Can you feel the shape of a manifold with Brownian motion? (Q796164)

Stochastic differential geometry can be thought of as a highway, with traffic moving in two opposing directions. In one direction the objective is to identify and to explore the influence of the geometry of a manifold on various stochastic processes living on it. The other direction attempts to discover properties of stochastic processes that force their ambient manifolds to take on particular geometric forms. This second direction has exposed a variety of such ''inverse problems''. The paper commences by reviewing basic facts about Brownian motion on a manifold. It uses the author's approach of homogenization (via semigroup methods) of a random walk approximation. The main part of the article expounds some results recently obtained by A. Gray and the author, characterising the local geometry of manifolds in terms of the exit times of Brownian motion from geodesic balls.