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Diffusions and random shadows in negatively curved manifolds. (English) Zbl 0877.58058
Author’s abstract: “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 $$\Delta /2-\beta\partial_r$$. 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”.

##### MSC:
 58J65 Diffusion processes and stochastic analysis on manifolds 60J65 Brownian motion
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