Brownian motion and negative curvature
by Marc Arnaudon and Anton Thalmaier

It is well-known that on a Riemannian manifold, there is a deep interplay between geometry, harmonic function theory, and the long-term behaviour of Brownian motion. Negative curvature amplifies the tendency of Brownian motion to exit compact sets and, if topologically possible, to wander out to infinity. On the other hand, non-trivial asymptotic properties of Brownian paths for large time correspond with non-trivial bounded harmonic functions on the manifold. We describe parts of this interplay in the case of negatively curved simply connected Riemannian manifolds. Recent results are related to known properties and old conjectures.

Random Walks, Boundaries and Spectra (D. Lenz, F. Sobieczky, W. Woess, eds)
Progress in Probability, Vol. 64, 145-163, Springer Basel, 2011

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Marc Arnaudon
Anton Thalmaier

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