Regularity of the Drift and Entropy of Random Walks on Groups

Random walks on a group G model many natural phenomena. A random walk
is defined by a probability measure p on G. We are interested in global asymptotic
properties of the random walks and in particular in the linear drift and the asymptotic
entropy. If the geometry of the group is rich, then these numbers are both positive and
the way of dependence on p is some global property of G. In this note, we review recent
results about the regularity of the drift and the entropy in some examples.