The fundamental inequality for random walks on cocompact Fuchsian groups

Abstract: Several stochastic processes are defined on the hyperbolic plane H^2. For instance, one can consider a Brownian motion, or a discretized version thereof, when one performs a random walk on the group of isometries of H^2.

It is a recurring question, going back to Furstenberg, Guivarc’h, Ledrappier, Kaimanovich, and others, whether the measures obtained from the random walks coincide with measures of geometric origin, such as the Lebesgue measure.

We prove that the hitting measure is singular with respect to Lebesgue measure for any random walk on a cocompact Fuchsian group generated by translations on opposite sides of a symmetric hyperbolic polygon. This addresses a question of Kaimanovich-Le Prince.

Joint with P. Kosenko.

dynamical systems

Audience: researchers in the topic

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