Diffusions and random shadows in negatively curved manifolds

作者：Lyons R^*

来源：Journal of Functional Analysis, 1996, 138(2): 426-448.

DOI：10.1006/jfan.1996.0071

Let M be a d-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 critical beta for the existence of a Green function for the operator Delta/2 - beta partial derivative(r). Second, for any d, 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.

• 出版日期1996-6-15