Brownian motion, exit times and stochastic riemannian geometry

Abstract

In this report we consider stochastic systems which are modelled by Brownian motion on a Riemannian manifold. Of special interest are the exit times from small geodesic balls. Our results may be viewed as quantitative refinements of the comparison theorems obtained by Debiard Gaveau-Mazet r31 which are of the followinq tvpe: Let M be a Riemannian manifold with sectional' curvature K, bounded by a constant in the form K< k. It is then shown that the exit time from a-geodesic ball is less than the exit time from a geodesic ball of the same radius in the space of constant curvature k. However no information is available in case we only have hypotheses on, say, the scalar curvature. Therefore it is important to develop estimates for the exit time with no hypotheses on the metric or the curvature tensor of the manifold. In this report we give results on the first and second moments of the exit time of Brownian motion. These are used to give stochastic characterizations of Euclidean and rank one symmetric spaces. Much of the work is in collaboration with Alfred Gray, to whom go many thanks for helpful conversations.