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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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