Harmonic Measures on Covers of Compact Surfaces of Nonpositive Curvature

Abstract

Let $M$ be the universal cover of a compact nonflat surface $N$ of nonpositive curvature. We show that on the average the Brownian motion on $M$ behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on $N$ is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.