Diffusions and Random Shadows in Negatively Curved Manifolds

LetMbe ad-dimensional complete simply-connected negatively-curved manifold. There is a natural notion of Hausdorff dimension for its boundary at infinity. This is shown to provide a notion of global curvature or average rate of growth in two probabilistic senses: First, on surfaces (d=2), it is twice the critical drift separating transience from recurrence for Brownian motion with constant-length radial drift. Equivalently, it is twice the criticalβfor the existence of a Green function for the operatorΔ/2−β∂r. Second, for anyd, it is the critical intensity for almost sure coverage of the boundary by random shadows cast by balls, appropriately scaled, produced from a constant-intensity Poisson point process.