theoretically and
experimentally investigate their relation by considering two observables: the
recoil after releasing a probe that was moved through the fluid and the
equilibrium mean squared displacement (MSD). Applying concepts of linear
response theory, we generalize Einstein's relation and thereby relate recoil
and MSD, which both provide access to the mobility kernel. With increasing
concentration, however, MSD and recoil show distinct behaviors, rooted in
different behaviors of the two kernels. Using two theoretical models, a linear
two-bath particle model and hard spheres treated by mode-coupling theory, we
find a Volterra relation between the two kernels, explaining differing
timescales in friction and mobility kernels under variation of concentration.
