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Diffusions and Random Shadows in Negatively Curved Manifolds

โœ Scribed by Russell Lyons

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
773 KB
Volume
138
Category
Article
ISSN
0022-1236

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โœฆ Synopsis

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 constantlength radial drift. Equivalently, it is twice the critical ; for the existence of a Green function for the operator 2ร‚2&; 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.

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