Instability of set recurrence and Green's function on groups with the Liouville property

Details Instability of set recurrence and Green's function on groups with the Liouville property Instability of set recurrence and Green's function on groups with the Liouville property

2003-10-17

arxiv.org/abs/math/0310258v1

Mathematics

Probability

Scientific paper

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.

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