Timeline for Probabilities of a random walk exiting a set
Current License: CC BY-SA 3.0
8 events
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| Apr 26, 2013 at 11:44 | comment | added | Mikael de la Salle | @Antoine: the fact that the Cayley graphs of non-amenable groups is non-Liouville is due to Furstenberg (at least according to Erschler). The converse (amenable groups carry a Liouville random walk) is due to Kaimanovich-Vershik and Rosenblatt. This is explained in ams.org/mathscinet-getitem?mr=2025301 | |
| Apr 26, 2013 at 11:00 | comment | added | Vincent Beffara | Not describing the space of limiting configuration, just stating that it is nontrivial - but indeed I saw things about that somewhere, not sure I could remember a reference though ... | |
| Apr 26, 2013 at 10:25 | comment | added | ARG | @Vincent: In the example of harmonic function you give, are you somehow describing the space of "limiting configuration"? I've seen reference for this but cannot access the paper of Erschler where she describes them. | |
| Apr 26, 2013 at 10:22 | comment | added | ARG | @Sean: the answer is yes (a result of Kaimanovich, I believe, the expert will confirm/infirm). Actually, the proper formulation is that an amenable group always has a measure for which the RW has trivial Poisson boundary, but in the case of wreath products, this measure is never finitely supported (results of Erschler? again, my memory is vague). | |
| Apr 26, 2013 at 10:11 | comment | added | ARG | Thankd for this nice and detailed answer! I really did not expect an "iff"... | |
| Apr 26, 2013 at 10:09 | comment | added | Sean Eberhard | @Vincent That's a nice example. What about the converse: is every Liouville group amenable? | |
| Apr 26, 2013 at 10:04 | vote | accept | ARG | ||
| Apr 26, 2013 at 9:47 | history | answered | Vincent Beffara | CC BY-SA 3.0 |