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Some asymptotic properties of random walks on free groups. (English) Zbl 0994.60073
Taylor, J. C. (ed.), Topics in probability and Lie groups: boundary theory. Providence, RI: American Mathematical Society (AMS). CRM Proc. Lect. Notes. 28, 117-152 (2001).
The present survey is dealing with properties of the harmonic measure at infinity associated with a random walk on a free group with a finite number of generators, emphasising in particular results about the measure of small balls around typical points on the boundary. One purpose of the article is to illustrate that potential theory and probabilistic objects are useful to describe the behaviour at infinity of hyperbolic or semi-hyperbolic spaces. Quite a lot is known in the case of nearest neighbour random walks, where the harmonic measure can be described explicitly. The paper moves on to discuss random walks with increments of finite support, where many qualitative aspects of the nearest neighbour situation are retained. Finally, much less is known when the support may be infinite and the paper gives some results based on moment conditions. This section also contains new results including a formula for the Hausdorff dimension of the harmonic measure assuming a finite first moment. In all sections of the paper important proofs are carried out and many references and links to related situations are given.
For the entire collection see [Zbl 0970.00015].

60J50 Boundary theory for Markov processes
60G50 Sums of independent random variables; random walks