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It is proved that the Green’s function of a symmetric finite range random walk on a co-compact Fuchsian group decays exponentially in distance at the radius of convergence . It is also shown that Ancona’s inequalities extend to , and therefore that the Martin boundary for -potentials coincides with the natural geometric boundary , and that the Martin kernel is uniformly Hölder continuous. Finally, this implies a local limit theorem for the transition probabilities: in the aperiodic case, .
We consider the reducibility and unitary equivalence of multiplication operators on the Dirichlet space. We first characterize reducibility of a multiplication operator induced by a finite Blaschke product and, as an application, we show that a multiplication operator induced by a Blaschke product with two zeros is reducible only in an obvious case. Also, we prove that a multiplication operator induced by a multiplier ϕ is unitarily equivalent to a weighted shift of multiplicity 2 if and only if...
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