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Poisson boundaries of random walks on discrete solvable groups. (English) Zbl 0823.60006

Heyer, Herbert (ed.), Probability measures on groups X. Proceedings of the tenth Oberwolfach conference, held November 4-10, 1990 in Oberwolfach, Germany. New York, NY: Plenum Publishing Corporation. 205-238 (1991).
We use the entropy approach [see author, Sov. Math., Dokl. 31, 193-197 (1985), translation from Dokl. Akad. Nauk SSSR 280, 1051-1054 (1985; Zbl 0611.60060); and author and A. M. Vershik, Ann. Probab. 11, 457-490 (1983; Zbl 0641.60009)] for obtaining a description of the Poisson boundary for random walks on discrete solvable groups. It permits to impose only relatively mild condition on the group \(G\) and measure \(\mu\) (e.g., finiteness of the first moment). The solvable groups are a natural intermediate class between nilpotent groups (for which the Poisson boundary is always trivial) and non-amenable groups (for which the Poisson boundary is non-trivial for an arbitrary non-degenerate measure), which makes studying the Poisson boundary for these groups especially interesting. We show that the situation with the Poisson boundary for discrete solvable groups is similar to the Lie group case only for polycyclic groups, which are a natural discrete analogue of solvable Lie groups and in a sense can be characterized as “finite-dimensional” discrete solvable groups. For other discrete solvable groups the behaviour of the Poisson boundary strikingly differs from the Lie case.
For the entire collection see [Zbl 0813.00009].

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
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