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Brownian motion and harmonic functions on covering manifolds. An entropy approach. (English. Russian original) Zbl 0615.60074
Sov. Math., Dokl. 33, 812-816 (1986); translation from Dokl. Akad. Nauk SSSR 288, 1045-1049 (1986).
Let \(\tilde M\) be a regular covering of a compact Riemannian manifold M with covering transformation group G. (That is, G is a discrete group of isometries of \(\tilde M\) such that \(M=\tilde M/G.)\) After defining entropy \(h(\tilde M)\) of Brownian motion on \(\tilde M\) in terms of boundary measures, various expressions for \(h(\tilde M)\) are presented in two theorems. Another eight theorems are concerned with relations between \(h(\tilde M)\), properties of G (amenability, polycyclicity), and geometric properties of M and \(\tilde M,\) in particular the Liouville property of \(\tilde M.\)
Most proofs are omitted. The paper rests on results of A. M. Vershik and the author, Random walks on discrete groups: boundary and entropy. Ann. Probab. 11, 457-490 (1983).
Reviewer: H.Crauel

60J65 Brownian motion
58J65 Diffusion processes and stochastic analysis on manifolds
31C12 Potential theory on Riemannian manifolds and other spaces
20P05 Probabilistic methods in group theory