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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

##### MSC:
 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