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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

The paper is available here:

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