  We study the response of probe particles to weak constant driving in
kinetically constrained models of glassy systems, and show that the probe's
response can be non-monotonic and give rise to negative differential mobility:
increasing the applied force can reduce the probe's drift velocity in the force
direction. Oth er significant non-linear effects are also demonstrated, such as
the enhancement with increasing force of the probe's fluctuations away from the
average path, a phenomenon known in other contexts as giant diffusivity. We
show that these res ults can be explained analytically by a continuous-time
random walk approximatio n where there is decoupling between persistence and
exchange times for local dis placements of the probe. This decoupling is due to
dynamic heterogeneity in the glassy system, which also leads to bimodal
distributions of probe particle disp lacements. We discuss the relevance of our
results to experiments.
