zbMATH — the first resource for mathematics

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.

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
Full Text: Link