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Progress in Probability, Vol. 64, 145163 c 2011 Springer Basel AG

Summary: Progress in Probability, Vol. 64, 145­163
c 2011 Springer Basel AG
Brownian Motion and Negative Curvature
Marc Arnaudon and Anton Thalmaier
Abstract. It is well known that on a Riemannian manifold, there is a deep
interplay between geometry, harmonic function theory, and the long-term be-
haviour of Brownian motion. Negative curvature amplifies the tendency of
Brownian motion to exit compact sets and, if topologically possible, to wan-
der 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.
Mathematics Subject Classification (2000). Primary 58J65; Secondary 60H30,
31C12, 31C35.
Keywords. Harmonic function, Poisson boundary, Cartan-Hadamard mani-
fold, Conjecture of Greene-Wu, Dirichlet problem at infinity.
1. Introduction
In complex analysis the desire to understand how geometry of a complex manifold
influences its complex structure has been a guiding inspiration for decades. A


Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers
Thalmaier, Anton - Laboratoire de Mathématiques, Université du Luxembourg


Collections: Mathematics