  Let $\mu$ and $\nu$ be probability measures on a group \Gamma and let G_\mu
and G_\nu denote Green's function with respect to \mu and \nu . The group
\Gamma is said to admit instability of Green's function if there are symmetric,
finitely supported measures $\mu$ and \nu and a sequence \{x_n\} such that
G_\mu(e, x_n)/G_\nu(e,x_n) \to 0, and \Gamma admits instability of recurrence
if there is a set S that is recurrent with respect to \nu but transient with
respect to \mu . We give a number of examples of groups that have the Liouville
property but have both types of instabilities. Previously known groups with
these instabilities did not have the Liouville property.
