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Random walks on co-compact Fuchsian groups. (Marches aléatoires sur des groupes fuchsiens co-compacts.) (English. French summary) Zbl 1277.60012
The authors study a random walk \(X_n=x\xi_1\xi_2\dots \xi_n\) on a co-compact Fuchsian group \(\Gamma\). Here, \(\xi_1,\xi_2,\dots\) are independent identically distributed \(\Gamma\)-valued random variables invariant under the mapping \(x\mapsto x^{-1}\). It is assumed that the distribution of each \(\xi_n\) has a finite support. The Green function of the random walk is defined as \[ G_r(x,y)=\sum_{n=0}^\infty \operatorname{P}^x\{ X_n=y\} r^n. \] Let \(R\) be the radius of convergence of the above series. In fact, \(G_r\) exists also for \(r=R\). The authors prove that \(G_R(1,x)\) decays exponentially in \(\operatorname {dist}(1,x)\). The inequalities for \(G_r, r<R\), proved by A. Ancona [Lect. Notes Math. 1344, 1–23 (1988; Zbl 0677.31006)], are shown to hold for \(r=R\). The Martin boundary for \(R\)-potentials coincides with the natural geometric boundary, and the Martin kernel is uniformly Hölder continuous. A local limit theorem for the transition probabilities is proved.

MSC:
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
20F67 Hyperbolic groups and nonpositively curved groups
31C35 Martin boundary theory
60J50 Boundary theory for Markov processes
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