Noise sensitivity and stability of random walks on groups

The general noise sensitivity problem asks:  
Given a stochastic process X defined in terms of i.i.d. variables,
does resampling a small fraction of these variables produce an
almost independent copy of X ? or a highly correlated copy?
In this talk, we discuss the following questions:
Can a random walk on a group be noise sensitive?
Which groups admit a noise sensitive random walk?