Summary
Kallenberg and Sztencel have recently discovered exponential upper bounds, independent of dimension, on the probability that a vector martingale will exit from a ball in Euclidean space by timet. This article extends their results to martingales on Riemannian manifolds, including Brownian motion, and shows how exit probabilities depend on curvature. Using comparison with rotationally symmetric manifolds, these estimates are easily computable, and are sharp up to a constant factor in certain cases.
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Darling, R.W.R. Exit probability estimates for martingales in geodesic balls, using curvature. Probab. Th. Rel. Fields 93, 137–152 (1992). https://doi.org/10.1007/BF01195225
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DOI: https://doi.org/10.1007/BF01195225